text: Constraints on Feedback in the Local Universe 9 Fig. 8.— Normalized cumulative distribution function of MKs at diﬀerent radio power bins and at diﬀerent distance bins. The sample is divided into four equal distance bins. Each panel corresponds to one of the distance bins and increasing from left to right. Each panel shows the cummulative distribution function (CDF) of MKs for galaxies with radio power greater than the median radio power in that bin (shown in cyan) and for galaxies with radio power less than the median radio power (shown in pink). The cyan and the pink dashed lines represent the median MKs for high radio bin and low radio bin respectively. tion going supernovae, whereas for massive galaxies, the with PAHs that are heated by young stars and AGN, radio power may be dominated by an AGN (see section ). as well as Gyr old evolved stellar population (Jarrett et al. 2013). Shown in Figure 9 is a plot of 1.4 GHz This type of relation between galaxy mass and ra- radio power against [3.4μm – 4.6μm] infrared color. The dio power is also found to exist in high redshift quasars [3.4μm – 4.6μm] color is sensitive to warm/hot dust and (Browne & Murphy 1987; Carballo et al. 1998; Serjeant thus to optical/UV nuclear activity. An excess in this et al. 1998; Willott et al. 1998; S´anchez & Gonz´alez- color identiﬁes galaxies in which hot dust surrounding Serrano 2003) and in radio galaxies (Yates et al. 1986; the AGNs produces a strong mid-IR continuum that van Velzen et al. 2014). Galaxies have to be massive dominates the host galaxy emission (Stern et al. 2012). enough to be powerful radio sources. The fact that such This enhanced mid-IR continuum may be associated a correlation exists for AGN and non-AGN population, with the dusty torus heated by the radiation from an from faint to bright galaxies is interesting. Since the accretion disk. About 83% of the galaxies in our sample black hole mass scales with the bulge mass (e.g., H¨aring lie in a narrow color range (between –0.3 and 0.1 mag), & Rix 2004), the correlation also suggests that the black and only a few galaxies show excess IR emission. The hole mass closely relates to the maximum radio power fact that most of the galaxies do not show a color excess (Franceschini et al. 1998; Laor 2000; Liu et al. 2006). The indicates that the galaxies in our sample are not associ- plot also shows that there is a broad dispersion in radio ated with bright accretion disks, and if they are AGN, power at a given absolute magnitude even for the most they are accreting in a radiatively ineﬃcient process(e.g. massive galaxies in our sample. This is consistent with ADAF Narayan & Yi 1994). the hypothesis that high black hole mass is necessary but not suﬃcient to produce a powerful radio source. Other 4.3.3. Radio power and SFR relation physical parameters such as the spin of the black hole, accretion eﬃciency and other large-scale environmental The supply of cold gas to fuel star formation and AGN eﬀects may be responsible for the broad dispersion in ra- activity can include galaxy mergers/interactions, the dio power (e.g., Baum et al. 1995; Meier 1999; Wold et al. cooling of gas in the ISM or surrounding hot halo, and 2007). the mass loss from stars (e.g., Heckman & Best 2014). 4.3.2. Radio power and nuclear activity The relation between radio power and SFR can provide The origin of WISE mid-IR emission is associated with some insight into the source of the fueling (Figure 10). a combination of continuum emission from dust, atomic and molecular emission lines and features associated We see a clear correlation (spearman correlation co- ##LTFigure## ##LTFigure##

text: Constraints

text: on

text: Feedback

text: in

text: the

text: Local

text: Universe

text: Fig.

text: Normalized

text: cumulative

text: distribution

text: function

text: of

text: MKs

text: at

text: diﬀerent

text: radio

text: power

text: bins

text: and

text: at

text: diﬀerent

text: distance

text: bins.

text: The

text: sample

text: is

text: divided

text: into

text: four

text: equal

text: distance

text: bins.

text: Each

text: panel

text: corresponds

text: to

text: one

text: of

text: the

text: distance

text: bins

text: and

text: increasing

text: from

text: left

text: to

text: right.

text: Each

text: panel

text: shows

text: the

text: cummulative

text: distribution

text: function

text: (CDF)

text: of

text: MKs

text: for

text: galaxies

text: with

text: radio

text: power

text: greater

text: than

text: the

text: median

text: radio

text: power

text: in

text: that

text: bin

text: (shown

text: in

text: cyan)

text: and

text: for

text: galaxies

text: with

text: radio

text: power

text: less

text: than

text: the

text: median

text: radio

text: power

text: (shown

text: in

text: pink).

text: The

text: cyan

text: and

text: the

text: pink

text: dashed

text: lines

text: represent

text: the

text: median

text: MKs

text: for

text: high

text: radio

text: bin

text: and

text: low

text: radio

text: bin

text: respectively.

text: tion

text: going

text: supernovae,

text: whereas

text: for

text: massive

text: galaxies,

text: the

text: with

text: PAHs

text: that

text: are

text: heated

text: by

text: young

text: stars

text: and

text: AGN,

text: radio

text: power

text: may

text: be

text: dominated

text: by

text: an

text: AGN

text: (see

text: section

text: as

text: well

text: as

text: Gyr

text: old

text: evolved

text: stellar

text: population

text: (Jarrett

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text: al.

text: Shown

text: in

text: Figure

text: is

text: a

text: plot

text: of

text: GHz

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text: relation

text: between

text: galaxy

text: mass

text: and

text: ra-

text: radio

text: power

text: against

text: infrared

text: color.

text: The

text: dio

text: power

text: is

text: also

text: found

text: to

text: exist

text: in

text: high

text: redshift

text: quasars

text: color

text: is

text: sensitive

text: to

text: warm/hot

text: dust

text: and

text: (Browne

text: Murphy

text: Carballo

text: et

text: al.

text: Serjeant

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text: to

text: optical/UV

text: nuclear

text: activity.

text: An

text: excess

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text: et

text: al.

text: Willott

text: et

text: al.

text: S´anchez

text: Gonz´alez-

text: color

text: identiﬁes

text: galaxies

text: in

text: which

text: hot

text: dust

text: surrounding

text: Serrano

text: and

text: in

text: radio

text: galaxies

text: (Yates

text: et

text: al.

text: the

text: AGNs

text: produces

text: a

text: strong

text: mid-IR

text: continuum

text: that

text: van

text: Velzen

text: et

text: al.

text: Galaxies

text: have

text: to

text: be

text: massive

text: dominates

text: the

text: host

text: galaxy

text: emission

text: (Stern

text: et

text: al.

text: enough

text: to

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text: powerful

text: radio

text: sources.

text: The

text: fact

text: that

text: such

text: This

text: enhanced

text: mid-IR

text: continuum

text: may

text: be

text: associated

text: a

text: correlation

text: exists

text: for

text: AGN

text: and

text: non-AGN

text: population,

text: with

text: the

text: dusty

text: torus

text: heated

text: by

text: the

text: radiation

text: from

text: an

text: from

text: faint

text: to

text: bright

text: galaxies

text: is

text: interesting.

text: Since

text: the

text: accretion

text: disk.

text: About

text: of

text: the

text: galaxies

text: in

text: our

text: sample

text: black

text: hole

text: mass

text: scales

text: with

text: the

text: bulge

text: mass

text: (e.g.,

text: H¨aring

text: lie

text: in

text: a

text: narrow

text: color

text: range

text: (between

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text: mag),

text: Rix

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text: correlation

text: also

text: suggests

text: that

text: the

text: black

text: and

text: only

text: a

text: few

text: galaxies

text: show

text: excess

text: IR

text: emission.

text: The

text: hole

text: mass

text: closely

text: relates

text: to

text: the

text: maximum

text: radio

text: power

text: fact

text: that

text: most

text: of

text: the

text: galaxies

text: do

text: not

text: show

text: a

text: color

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text: (Franceschini

text: et

text: al.

text: Laor

text: Liu

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text: al.

text: The

text: indicates

text: that

text: the

text: galaxies

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text: our

text: sample

text: are

text: not

text: associ-

text: plot

text: also

text: shows

text: that

text: there

text: is

text: a

text: broad

text: dispersion

text: in

text: radio

text: ated

text: with

text: bright

text: accretion

text: disks,

text: and

text: if

text: they

text: are

text: AGN,

text: power

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text: a

text: given

text: absolute

text: magnitude

text: even

text: for

text: the

text: most

text: they

text: are

text: accreting

text: in

text: a

text: radiatively

text: ineﬃcient

text: process(e.g.

text: massive

text: galaxies

text: in

text: our

text: sample.

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text: is

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text: with

text: ADAF

text: Narayan

text: Yi

text: the

text: hypothesis

text: that

text: high

text: black

text: hole

text: mass

text: is

text: necessary

text: but

text: not

text: suﬃcient

text: to

text: produce

text: a

text: powerful

text: radio

text: source.

text: Other

text: Radio

text: power

text: and

text: SFR

text: relation

text: physical

text: parameters

text: such

text: as

text: the

text: spin

text: of

text: the

text: black

text: hole,

text: accretion

text: eﬃciency

text: and

text: other

text: large-scale

text: environmental

text: The

text: supply

text: of

text: cold

text: gas

text: to

text: fuel

text: star

text: formation

text: and

text: AGN

text: eﬀects

text: may

text: be

text: responsible

text: for

text: the

text: broad

text: dispersion

text: in

text: ra-

text: activity

text: can

text: include

text: galaxy

text: mergers/interactions,

text: the

text: dio

text: power

text: (e.g.,

text: Baum

text: et

text: al.

text: Meier

text: Wold

text: et

text: al.

text: cooling

text: of

text: gas

text: in

text: the

text: ISM

text: or

text: surrounding

text: hot

text: halo,

text: and

text: the

text: mass

text: loss

text: from

text: stars

text: (e.g.,

text: Heckman

text: Best

text: Radio

text: power

text: and

text: nuclear

text: activity

text: The

text: relation

text: between

text: radio

text: power

text: and

text: SFR

text: can

text: provide

text: The

text: origin

text: of

text: WISE

text: mid-IR

text: emission

text: is

text: associated

text: with

text: some

text: insight

text: into

text: the

text: source

text: of

text: the

text: fueling

text: (Figure

text: a

text: combination

text: of

text: continuum

text: emission

text: from

text: dust,

text: atomic

text: and

text: molecular

text: emission

text: lines

text: and

text: features

text: associated

text: We

text: see

text: a

text: clear

text: correlation

text: (spearman

text: correlation

text: co-

text: ##LTFigure##

text: ##LTFigure##


text: 2 positive correction K = K +K to the membrane bend- el m DL ing rigidity κ: bending brings the charges closer and increases the electrostatic repulsion (see Fig.1.A, bottom part). Consistently, the largest correction is now due to − the thick double layer a(cid:32)nd is(cid:33)proportional to κ 1: D − k T 2 − c2(3 c2) K = 4(cid:15) B κ 1 , (2) DL w e D 1 + c2 where (cid:15) is the permittivity of water. w A B FIG. 1: Schematic representation of the eﬀect of electric ﬁeld (A). Top, freely ﬂuctuating bilayer. Middle, interaction be- tween the induced charges and the external electric ﬁeld leads FIG. 2: Schematic view of the experiment and electrokinetic to an electric force which is amplifying the undulation, acting model of the supported bilayers (A). The incident beam at as a destabilizing negative surface tension. Bottom, bending grazing incidence (direction k ) is scattered in direction k , in sc brings the charges closer increasing the electrostatic interac- giving access to in-plane ﬂuctuations. The oﬀ-specular reﬂec- tions, mainly in the electric double layer (black arrows) and tivity curves and the associated best ﬁts (B). leads to an increase in bending rigidity. Calculated electric po- tential as a function of the distance for 10 Hz (black solid curve) and 50 Hz (red dotted line) (B). The inset of (B) shows the local as this allows easy background subtraction, and θ is voltage at the membrane boundaries as a function of frequency sc and for diﬀerent Debye lengths κ−1 = 800 nm (red curve), 300 scanned in the plane of incidence [24, 26×]. In all exper- nm (green curve) and 150 nm (blaDck curve) using d = 5 nm. iments, the incident beam was 500µm 20µm×and the m z = 0 corresponds to the middle plane of the ﬂoating bilayer. reﬂected intensity was deﬁned using a 20 m×m 200µm slit at 210 mm from the sample and a 20 mm 200µm slit The model system we have used consisted of two at 815 mm from the sample and recorded using a NaI supported bilayers of DSPC (L−α 1,2-distearoyl-sn- (Tl) scintillator. glycero-3-phosphocholine, Avanti Polar Lipids, Lan- Oﬀ-specular scattering is sensitive to both static de- caster, Alabama) deposited on ultra-ﬂat silicon sub- formation and thermal ﬂuctuations of bilayers. In the strates (Fig. 2.A) [22]. All the experiments were per- limit of small amplitudes, and the simple case of a sin- ◦ formed in ﬂuid phase at 58 C. The ﬁrst bilayer serves as gle bilayer in a∝n (cid:104)intera(cid:105)ction potential U, the scattered a spacer to reduce the interaction between the ﬂoating i(cid:104)ntens(cid:105)ity is Isc zqz−q , with the ﬂuctuation spectrum second bilayer and the substrate and keeps it free to ﬂuc- zqz−q = kBT/h(q)(cid:0)an(cid:1)d tuate [23, 24]. Potential was applied between a Cu layer deposited at the back of the thick Si substrate and an ITO (cid:48)(cid:48) coated glass plate mounted parallel to the substrate, 0.5 h q = U + γq2 + κq4. (3) cm from the membrane. (cid:80) (cid:80) (cid:0) (cid:1) (cid:12)(cid:12) (cid:12)(cid:12) We used a 27 keV x-ray beam (wavelength λ = 0.0459 (cid:12) (cid:12) H The Hamiltonian of the system is given by = nm) at the CRG-IF beamline of the European Syn- H 2 chrotron Radiation Facility (ESRF) in an oﬀ-specular ge- (q) = h q z . Fitting of the scattering curves, q q q ometry described in Fig. 2.A [25]. The grazing angle accounting for both thermal ﬂuctuations and the static of incidence is kept ﬁxed (θin = 0.7 mrad)(cid:39), below the roughness induced by the substrate [27] (following the critical angle for total external reﬂection θ 0.85 mrad procedure described in details in [23, 26]), gives access to c ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine##

label: paragraph

font: RNHHWK+URWPalladioL-Roma

text: positive

text: correction

text: K

font: RQOBMA+URWPalladioL-Ital

font: YHZAWB+Rpxr

text: K

text: +K

text: to

text: the

text: membrane

text: bend-

text: el

text: m

text: DL

text: ing

text: rigidity

text: κ:

font: LUXAUR+Rpxmi

text: bending

text: brings

text: the

text: charges

text: closer

text: and

text: increases

text: the

text: electrostatic

text: repulsion

text: (see

text: bottom

text: part).

text: Consistently,

text: the

text: largest

text: correction

text: is

text: now

text: due

text: to

font: UOYLWJ+Pxsy

text: the

text: thick

text: double

text: layer

label: equation

text: to

text: κ

text: D

text: k

text: T

text: K

text: B

text: κ

text: DL

text: w

text: e

text: D

text: where

text: is

text: the

text: permittivity

text: of

text: water.

text: w

text: A

text: B

text: FIG.

text: Schematic

text: representation

text: of

text: the

text: eﬀect

text: of

text: electric

text: ﬁeld

text: (A).

text: Top,

text: freely

text: ﬂuctuating

text: bilayer.

text: Middle,

text: interaction

text: be-

text: tween

text: the

text: induced

text: charges

text: and

text: the

text: external

text: electric

text: ﬁeld

text: leads

text: FIG.

text: Schematic

text: view

text: of

text: the

text: experiment

text: and

text: electrokinetic

text: to

text: an

text: electric

text: force

text: which

text: is

text: amplifying

text: the

text: undulation,

text: acting

text: model

text: of

text: the

text: supported

text: bilayers

text: (A).

text: The

text: incident

text: beam

text: at

text: as

text: a

text: destabilizing

text: negative

text: surface

text: tension.

text: Bottom,

text: bending

text: grazing

text: incidence

text: (direction

text: k

text: is

text: scattered

text: in

text: direction

text: k

text: in

text: sc

text: brings

text: the

text: charges

text: closer

text: increasing

text: the

text: electrostatic

text: interac-

text: giving

text: access

text: to

text: in-plane

text: ﬂuctuations.

text: The

text: oﬀ-specular

text: reﬂec-

text: tions,

text: mainly

text: in

text: the

text: electric

text: double

text: layer

text: (black

text: arrows)

text: and

text: tivity

text: curves

text: and

text: the

text: associated

text: best

text: ﬁts

text: (B).

text: leads

text: to

text: an

text: increase

text: in

text: bending

text: rigidity.

text: Calculated

text: electric

text: po-

text: tential

text: as

text: a

text: function

text: of

text: the

text: distance

text: for

text: Hz

text: (black

text: solid

text: curve)

text: and

text: Hz

text: (red

text: dotted

text: line)

text: (B).

text: The

text: inset

text: of

text: (B)

text: shows

text: the

text: local

text: as

text: this

text: allows

text: easy

text: background

text: subtraction,

text: and

text: θ

text: is

text: voltage

text: at

text: the

text: membrane

text: boundaries

text: as

text: a

text: function

text: of

text: frequency

text: sc

text: and

text: for

text: diﬀerent

text: Debye

text: lengths

text: κ−1

text: nm

text: (red

text: curve),

text: scanned

text: in

text: the

text: plane

text: of

text: incidence

text: In

text: all

text: exper-

text: nm

text: (green

text: curve)

text: and

text: nm

text: (blaDck

text: curve)

text: using

text: d

text: nm.

text: iments,

text: the

text: incident

text: beam

text: was

text: the

text: m

text: z

text: corresponds

text: to

text: the

text: middle

text: plane

text: of

text: the

text: ﬂoating

text: bilayer.

text: reﬂected

text: intensity

text: was

text: deﬁned

text: using

text: a

text: m×m

text: slit

text: at

text: mm

text: from

text: the

text: sample

text: and

text: a

text: mm

text: slit

text: The

text: model

text: system

text: we

text: have

text: used

text: consisted

text: of

text: two

text: at

text: mm

text: from

text: the

text: sample

text: and

text: recorded

text: using

text: a

text: NaI

text: supported

text: bilayers

text: of

text: DSPC

text: (L−α

text: (Tl)

text: scintillator.

text: Avanti

text: Polar

text: Lipids,

text: Lan-

text: Oﬀ-specular

text: scattering

text: is

text: sensitive

text: to

text: both

text: static

text: de-

text: caster,

text: Alabama)

text: deposited

text: on

text: ultra-ﬂat

text: silicon

text: sub-

text: formation

text: and

text: thermal

text: ﬂuctuations

text: of

text: bilayers.

text: In

text: the

text: strates

text: (Fig.

text: All

text: the

text: experiments

text: were

text: per-

text: limit

text: of

text: small

text: amplitudes,

text: and

text: the

text: simple

text: case

text: of

text: a

text: sin-

text: formed

text: in

text: ﬂuid

text: phase

text: at

text: C.

text: The

text: ﬁrst

text: bilayer

text: serves

text: as

text: gle

text: bilayer

text: in

text: a∝n

text: potential

text: U,

text: the

text: scattered

text: a

text: spacer

text: to

text: reduce

text: the

text: interaction

text: between

text: the

text: ﬂoating

text: is

text: Isc

text: zqz−q

text: with

text: the

text: ﬂuctuation

text: spectrum

text: second

text: bilayer

text: and

text: the

text: substrate

text: and

text: keeps

text: it

text: free

text: to

text: ﬂuc-

text: zqz−q

text: tuate

text: Potential

text: was

text: applied

text: between

text: a

text: Cu

text: layer

text: deposited

text: at

text: the

text: back

text: of

text: the

text: thick

text: Si

text: substrate

text: and

text: an

text: ITO

text: coated

text: glass

text: plate

text: mounted

text: parallel

text: to

text: the

text: substrate,

text: h

text: q

text: U

text: cm

text: from

text: the

text: membrane.

text: We

text: used

text: a

text: keV

text: x-ray

text: beam

text: (wavelength

text: λ

text: H

text: The

text: Hamiltonian

text: of

text: the

text: system

text: is

text: given

text: by

text: nm)

text: at

text: the

text: CRG-IF

text: beamline

text: of

text: the

text: European

text: Syn-

text: H

text: chrotron

text: Radiation

text: Facility

text: (ESRF)

text: in

text: an

text: oﬀ-specular

text: ge-

text: (q)

text: h

text: q

text: z

text: Fitting

text: of

text: the

text: scattering

text: curves,

text: q

text: q

text: q

text: ometry

text: described

text: in

text: Fig.

text: The

text: grazing

text: angle

text: accounting

text: for

text: both

text: thermal

text: ﬂuctuations

text: and

text: the

text: static

text: of

text: incidence

text: is

text: kept

text: ﬁxed

text: (θin

text: below

text: the

text: roughness

text: induced

text: by

text: the

text: substrate

text: (following

text: the

text: critical

text: angle

text: for

text: total

text: external

text: reﬂection

text: θ

text: mrad

text: procedure

text: described

text: in

text: details

text: in

text: gives

text: access

text: to

text: c

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: 3 (cid:48)(cid:48) the bilayer electron density proﬁle, γ, κ and U . Diﬀer- ent scattering curves are presented in Figure 2.B and in 102 102 A B Supplemental Material, showing that high voltage dif- 101 ferences, up to 10 V can be applied to the cell without 100 destroying the membrane, but strongly aﬀecting its ﬂuc- tuations. Fig. 3 summarizes the main ﬁndings of this 10−1 Π = 0 101 paper. We clearly observe an electrostriction eﬀect on 10−2 structural properties. The thickness d of the water layer w 10−3 in between the two lipid bilayers decreases with the elec- tric ﬁeld (Fig. 3.A). Depending on voltage and frequency, 10−40 1 2 3 10−2 10−1 100 101 100 we also obse−rve large negative corrections to the tension dw (nm) Π (MPa) 101 Γel = γ(V=0) γ (up to 1 mN/m, Fig. 3.C,D)−and positive C D corrections to the bending stiﬀness K = κ κ (up to el (V=0) V = 5 V 0 a few hundr−eds of kBT, Fig. 3.E,F). Th−e measured values 100 100 γ = 0.5 1 mN/m and κ = 15 20k T are in good (V=0) (V=0) B agreement with known values for DSPC bilayer [26]. 10−1 10−1 Analyzing our results ﬁrst requires to determine the local voltage drop V seen by the bilayer. To this loc end, we model the system electrokinetics by solving 10−2 10−2 the Poisson-Nernst-Planck equations, generalizing the E F model of Ziebert et al [16] to the double supported bi- V = 5 V 0 layer (Fig. 2.A) [19]. The only unknown parameter is 100 − the Debye screening length κ 1, which might slightly 102 D depend on the dissolved carbon dioxide and ﬁxes the − 10−1 conductivity of the solution [28]. Whereas κ 1 = 960 nm D in pure water, it is reduced to 150 nm for normal atmo- spheric conditions. As scattering curves were recorded 101 10−2 100 101 102 0.0 0.1 1.0 5 to 10 hours after sample preparation, which can inﬂu- f (Hz) Vloc (V) − ence the Debye length, κ 1 = 150 nm, 300 nm and 800 nm D were used in the analysis. With ×these− values and a sin- gle diﬀusion coeﬃcient D = 7.5 10 9 m2/s for all ions FIG. 3: Eﬀect of an AC ﬁeld on a supported bilayer. Filled [29], the eﬀective membrane resistance, lower than its symbols• are data from this work. Black squares ((cid:4)), green intrinsic resistance, ranges from 20 Ω.cm2 to 300 Ω.cm2. circles ( ) and red triangles ((cid:78)) correspond to diﬀerent experi- The system behaves as a low-pass ﬁlter with a cut-oﬀ ments. Solid lines correspond to the full electrostatic contribu- fr−equency determined by the bulk solution conductance tion (Poisson-Boltzmann) to Γel and Kel which can be decom- R 1 and the electric double layer capacitance per unit posed in a membrane contribution (dotted line) and an electric B double layer contribution (dashed line). Linear Debye-Hu¨ckel area C (inset of Fig. 1.B), the highest resistance and ca- − DL theory is shown as dashed-dotted lines. κ 1 = 800 nm (red pacitance in the syste−m respectively. Depending o−n De- curves), κ−1 = 300 nm (green curves) and κ−D1 = 150 nm (black bye length, R = 0.5 10 MΩ.cm2 and C = 0.04 0.18 D D B DL − curves). Mechanical pressure as a function of distance (A). Blue µF/cm2, leading to cutoﬀ− frequencies of 0.2 Hz for κD1 stars are from Ref. [24] where pressure was applied by osmotic = 150 nm to 3 Hz for κ 1 = 800 nm. Accordingly, the stress on similar double bilayers. Empty symbols are from D voltage drop at th≈e membrane increases from less than Ref. [30] (osmotic stress on multilayer stacks). The solid line is 0.01V at 50 Hz to 0.04V at 10 Hz, where V is the AC calculated after Ref. [24] using dispersive, electrostatic and en- 0 0 0 tropic contributions to the inter-bilayer potential. Number of ﬁeld applied to the membrane (Fig. 2.A). First discussing electrostriction, the most compress- water molecules per lip◦id nw as a function of electrostatic pres- sure Π (B). Blue circles ( ) were obtained by NMR spectroscopy ible part in the system is the water layer in between for osmotically stressed DMPC multilayer stacks[31]. Electro- the two lipid bilayers and the electromagnetic stress static contribution to the membrane tension Γ as a function of el is balanced by the interbilayer potential. By plotting frequency for a ﬁxed voltage V = 5 V (C) and as a function of 0 the electrostatic pressure Π [19] as a function of the the local voltage Vloc at the membrane (D). Electrostatic contri- interbilayer water thickness d (Fig. 3.A), all points fall bution to the membrane rigidity Kel as a function of frequency w for a ﬁxed voltage V = 5 V (E) and K κ /k T as a function of on a master curve obtained for both the natural entropic 0 el D B the local potential diﬀerence V (F). repulsion between bilayers [24] and osmotic stress, loc either applied on ﬂoating bilayers [24] or multilayer stacks [30], demonstrating that the local electromagnetic stress is well described by our model. We also report n [32] as a function of the pressure Π. Similar curves w in Fig. 3.B the number of water molecules per lipids obtained when the pressure is osmotically applied on )aPM(Π )m/Nm( leΓ ΤΒk/leK nw Γel (mN/m) KelκD/(kΒΤ ) (nm−1) ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: the

text: bilayer

text: electron

text: density

text: proﬁle,

text: γ,

text: κ

text: and

text: U

text: Diﬀer-

text: ent

text: scattering

text: curves

text: are

text: presented

text: in

text: Figure

text: and

text: in

font: default

text: A

text: B

text: Supplemental

text: Material,

text: showing

text: that

text: high

text: voltage

text: dif-

text: ferences,

text: up

text: to

text: V

text: can

text: be

text: applied

text: to

text: the

text: cell

text: without

text: destroying

text: the

text: membrane,

text: but

text: strongly

text: aﬀecting

text: its

text: ﬂuc-

text: tuations.

text: Fig.

text: summarizes

text: the

text: main

text: ﬁndings

text: of

text: this

text: Π

text: paper.

text: We

text: clearly

text: observe

text: an

text: electrostriction

text: eﬀect

text: on

text: structural

text: properties.

text: The

text: thickness

text: d

text: of

text: the

text: water

text: layer

text: w

text: in

text: between

text: the

text: two

text: lipid

text: bilayers

text: decreases

text: with

text: the

text: elec-

text: tric

text: ﬁeld

text: (Fig.

text: Depending

text: on

text: voltage

text: and

text: frequency,

text: we

text: also

text: obse−rve

text: large

text: negative

text: corrections

text: to

text: the

text: tension

text: dw

text: (nm)

text: Π

text: (MPa)

text: Γel

text: γ

text: (up

text: to

text: mN/m,

text: Fig.

text: positive

text: C

text: D

text: corrections

text: to

text: the

text: bending

text: stiﬀness

text: K

text: κ

text: κ

text: (up

text: to

text: el

text: V

text: V

text: a

text: few

text: hundr−eds

text: of

text: kBT,

text: Fig.

text: Th−e

text: measured

text: values

text: γ

text: mN/m

text: and

text: κ

text: T

text: are

text: in

text: good

text: B

text: agreement

text: with

text: known

text: values

text: for

text: DSPC

text: bilayer

text: Analyzing

text: our

text: results

text: ﬁrst

text: requires

text: to

text: determine

text: the

text: local

text: voltage

text: drop

text: V

text: seen

text: by

text: the

text: bilayer.

text: To

text: this

text: loc

text: end,

text: we

text: model

text: the

text: system

text: electrokinetics

text: by

text: solving

text: the

text: Poisson-Nernst-Planck

text: equations,

text: generalizing

text: the

text: E

text: F

text: model

text: of

text: Ziebert

text: et

text: al

text: to

text: the

text: double

text: supported

text: bi-

text: V

text: V

text: layer

text: (Fig.

text: The

text: only

text: unknown

text: parameter

text: is

text: the

text: Debye

text: screening

text: length

text: κ

text: which

text: might

text: slightly

text: D

text: depend

text: on

text: the

text: dissolved

text: carbon

text: dioxide

text: and

text: ﬁxes

text: the

text: conductivity

text: of

text: the

text: solution

text: Whereas

text: κ

text: nm

text: D

text: in

text: pure

text: water,

text: it

text: is

text: reduced

text: to

text: nm

text: for

text: normal

text: atmo-

text: spheric

text: conditions.

text: As

text: scattering

text: curves

text: were

text: recorded

text: to

text: hours

text: after

text: sample

text: preparation,

text: which

text: can

text: inﬂu-

text: f

text: (Hz)

text: Vloc

text: (V)

text: ence

text: the

text: Debye

text: length,

text: κ

text: nm,

text: nm

text: and

text: nm

text: D

text: were

text: used

text: in

text: the

text: analysis.

text: With

text: ×these−

text: values

text: and

text: a

text: sin-

text: gle

text: diﬀusion

text: coeﬃcient

text: D

text: for

text: all

text: ions

text: FIG.

text: Eﬀect

text: of

text: an

text: AC

text: ﬁeld

text: on

text: a

text: supported

text: bilayer.

text: Filled

text: the

text: eﬀective

text: membrane

text: resistance,

text: lower

text: than

text: its

text: symbols•

text: are

text: data

text: from

text: this

text: work.

text: Black

text: squares

text: green

text: intrinsic

text: resistance,

text: ranges

text: from

text: to

text: circles

text: and

text: red

text: triangles

text: correspond

text: to

text: diﬀerent

text: experi-

text: The

text: system

text: behaves

text: as

text: a

text: low-pass

text: ﬁlter

text: with

text: a

text: cut-oﬀ

text: ments.

text: Solid

text: lines

text: correspond

text: to

text: the

text: full

text: electrostatic

text: contribu-

text: fr−equency

text: determined

text: by

text: the

text: bulk

text: solution

text: conductance

text: tion

text: (Poisson-Boltzmann)

text: to

text: Γel

text: and

text: Kel

text: which

text: can

text: be

text: decom-

text: R

text: and

text: the

text: electric

text: double

text: layer

text: capacitance

text: per

text: unit

text: posed

text: in

text: a

text: membrane

text: contribution

text: (dotted

text: line)

text: and

text: an

text: electric

text: B

text: double

text: layer

text: contribution

text: (dashed

text: line).

text: Linear

text: Debye-Hu¨ckel

text: area

text: C

text: (inset

text: of

text: Fig.

text: the

text: highest

text: resistance

text: and

text: ca-

text: DL

text: theory

text: is

text: shown

text: as

text: dashed-dotted

text: lines.

text: κ

text: nm

text: (red

text: pacitance

text: in

text: the

text: syste−m

text: respectively.

text: Depending

text: o−n

text: De-

text: curves),

text: κ−1

text: nm

text: (green

text: curves)

text: and

text: κ−D1

text: nm

text: (black

text: bye

text: length,

text: R

text: and

text: C

text: D

text: D

text: B

text: DL

text: curves).

text: Mechanical

text: pressure

text: as

text: a

text: function

text: of

text: distance

text: (A).

text: Blue

text: leading

text: to

text: cutoﬀ−

text: frequencies

text: of

text: Hz

text: for

text: stars

text: are

text: from

text: Ref.

text: where

text: pressure

text: was

text: applied

text: by

text: osmotic

text: nm

text: to

text: Hz

text: for

text: κ

text: nm.

text: Accordingly,

text: the

text: stress

text: on

text: similar

text: double

text: bilayers.

text: Empty

text: symbols

text: are

text: from

text: D

text: voltage

text: drop

text: at

text: th≈e

text: membrane

text: increases

text: from

text: less

text: than

text: Ref.

text: (osmotic

text: stress

text: on

text: multilayer

text: stacks).

text: The

text: solid

text: line

text: is

text: at

text: Hz

text: to

text: at

text: Hz,

text: where

text: V

text: is

text: the

text: AC

text: calculated

text: after

text: Ref.

text: using

text: dispersive,

text: electrostatic

text: and

text: en-

text: tropic

text: contributions

text: to

text: the

text: inter-bilayer

text: potential.

text: Number

text: of

text: ﬁeld

text: applied

text: to

text: the

text: membrane

text: (Fig.

text: First

text: discussing

text: electrostriction,

text: the

text: most

text: compress-

text: water

text: molecules

text: per

text: lip◦id

text: nw

text: as

text: a

text: function

text: of

text: electrostatic

text: pres-

text: sure

text: Π

text: (B).

text: Blue

text: circles

text: were

text: obtained

text: by

text: NMR

text: spectroscopy

text: ible

text: part

text: in

text: the

text: system

text: is

text: the

text: water

text: layer

text: in

text: between

text: for

text: osmotically

text: stressed

text: DMPC

text: multilayer

text: Electro-

text: the

text: two

text: lipid

text: bilayers

text: and

text: the

text: electromagnetic

text: stress

text: static

text: contribution

text: to

text: the

text: membrane

text: tension

text: Γ

text: as

text: a

text: function

text: of

text: el

text: is

text: balanced

text: by

text: the

text: interbilayer

text: potential.

text: By

text: plotting

text: frequency

text: for

text: a

text: ﬁxed

text: voltage

text: V

text: V

text: (C)

text: and

text: as

text: a

text: function

text: of

text: the

text: electrostatic

text: pressure

text: Π

text: as

text: a

text: function

text: of

text: the

text: the

text: local

text: voltage

text: Vloc

text: at

text: the

text: membrane

text: (D).

text: Electrostatic

text: contri-

text: interbilayer

text: water

text: thickness

text: d

text: (Fig.

text: all

text: points

text: fall

text: bution

text: to

text: the

text: membrane

text: rigidity

text: Kel

text: as

text: a

text: function

text: of

text: frequency

text: w

text: for

text: a

text: ﬁxed

text: voltage

text: V

text: V

text: (E)

text: and

text: K

text: κ

text: /k

text: T

text: as

text: a

text: function

text: of

text: on

text: a

text: master

text: curve

text: obtained

text: for

text: both

text: the

text: natural

text: entropic

text: el

text: D

text: B

text: the

text: local

text: potential

text: diﬀerence

text: V

text: (F).

text: repulsion

text: between

text: bilayers

text: and

text: osmotic

text: stress,

text: loc

text: either

text: applied

text: on

text: ﬂoating

text: bilayers

text: or

text: multilayer

text: stacks

text: demonstrating

text: that

text: the

text: local

text: electromagnetic

text: stress

text: is

text: well

text: described

text: by

text: our

text: model.

text: We

text: also

text: report

text: n

text: as

text: a

text: function

text: of

text: the

text: pressure

text: Π.

text: Similar

text: curves

text: w

text: in

text: Fig.

text: the

text: number

text: of

text: water

text: molecules

text: per

text: lipids

text: obtained

text: when

text: the

text: pressure

text: is

text: osmotically

text: applied

text: on

text: )aPM(Π

text: )m/Nm(

text: leΓ

text: ΤΒk/leK

text: nw

text: Γel

text: (mN/m)

text: KelκD/(kΒΤ

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Theorem 1.1 (Odd Degree Theorem, [5], cf. [2, Corollary 9.4]). Let F be a number ﬁeld of odd degree, let E be a K-CM elliptic curve, and let T = E(F )[tors]. Then: /F a) One of the following occurs: {•} Z Z Z Z Z Z ⊕√Z Z (1) T ∼is isomorphic to the trivial group , /2 , /4 , or /2 /2 ; (2) T =∼ Z/(cid:96)nZ for a prime (cid:96) ≡ 3 (mod 8) and n ∈ Z+ and K = Q( √−(cid:96)); Z Z ≡ ∈ Z Q − (3) T = /2(cid:96)∼n for a prime (cid:96) 3 (mod 4) and n + and K = ( (cid:96)). Z Z ⊕ Z Z − b) If E(F )[tors] =∼ /2 /2 , then End E has discriminant ∆ = 4. Z Z ∈ {− − } c) If E(F )[tors] = /4 , then End E has discriminant ∆ 4, 16 . d) Each of the groups listed in part a) arises up to isomorphism as the torsion subgroup E(F ) of a CM elliptic curve E deﬁned over an odd degree number ﬁeld F . However, given a particular subgroup that does arise, the argument of [5] does not identify the √ degrees d in which it occurs. The main theorem of this paper is precisely such a result. Here, √ Q − hQ − denotes the class number of ( (cid:96)). ( (cid:96)) (cid:4) (cid:5) Theorem 1.2 (Strong Odd Degree Theorem). Let (cid:96) ≡ 3 (mod 4) and n ∈ Z+. Deﬁne δ as follows: − (cid:4)3n(cid:5) 1, (cid:96) > 3, 2 δ = 0, (cid:96) = 3 and n = 1, − ≥ 3n 2, (cid:96) = 3 and n 2. 2 Then: {•} Z Z Z Z Z Z ⊕ Z Z (1) For any odd positive integer d, the groups , /2 , /4 , and /2 /2 appear as the torsion subgroup of a CM elliptic curve deﬁned over a number ﬁeld of degree d. Z Z (2) /(cid:96)n appears as the torsion subgroup of a CM elliptic curve deﬁned over a number ﬁeld of ≡ √ · − · odd degree d if and only if (cid:96) 3 (mod 8) and d is a multiple of hQ − (cid:96) 1 (cid:96)δ. Z Z ( (cid:96)) 2 (3) /2(cid:96)n appears as the torsion subgroup of a CM elliptic curve deﬁned over a number ﬁeld of odd degree d if and only if one of the following holds: ≡ ≥ · √ · − · a. (cid:96) 3 (mod 8), where n 2 if (cid:96) = 3, and d is a multiple of 3 hQ − (cid:96) 1 (cid:96)δ, or ( (cid:96)) 2 b. (cid:96) = 3 and n = 1 and d is any odd positive integer, or ≡ √ · − · c. (cid:96) 7 (mod 8) and d is a multiple of hQ − (cid:96) 1 (cid:96)δ. ( (cid:96)) 2 This theorem can be used to algorithmically determine G (d) for any odd degree d. See section 7 ≤ CM for a table of the groups which arise for odd d 99. The CM elliptic curves with a point of order (cid:96)n in lowest possible odd degree are unexpectedly varied. In the case of even degrees, points of order (cid:96) often appear for the ﬁrst time on a CM elliptic ≡ curve with a rational j-invariant. For example, any prime (cid:96) 1 (mod 3) appears for the ﬁrst time − in even degree (cid:96) 1 on an elliptic curve E with j(E) = 0 by [8, Theorem 1]. However, we see already 3 from the Odd Degree Theorem that a CM elliptic curve with a rational point of order (cid:96) in odd degree will not have a rational j-invariant once (cid:96) > 163. Moreover, through the proof of the Strong Odd Degree Theorem, we ﬁnd that the algebraic structure of these optimal examples is surprisingly ≥ complex. Speciﬁcally, we ﬁnd that if n 3, an elliptic curve with a point of order (cid:96)n in lowest possible odd degree necessarily has CM by a non-maximal order, and the size of the conductor increases with n. See Remark 2.7 for a precise statement along these lines. Non-maximal orders provide considerable technical complications, due in part to the fact that ideals do not necessarily factor uniquely into prime ideals, and so many results in the literature are formulated only to address the case of CM by the full ring of integers. However, we know now that non-maximal orders play a crucial role in the extremal behavior of rational torsion points of elliptic curves. Theorem 1.2 opens the door to establishing new statistical properties of G (d) and T (d), as ≤ CM CM d ranges over odd integers. The mean of T (d) for odd d x is studied already in [4], where it is CM 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Theorem

text: (Odd

text: Degree

text: Theorem,

text: cf.

text: Corollary

text: Let

text: F

text: be

text: a

text: number

text: ﬁeld

text: of

text: odd

text: degree,

text: let

text: E

text: be

text: a

text: K-CM

text: elliptic

text: curve,

text: and

text: let

text: T

text: E(F

text: )[tors].

text: Then:

text: /F

text: a)

label: list

text: One

text: of

text: the

text: following

text: occurs:

text: Z

text: Z

text: Z

text: Z

text: Z

text: Z

text: ⊕√Z

text: Z

text: T

text: ∼is

text: isomorphic

text: to

text: the

text: trivial

text: group

text: or

text: T

text: for

text: a

text: prime

text: (mod

text: and

text: n

text: Z+

text: and

text: K

text: Q(

text: Z

text: Z

text: Z

text: Q

text: T

text: for

text: a

text: prime

text: (mod

text: and

text: n

text: and

text: K

text: Z

text: Z

text: Z

text: Z

text: b)

text: If

text: E(F

text: )[tors]

text: then

text: End

text: E

text: has

text: discriminant

text: Z

text: Z

text: c)

text: If

text: E(F

text: )[tors]

text: then

text: End

text: E

text: has

text: discriminant

text: d)

text: Each

text: of

text: the

text: groups

text: listed

text: in

text: part

text: a)

text: arises

text: up

text: to

text: isomorphism

text: as

text: the

text: torsion

text: subgroup

text: E(F

text: of

text: a

text: CM

text: elliptic

text: curve

text: E

text: deﬁned

text: over

text: an

text: odd

text: degree

text: number

text: ﬁeld

text: F

text: However,

text: given

text: a

text: particular

text: subgroup

text: that

text: does

text: arise,

text: the

text: argument

text: of

text: does

text: not

text: identify

text: the

text: degrees

text: d

text: in

text: which

text: it

text: occurs.

text: The

text: main

text: theorem

text: of

text: this

text: paper

text: is

text: precisely

text: such

text: a

text: result.

text: Here,

text: Q

text: hQ

text: denotes

text: the

text: class

text: number

text: of

text: Theorem

text: (Strong

text: Odd

text: Degree

text: Theorem).

text: Let

text: (mod

text: and

text: n

text: Z+.

text: Deﬁne

text: δ

text: as

text: follows:

text: δ

text: and

text: n

text: and

text: n

text: Then:

text: Z

text: Z

text: Z

text: Z

text: Z

text: Z

text: Z

text: Z

text: For

text: any

text: odd

text: positive

text: integer

text: d,

text: the

text: groups

text: and

text: appear

text: as

text: the

text: torsion

text: subgroup

text: of

text: a

text: CM

text: elliptic

text: curve

text: deﬁned

text: over

text: a

text: number

text: ﬁeld

text: of

text: degree

text: d.

text: Z

text: Z

text: appears

text: as

text: the

text: torsion

text: subgroup

text: of

text: a

text: CM

text: elliptic

text: curve

text: deﬁned

text: over

text: a

text: number

text: ﬁeld

text: of

text: odd

text: degree

text: d

text: if

text: and

text: only

text: if

text: (mod

text: and

text: d

text: is

text: a

text: multiple

text: of

text: hQ

text: Z

text: Z

text: appears

text: as

text: the

text: torsion

text: subgroup

text: of

text: a

text: CM

text: elliptic

text: curve

text: deﬁned

text: over

text: a

text: number

text: ﬁeld

text: of

text: odd

text: degree

text: d

text: if

text: and

text: only

text: if

text: one

text: of

text: the

text: following

text: holds:

text: a.

text: (mod

text: where

text: n

text: if

text: and

text: d

text: is

text: a

text: multiple

text: of

text: hQ

text: or

text: b.

text: and

text: n

text: and

text: d

text: is

text: any

text: odd

text: positive

text: integer,

text: or

text: c.

text: (mod

text: and

text: d

text: is

text: a

text: multiple

text: of

text: hQ

text: This

text: theorem

text: can

text: be

text: used

text: to

text: algorithmically

text: determine

text: G

text: (d)

text: for

text: any

text: odd

text: degree

text: d.

text: See

text: section

text: CM

text: for

text: a

text: table

text: of

text: the

text: groups

text: which

text: arise

text: for

text: odd

text: d

text: The

text: CM

text: elliptic

text: curves

text: with

text: a

text: point

text: of

text: order

text: in

text: lowest

text: possible

text: odd

text: degree

text: are

text: unexpectedly

text: varied.

text: In

text: the

text: case

text: of

text: even

text: degrees,

text: points

text: of

text: order

text: often

text: appear

text: for

text: the

text: ﬁrst

text: time

text: on

text: a

text: CM

text: elliptic

text: curve

text: with

text: a

text: rational

text: j-invariant.

text: For

text: example,

text: any

text: prime

text: (mod

text: appears

text: for

text: the

text: ﬁrst

text: time

text: in

text: even

text: degree

text: on

text: an

text: elliptic

text: curve

text: E

text: with

text: j(E)

text: by

text: Theorem

text: However,

text: we

text: see

text: already

text: from

text: the

text: Odd

text: Degree

text: Theorem

text: that

text: a

text: CM

text: elliptic

text: curve

text: with

text: a

text: rational

text: point

text: of

text: order

text: in

text: odd

text: degree

text: will

text: not

text: have

text: a

text: rational

text: j-invariant

text: once

text: Moreover,

text: through

text: the

text: proof

text: of

text: the

text: Strong

text: Odd

text: Degree

text: Theorem,

text: we

text: ﬁnd

text: that

text: the

text: algebraic

text: structure

text: of

text: these

text: optimal

text: examples

text: is

text: surprisingly

text: complex.

text: Speciﬁcally,

text: we

text: ﬁnd

text: that

text: if

text: n

text: an

text: elliptic

text: curve

text: with

text: a

text: point

text: of

text: order

text: in

text: lowest

text: possible

text: odd

text: degree

text: necessarily

text: has

text: CM

text: by

text: a

text: non-maximal

text: order,

text: and

text: the

text: size

text: of

text: the

text: conductor

text: increases

text: with

text: n.

text: See

text: Remark

text: for

text: a

text: precise

text: statement

text: along

text: these

text: lines.

text: Non-maximal

text: orders

text: provide

text: considerable

text: technical

text: complications,

text: due

text: in

text: part

text: to

text: the

text: fact

text: that

text: ideals

text: do

text: not

text: necessarily

text: factor

text: uniquely

text: into

text: prime

text: ideals,

text: and

text: so

text: many

text: results

text: in

text: the

text: literature

text: are

text: formulated

text: only

text: to

text: address

text: the

text: case

text: of

text: CM

text: by

text: the

text: full

text: ring

text: of

text: integers.

text: However,

text: we

text: know

text: now

text: that

text: non-maximal

text: orders

text: play

text: a

text: crucial

text: role

text: in

text: the

text: extremal

text: behavior

text: of

text: rational

text: torsion

text: points

text: of

text: elliptic

text: curves.

text: Theorem

text: opens

text: the

text: door

text: to

text: establishing

text: new

text: statistical

text: properties

text: of

text: G

text: (d)

text: and

text: T

text: (d),

text: as

text: CM

text: CM

text: d

text: ranges

text: over

text: odd

text: integers.

text: The

text: mean

text: of

text: T

text: (d)

text: for

text: odd

text: d

text: x

text: is

text: studied

text: already

text: in

text: where

text: it

text: is

text: CM

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Lemmas 4.1 and 4.2 make pleasant elementary exercises, and we omit the proofs. The more diﬃcult of the two, Lemma 4.2, appears in more general form in work of Behrend [3, Lemma, p. 681]. G Proof of the upper bound in Theorem 1.5. We begin by computing a large list of elements of H { } which eventually will be truncated to form our . Speciﬁcally, we start with the singleton set 2 . ≤ We then successively go through the primes 3 < (cid:96) 100000 from the congruence class 3 mod 4, throwing g into our set whenever g is not divisible by a preexisting element. (If g is divisible (cid:96) (cid:96) (cid:96) by such an element, then there is no need to throw it in, as this would not lead to a larger set of multiples.) At the end of this process, we sort the resulting list; this leaves us with a set the ﬁrst several elements of which are 2, 3, 5, 913, 1631, 1703, 2051, 2891, 3247, 3401, 3619, 4067, 5327, 6251, 6617, 7051, 7183, 7429, 9737, 10829, 11129, 11143, 12389, 12463, 12673, 12847, 17611, 18403, 19253, 19931, 20033, 22211, 22747, 23351, 27491, 28237, 30173, 32927, 33541, 38171, 38641, 39311, 39689, 40687, 42601, 45103, . . . . H We let consist of the ﬁrst 38 elements of this list, so that H { } = 2, 3, 5, 913, . . . , 32927 . We will show that − H 1 d(M ( )) < 0.265. This implies the same upper bound 0.265 for the density of Olson degrees. H H { } (cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19) Apply Lemma 4.1 to . In this case, one computes that = 2, 3, 5, 11129, 27491 . Thus, H(cid:48) H \ H rel puting = , rel − H − H(cid:48) − 1 − 1 − 1 − 1 − 1 1 d(M ( )) = (1 d(M ( ))) 1 1 1 1 1 . 2 (cid:18)3 (cid:19)5 11129 27491 H(cid:48) To estimate d(M ( )), we apply Lemma 4.2 with p = 11: H(cid:48) 1 H(cid:48) − 1 H(cid:48) d(M ( )) = d(M ( )) + 1 d(M ( (11))). 11 (11) 11 H(cid:48) H(cid:48) The set (11) has only 23 elements, and so d(M ( (11))) can be computed without fuss by inclusion-exclusion. We ﬁnd that H(cid:48) d(M ( (11))) = 0.004217267361708 . . . . H(cid:48)(cid:48) H(cid:48) H(cid:48)(cid:48) Let = . Then has 33 elements; 33 is large enough that a direct inclusion-exclusion (11) H(cid:48)(cid:48) computation is best avoided. So we (cid:18)make ano(cid:19)th(cid:18)er applic(cid:19)at(cid:18)ion of Lem(cid:19)m(cid:18)a 4.1, this(cid:19)ti(cid:18)me to .(cid:19)We H(cid:48)(cid:48) { } H(cid:48)(cid:48)(cid:48) H(cid:48)(cid:48) \ H(cid:48)(cid:48) compute that = 641, 653, 1013, 1133, 1601 . So with = , rel rel − H(cid:48)(cid:48) − H(cid:48)(cid:48)(cid:48) − 1 − 1 − 1 − 1 − 1 1 d(M ( )) = (1 d(M ( )) 1 1 1 1 1 . 641 653 1013 1133 1601 H(cid:48)(cid:48)(cid:48) · The set has 28 elements. However, it contains both 83 and 4067 = 83 49. So we may remove H(cid:48)(cid:48)(cid:48) H(cid:48)(cid:48)(cid:48) 4067 from without changing the corresponding set of multiples. Similarly, contains both · H(cid:48)(cid:48)(cid:48) 329 and 6251 = 19 329, and so 6251 can also be removed. This brings # down to 26, which is small enough that the inclusion-exclusion computation is manageable. We ﬁnd that − H(cid:48)(cid:48)(cid:48) 1 d(M ( )) = 0.979914305743609 . . . . Working back through the chain of equalities, − H(cid:48)(cid:48) 1 d(M ( )) = 0.974452539520107 . . . , H(cid:48) d(M ( )) = 0.006156375826997 . . . , 15 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Lemmas

text: and

text: make

text: pleasant

text: elementary

text: exercises,

text: and

text: we

text: omit

text: the

text: proofs.

text: The

text: more

text: diﬃcult

text: of

text: the

text: two,

text: Lemma

text: appears

text: in

text: more

text: general

text: form

text: in

text: work

text: of

text: Behrend

text: Lemma,

text: p.

text: G

text: Proof

text: of

text: the

text: upper

text: bound

text: in

text: Theorem

text: We

text: begin

text: by

text: computing

text: a

text: large

text: list

text: of

text: elements

text: of

text: H

text: which

text: eventually

text: will

text: be

text: truncated

text: to

text: form

text: our

text: Speciﬁcally,

text: we

text: start

text: with

text: the

text: singleton

text: set

text: We

text: then

text: successively

text: go

text: through

text: the

text: primes

text: from

text: the

text: congruence

text: class

text: mod

text: throwing

text: g

text: into

text: our

text: set

text: whenever

text: g

text: is

text: not

text: divisible

text: by

text: a

text: preexisting

text: element.

text: (If

text: g

text: is

text: divisible

text: by

text: such

text: an

text: element,

text: then

text: there

text: is

text: no

text: need

text: to

text: throw

text: it

text: in,

text: as

text: this

text: would

text: not

text: lead

text: to

text: a

text: larger

text: set

text: of

text: multiples.)

text: At

text: the

text: end

text: of

text: this

text: process,

text: we

text: sort

text: the

text: resulting

text: list;

text: this

text: leaves

text: us

text: with

text: a

text: set

text: the

text: ﬁrst

text: several

text: elements

text: of

text: which

text: are

text: H

text: We

text: let

text: consist

text: of

text: the

text: ﬁrst

text: elements

text: of

text: this

text: list,

text: so

text: that

text: H

text: We

text: will

text: show

text: that

text: H

text: d(M

text: This

text: implies

text: the

text: same

text: upper

text: bound

text: for

text: the

text: density

text: of

text: Olson

text: degrees.

text: H

text: H

text: Apply

text: Lemma

text: to

text: In

text: this

text: case,

text: one

text: computes

text: that

text: Thus,

text: H

text: H

text: rel

text: puting

text: rel

text: H

text: d(M

text: d(M

text: To

text: estimate

text: d(M

text: we

text: apply

text: Lemma

text: with

text: p

text: d(M

text: d(M

text: d(M

text: The

text: set

text: has

text: only

text: elements,

text: and

text: so

text: d(M

text: can

text: be

text: computed

text: without

text: fuss

text: by

text: inclusion-exclusion.

text: We

text: ﬁnd

text: that

text: d(M

text: Let

text: Then

text: has

text: elements;

text: is

text: large

text: enough

text: that

text: a

text: direct

text: inclusion-exclusion

text: computation

text: is

text: best

text: avoided.

text: So

text: we

text: of

text: to

text: compute

text: that

text: So

text: with

text: rel

text: rel

text: d(M

text: d(M

text: The

text: set

text: has

text: elements.

text: However,

text: it

text: contains

text: both

text: and

text: So

text: we

text: may

text: remove

text: from

text: without

text: changing

text: the

text: corresponding

text: set

text: of

text: multiples.

text: Similarly,

text: contains

text: both

text: and

text: and

text: so

text: can

text: also

text: be

text: removed.

text: This

text: brings

text: down

text: to

text: which

text: is

text: small

text: enough

text: that

text: the

text: inclusion-exclusion

text: computation

text: is

text: manageable.

text: We

text: ﬁnd

text: that

text: d(M

text: Working

text: back

text: through

text: the

text: chain

text: of

text: equalities,

text: d(M

text: d(M

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:90) (cid:90) ∞ 2 (cid:90) 10115 log(cid:18)t√ (cid:19) 2 (cid:90) 1 (cid:18) (cid:19) = − dΠ(t) + · − · − dΠ(t) 0.041 2.8·109 (t 1) t 3 104 10115 (t 1) t1/2 (cid:15)0 ∞ (cid:48) ∞ (cid:48) 2 − log t√ 2 − 1 < Π(t) − dt + · Π(t) − − dt. 0.041 2.8·109 (t 1) t 3 104 10115 (t 1)t1/2 (cid:15)0 (cid:88) Inserting the above upper bound for Π(t) and using Mathematica to bound the resulting integrals from above, we ﬁnd that (cid:48)(cid:48) < 0.001220. (cid:16)(cid:88) 3 (cid:88) (cid:88) (cid:17) Putting everything together, G \ { } ≤ 4 (cid:0) (cid:1) d(M ( ) M ( 2, 3, 5 )) + + 15 1 2 3 4 − < 0.00788 + 0.0001819 + 10 11 + 0.001220 < 0.00248. 15 (cid:3) This establishes (9) and so completes the proof of the lower bound in Theorem 1.5. 5. Stratification of torsion in odd degrees: Proof of Theorem 1.7 G We begin with an analytic lemma concerning integers with prescribed sets of divisors. Let be a D G { ∈ G | } set of positive integers. For each positive integer n, put (n, ) = g : g n . G Lemma 5.1. Let be a set of odd positive integers, and suppose that the sum of the reciprocals of G H G D G H the elements of converges. Let be any ﬁnite subset of . The set of odd n with (n, ) = possesses a well-deﬁned asymptotic density; this density is positive as long as there is at least one such n. Proof. We prove the lemma in two steps. First, we show that the density exists, and then we show A D G H H positivity. Let be the set of odd n with (n, ) = . For each real z > max , put A { D G ∩ H} = n : (n, [1, z]) = . z (cid:48) H Notice that whenever z > z > max , A ⊂ A ⊂ A (10) (cid:48) . (cid:81) z z A A Now whether or not n belongs to depends only on n modulo 2 ∈G∩ g. Thus, is a z A A g [1,z] z ﬁnite union of congruence classes, and so d( ) exists. From (10), d( ) is a nonincreasing function z z of z, and so we we may deﬁne A δ = lim d( ). →∞ z z A We will show that has asymptotic density δ. · · In what follows, we continue to use d( ) for upper density, and we use d( ) for lower density. A ≤ A → ∞ A From the ﬁrst inclusion in (10), d( ) d( ) for all z; now letting z shows that has z A ∈ A ∈ A upper density at most δ. Now consider the lower density of . If n but n / , then n is z 18 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: dΠ(t)

text: dΠ(t)

text: (t

text: t

text: (t

text: log

text: t√

text: Π(t)

text: dt

text: Π(t)

text: dt.

text: (t

text: t

text: (t

text: Inserting

text: the

text: above

text: upper

text: bound

text: for

text: Π(t)

text: and

text: using

text: Mathematica

text: to

text: bound

text: the

text: resulting

text: integrals

text: from

text: above,

text: we

text: ﬁnd

text: that

text: Putting

text: everything

text: together,

text: G

text: d(M

text: M

text: This

text: establishes

text: and

text: so

text: completes

text: the

text: proof

text: of

text: the

text: lower

text: bound

text: in

text: Theorem

text: Stratification

text: of

text: torsion

text: in

text: odd

text: degrees:

text: Proof

text: of

text: Theorem

text: G

text: We

text: begin

text: with

text: an

text: analytic

text: lemma

text: concerning

text: integers

text: with

text: prescribed

text: sets

text: of

text: divisors.

text: Let

text: be

text: a

text: D

text: G

text: G

text: set

text: of

text: positive

text: integers.

text: For

text: each

text: positive

text: integer

text: n,

text: put

text: (n,

text: g

text: g

text: n

text: G

text: Lemma

text: Let

text: be

text: a

text: set

text: of

text: odd

text: positive

text: integers,

text: and

text: suppose

text: that

text: the

text: sum

text: of

text: the

text: reciprocals

text: of

text: G

text: H

text: G

text: D

text: G

text: H

text: the

text: elements

text: of

text: converges.

text: Let

text: be

text: any

text: ﬁnite

text: subset

text: of

text: The

text: set

text: of

text: odd

text: n

text: with

text: (n,

text: possesses

text: a

text: well-deﬁned

text: asymptotic

text: density;

text: this

text: density

text: is

text: positive

text: as

text: long

text: as

text: there

text: is

text: at

text: least

text: one

text: such

text: n.

text: Proof.

text: We

text: prove

text: the

text: lemma

text: in

text: two

text: steps.

text: First,

text: we

text: show

text: that

text: the

text: density

text: exists,

text: and

text: then

text: we

text: show

text: A

text: D

text: G

text: H

text: H

text: positivity.

text: Let

text: be

text: the

text: set

text: of

text: odd

text: n

text: with

text: (n,

text: For

text: each

text: real

text: z

text: max

text: put

text: A

text: D

text: G

text: H}

text: n

text: (n,

text: z])

text: z

text: H

text: Notice

text: that

text: whenever

text: z

text: z

text: max

text: A

text: A

text: A

text: z

text: z

text: A

text: A

text: Now

text: whether

text: or

text: not

text: n

text: belongs

text: to

text: depends

text: only

text: on

text: n

text: modulo

text: ∈G∩

text: g.

text: Thus,

text: is

text: a

text: z

text: A

text: A

text: g

text: z

text: ﬁnite

text: union

text: of

text: congruence

text: classes,

text: and

text: so

text: d(

text: exists.

text: From

text: d(

text: is

text: a

text: nonincreasing

text: function

text: z

text: z

text: of

text: z,

text: and

text: so

text: we

text: we

text: may

text: deﬁne

text: A

text: δ

text: lim

text: d(

text: z

text: z

text: A

text: We

text: will

text: show

text: that

text: has

text: asymptotic

text: density

text: δ.

text: In

text: what

text: follows,

text: we

text: continue

text: to

text: use

text: d(

text: for

text: upper

text: density,

text: and

text: we

text: use

text: d(

text: for

text: lower

text: density.

text: A

text: A

text: A

text: From

text: the

text: ﬁrst

text: inclusion

text: in

text: d(

text: d(

text: for

text: all

text: z;

text: now

text: letting

text: z

text: shows

text: that

text: has

text: z

text: A

text: A

text: A

text: upper

text: density

text: at

text: most

text: δ.

text: Now

text: consider

text: the

text: lower

text: density

text: of

text: If

text: n

text: but

text: n

text: then

text: n

text: is

text: z

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: → ∞ shown to be x1/3+o(1), as x . This should be compared with the unrestricted average, which we recalled above was x/(log x)1+o(1). In the next theorem, we study the upper order of T (d) for CM odd d. Again, we ﬁnd that it is much smaller than the corresponding unrestricted statistic. Theorem 1.3 (Upper order of TCM(d(cid:18)) (cid:18)for odd d). (cid:19) (cid:19) → ∞ (1) There is an inﬁnite sequence of odd d, with d , where ≥ 12eγ 2/3 T (d) + o(1) d log log d . CM π (cid:18)(cid:18) (cid:19) (cid:19) → ∞ (2) Assume the Riemann Hypothesis for Dirichlet L-functions. Then as d through all odd integers, ≤ 24eγ 2/3 T (d) + o(1) d log log d . CM π (cid:28) Unconditionally, T (d) d2/3+(cid:15) for each ﬁxed (cid:15) > 0 and all odd d. CM (cid:15) We turn next to understanding the “stratiﬁcation” of torsion by degree. {{•} Z Z Z Z Z Z Z Z Z Z⊕Z Z} In 1974, Olson showed that G (1) = , /2 , /3 , /4 , /6 , /2 /2 [26]. Compu- CM tations carried out in [9] revealed that this same list reoccurs as G (d) for several other small values CM of d. This was one of the phenomena investigated in [5], where it was shown that G (d) = G (1) ≥ CM CM if d = p or p2 for some prime p 7. Call d an Olson degree if G (d) = G (1). The following complete classiﬁcation of Olson degrees CM CM G was proved in [4]. To state the result, we need one more piece of notation. Given a set of positive G G integers, we write M ( ) for the set of multiples of , meaning the collection of all positive integers G divisible by some element of(cid:26) . (cid:27) G Proposition 1.4 ([4]). The complement of the set of Olson degrees can be written as M ( ), where − G { } ∪ (cid:96) 1 · √ ≡ = 2 hQ − : (cid:96) prime, (cid:96) 3 (mod 4), (cid:96) > 3 . 2 ( (cid:96)) As a corollary of Proposition 1.4, it was proved in [5] that the set of Olson degrees possesses a § positive density. It is easy, for reasons recalled at the start of , to rigorously bound this density from above by 4/15 = 0.26666 . . . . However, no explicit lower bound is proved in [4]. We take the opportunity here to address this lacuna, estimating the density of Olson degrees to within 0.1%. Theorem 1.5. The density of Olson degrees lies in the open interval (0.264, 0.265). Thus, a little more than half of the odd numbers d are Olson degrees. It is natural to wonder if our results on Olson degrees are the tip of a larger iceberg. Generalizing (cid:48) (cid:48) the above, we say that d and d are CM-torsion-equivalent if G (d) = G (d ). In this case, we (cid:48) CM CM call d a d -Olson degree. The following questions were suggested by Pete L. Clark: Questions 1.6. Is it true that for every d, the set of d-Olson degrees possesses an asymptotic density? If so, is the sum of the densities of d-Olson degrees, taken over inequivalent d, equal to 1? To avoid a possible source of confusion, we remind the reader that asymptotic density is ﬁnitely additive but not countably additive. Thus, an aﬃrmative answer to the ﬁrst question does not immediately imply an aﬃrmative answer to the second. (cid:48) (cid:48) Note that if d and d are CM-torsion equivalent, then d and d share the same parity. Indeed, (cid:48) Z Z ⊕ Z Z (cid:48) if d is even, then /3 /3 is realizable in degree d (see [5, Theorems 1.4, 2.1]), whereas the Odd Degree Theorem guarantees that such a group is never realizable in any odd degree d, so that (cid:54) (cid:48) G (d) = G (d ). Consequently, each equivalence class consists entirely of even integers or entirely CM CM of odd integers. Using Theorem 1.2, we are able to answer aﬃrmatively the odd degree variants of Questions 1.6. 3 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: shown

text: to

text: be

text: as

text: x

text: This

text: should

text: be

text: compared

text: with

text: the

text: unrestricted

text: average,

text: which

text: we

text: recalled

text: above

text: was

text: x/(log

text: In

text: the

text: next

text: theorem,

text: we

text: study

text: the

text: upper

text: order

text: of

text: T

text: (d)

text: for

text: CM

text: odd

text: d.

text: Again,

text: we

text: ﬁnd

text: that

text: it

text: is

text: much

text: smaller

text: than

text: the

text: corresponding

text: unrestricted

text: statistic.

text: Theorem

text: (Upper

text: order

text: of

text: odd

text: d).

text: There

text: is

text: an

text: inﬁnite

text: sequence

text: of

text: odd

text: d,

text: with

text: d

text: where

text: T

text: (d)

text: d

text: log

text: log

text: d

text: CM

text: π

text: Assume

text: the

text: Riemann

text: Hypothesis

text: for

text: Dirichlet

text: L-functions.

text: Then

text: as

text: d

text: through

text: all

text: odd

text: integers,

text: T

text: (d)

text: d

text: log

text: log

text: d

text: CM

text: π

text: Unconditionally,

text: T

text: (d)

text: for

text: each

text: ﬁxed

text: and

text: all

text: odd

text: d.

text: CM

text: We

text: turn

text: next

text: to

text: understanding

text: the

text: “stratiﬁcation”

text: of

text: torsion

text: by

text: degree.

text: Z

text: Z

text: Z

text: Z

text: Z

text: Z

text: Z

text: Z

text: Z

text: Z⊕Z

text: Z}

text: In

text: Olson

text: showed

text: that

text: G

text: Compu-

text: CM

text: tations

text: carried

text: out

text: in

text: revealed

text: that

text: this

text: same

text: list

text: reoccurs

text: as

text: G

text: (d)

text: for

text: several

text: other

text: small

text: values

text: CM

text: of

text: d.

text: This

text: was

text: one

text: of

text: the

text: phenomena

text: investigated

text: in

text: where

text: it

text: was

text: shown

text: that

text: G

text: (d)

text: G

text: CM

text: CM

text: if

text: d

text: p

text: or

text: for

text: some

text: prime

text: p

text: Call

text: d

text: an

text: Olson

text: degree

text: if

text: G

text: (d)

text: G

text: The

text: following

text: complete

text: classiﬁcation

text: of

text: Olson

text: degrees

text: CM

text: CM

text: G

text: was

text: proved

text: in

text: To

text: state

text: the

text: result,

text: we

text: need

text: one

text: more

text: piece

text: of

text: notation.

text: Given

text: a

text: set

text: of

text: positive

text: G

text: G

text: integers,

text: we

text: write

text: M

text: for

text: the

text: set

text: of

text: multiples

text: of

text: meaning

text: the

text: collection

text: of

text: all

text: positive

text: integers

text: G

text: divisible

text: by

text: some

text: element

text: G

text: Proposition

text: The

text: complement

text: of

text: the

text: set

text: of

text: Olson

text: degrees

text: can

text: be

text: written

text: as

text: M

text: where

text: G

text: hQ

text: prime,

text: (mod

text: As

text: a

text: corollary

text: of

text: Proposition

text: it

text: was

text: proved

text: in

text: that

text: the

text: set

text: of

text: Olson

text: degrees

text: possesses

text: a

text: positive

text: density.

text: It

text: is

text: easy,

text: for

text: reasons

text: recalled

text: at

text: the

text: start

text: of

text: to

text: rigorously

text: bound

text: this

text: density

text: from

text: above

text: by

text: However,

text: no

text: explicit

text: lower

text: bound

text: is

text: proved

text: in

text: We

text: take

text: the

text: opportunity

text: here

text: to

text: address

text: this

text: lacuna,

text: estimating

text: the

text: density

text: of

text: Olson

text: degrees

text: to

text: within

text: Theorem

text: The

text: density

text: of

text: Olson

text: degrees

text: lies

text: in

text: the

text: open

text: interval

text: Thus,

text: a

text: little

text: more

text: than

text: half

text: of

text: the

text: odd

text: numbers

text: d

text: are

text: Olson

text: degrees.

text: It

text: is

text: natural

text: to

text: wonder

text: if

text: our

text: results

text: on

text: Olson

text: degrees

text: are

text: the

text: tip

text: of

text: a

text: larger

text: iceberg.

text: Generalizing

text: the

text: above,

text: we

text: say

text: that

text: d

text: and

text: d

text: are

text: CM-torsion-equivalent

text: if

text: G

text: (d)

text: G

text: (d

text: In

text: this

text: case,

text: we

text: CM

text: CM

text: call

text: d

text: a

text: d

text: -Olson

text: degree.

text: The

text: following

text: questions

text: were

text: suggested

text: by

text: Pete

text: L.

text: Clark:

text: Questions

text: Is

text: it

text: true

text: that

text: for

text: every

text: d,

text: the

text: set

text: of

text: d-Olson

text: degrees

text: possesses

text: an

text: asymptotic

text: density?

text: If

text: so,

text: is

text: the

text: sum

text: of

text: the

text: densities

text: of

text: d-Olson

text: degrees,

text: taken

text: over

text: inequivalent

text: d,

text: equal

text: to

text: To

text: avoid

text: a

text: possible

text: source

text: of

text: confusion,

text: we

text: remind

text: the

text: reader

text: that

text: asymptotic

text: density

text: is

text: ﬁnitely

text: additive

text: but

text: not

text: countably

text: additive.

text: Thus,

text: an

text: aﬃrmative

text: answer

text: to

text: the

text: ﬁrst

text: question

text: does

text: not

text: immediately

text: imply

text: an

text: aﬃrmative

text: answer

text: to

text: the

text: second.

text: Note

text: that

text: if

text: d

text: and

text: d

text: are

text: CM-torsion

text: equivalent,

text: then

text: d

text: and

text: d

text: share

text: the

text: same

text: parity.

text: Indeed,

text: Z

text: Z

text: Z

text: Z

text: if

text: d

text: is

text: even,

text: then

text: is

text: realizable

text: in

text: degree

text: d

text: (see

text: Theorems

text: whereas

text: the

text: Odd

text: Degree

text: Theorem

text: guarantees

text: that

text: such

text: a

text: group

text: is

text: never

text: realizable

text: in

text: any

text: odd

text: degree

text: d,

text: so

text: that

text: G

text: (d)

text: G

text: (d

text: Consequently,

text: each

text: equivalence

text: class

text: consists

text: entirely

text: of

text: even

text: integers

text: or

text: entirely

text: CM

text: CM

text: of

text: odd

text: integers.

text: Using

text: Theorem

text: we

text: are

text: able

text: to

text: answer

text: aﬃrmatively

text: the

text: odd

text: degree

text: variants

text: of

text: Questions

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:18) (cid:19) argument o(cid:88)f Prachar (cid:88)[30]. Cf. the proof of [1, Proposition 10].) Hence, some individual term r(d) must also be large. Now − { (cid:96) 1 √ } ≤ r(d) = ≤ # (m, (cid:96)) : m (cid:18) 2 (cid:19)hQ( −(cid:96)) = d (cid:18) (cid:19) d |x d |x M d M d − − { | (cid:96) 1 √ (cid:96) 1 √ ≤ } = # (m, (cid:96)) : M m hQ − , and m hQ − x . 2 ( (cid:96)) 2 ( (cid:96)) We partition the pairs (m, (cid:96)) counted above according to the value of gcd(2M, m). Since we seek only a lower bound on the partial sums of r(d), it is enough to consider pairs with gcd(2M, m) highly restricted. Let T = (log x)A. | ∈ Given g 2M with g (T, 2T ], we construct pairs (m, (cid:96)) with gcd(2M, m) = 2M/g as follows: First, ≤ − | − ≤ ﬁx (cid:96) T 2 δ0 with g/ gcd(g, 2) (cid:96) 1. Choose any m √ x· − with gcd(2M, m) = 2M/g. hQ − ((cid:96) 1)/2 ( (cid:96)) Then the pair (m, (cid:96)) is counted above. Given (cid:96), inclusion-exclusion shows that the number of → ∞ corresponding m is (as x ) ∼ x 1 ϕ(g) − √ . (cid:96) 1hQ − 2M/g g 2 ( (cid:96)) √ (cid:28) (cid:29) Using hQ − (cid:96)1/2 log (cid:96) and ϕ(g) g/ log log g, the preceding expression is ( (cid:96)) (cid:29) x · g . M(log log x)2 (cid:96)3/2 (cid:88) ≤ − Summing on (cid:96), and recalling that (cid:96) T 2 δ0, the number of pairs we construct for our given g is (cid:29) x T − 1. M(log log x)2 T 32(2 δ0) ≡(cid:96)≤T2−δ0 (cid:88) ≡ (cid:96) 3 (mod 4) (cid:96) 1 (mod g/gcd(g,2)) Now − − − 1 = π(T 2 δ0; g/ gcd(g, 2), 1) π(T 2 δ0; 4g/ gcd(g, 2), 1). ≤ − ≡(cid:96) T2 δ0 (cid:96) 3 (mod 4) ≡ (cid:96) 1 (mod g/gcd(g,2)) In the notation of Proposition 6.1, − − ≥ · T 2 δ0 π(T 2 δ0; g/ gcd(g, 2), 1) 0.85 − ϕ(g/ gcd(g, 2)) log(T 2 δ0) ∈ − ∈ − unless g is even and g/2 E1(T 2 δ0, T/2) or g is odd and g E1(T 2 δ0, T ); from Proposition 6.1, the number of g involved in these exceptional sets is O(T/ log log x). Similarly, as long as g avoids a certain set of size O(T/ log log x), we have − − ≤ · T 2 δ0 π(T 2 δ0; 4g/ gcd(g, 2), 1) 1.48 − . 2ϕ(g/ gcd(g, 2)) log(T 2 δ0) Inserting these prime counting estimates above (noting that 1.48/2 < 0.85) we ﬁnd that as long as g avoids a set of size O(T/ log log x), we construct (cid:29) x T 1 (cid:29) x 1 − − M(log log x)2 T 12(2 δ0) g log T M(log log x)3 T 1 12δ0 pairs for a given g. 22 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: argument

text: Prachar

text: Cf.

text: the

text: proof

text: of

text: Proposition

text: Hence,

text: some

text: individual

text: term

text: r(d)

text: must

text: also

text: be

text: large.

text: Now

text: r(d)

text: (m,

text: m

text: d

text: d

text: |x

text: d

text: |x

text: M

text: d

text: M

text: d

text: (m,

text: M

text: m

text: hQ

text: and

text: m

text: hQ

text: x

text: We

text: partition

text: the

text: pairs

text: (m,

text: counted

text: above

text: according

text: to

text: the

text: value

text: of

text: m).

text: Since

text: we

text: seek

text: only

text: a

text: lower

text: bound

text: on

text: the

text: partial

text: sums

text: of

text: r(d),

text: it

text: is

text: enough

text: to

text: consider

text: pairs

text: with

text: m)

text: highly

text: restricted.

text: Let

text: T

text: (log

text: x)A.

text: Given

text: g

text: with

text: g

text: (T,

text: we

text: construct

text: pairs

text: (m,

text: with

text: m)

text: as

text: follows:

text: First,

text: ﬁx

text: T

text: δ0

text: with

text: g/

text: gcd(g,

text: Choose

text: any

text: m

text: x·

text: with

text: m)

text: hQ

text: Then

text: the

text: pair

text: (m,

text: is

text: counted

text: above.

text: Given

text: inclusion-exclusion

text: shows

text: that

text: the

text: number

text: of

text: corresponding

text: m

text: is

text: (as

text: x

text: x

text: ϕ(g)

text: g

text: Using

text: hQ

text: log

text: and

text: ϕ(g)

text: g/

text: log

text: log

text: g,

text: the

text: preceding

text: expression

text: is

text: x

text: g

text: M(log

text: log

text: Summing

text: on

text: and

text: recalling

text: that

text: T

text: the

text: number

text: of

text: pairs

text: we

text: construct

text: for

text: our

text: given

text: g

text: is

text: x

text: T

text: M(log

text: log

text: T

text: (mod

text: (mod

text: Now

text: π(T

text: g/

text: gcd(g,

text: π(T

text: gcd(g,

text: δ0

text: (mod

text: (mod

text: In

text: the

text: notation

text: of

text: Proposition

text: T

text: δ0

text: π(T

text: g/

text: gcd(g,

text: ϕ(g/

text: gcd(g,

text: log(T

text: unless

text: g

text: is

text: even

text: and

text: or

text: g

text: is

text: odd

text: and

text: g

text: T

text: from

text: Proposition

text: the

text: number

text: of

text: g

text: involved

text: in

text: these

text: exceptional

text: sets

text: is

text: O(T/

text: log

text: log

text: x).

text: Similarly,

text: as

text: long

text: as

text: g

text: avoids

text: a

text: certain

text: set

text: of

text: size

text: O(T/

text: log

text: log

text: x),

text: we

text: have

text: T

text: δ0

text: π(T

text: gcd(g,

text: gcd(g,

text: log(T

text: Inserting

text: these

text: prime

text: counting

text: estimates

text: above

text: (noting

text: that

text: we

text: ﬁnd

text: that

text: as

text: long

text: as

text: g

text: avoids

text: a

text: set

text: of

text: size

text: O(T/

text: log

text: log

text: x),

text: we

text: construct

text: x

text: T

text: x

text: M(log

text: log

text: T

text: g

text: log

text: T

text: M(log

text: log

text: T

text: pairs

text: for

text: a

text: given

text: g.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:3) Proof. This follows from Corollary 2.8. Let T be a group which appears as the torsion subgroup of a CM elliptic curve deﬁned over | Q a number ﬁeld of odd degree. We will identify an od∼d positive integer dT such that dT [F : ] whenever E/F is a CM elliptic curve with E(F )[tors] = T and F is of odd deg∼ree. Once we exhibit a number ﬁeld F of degree d and a CM elliptic curve E with E(F )[tors] = T , the Strong Odd T /F Degree Theorem will follow by a result of [5]: ≥ Theorem 2.10. Let A be an abelian variety over a number ﬁeld, and let d 2. There are /F inﬁnitely many L/F such that [L : F ] = d and A(L)[tors] = A(F )[tors]. (cid:3) Proof. See Theorem 2.1 of [5]. We isolate the more involved case in the following lemma. Here, δ is as deﬁned in the statement of the Strong Odd Degree Theorem. ∼ Lemma 2.11. Let F be a number ﬁeld of odd degree. If E is a CM elliptic curve with E(F )[tors] = Z Z ≡ ≥ ·/F √ · − · | Q /2(cid:96)n for (cid:96) 3 (mod 8), where n 2 if (cid:96) = 3, then 3 hQ − (cid:96) 1 (cid:96)δ [F : ]. ( (cid:96)) 2 ≡ (cid:54) wPritohofC. MWebywiallnﬁorrsdt ecronOsiodferditshcericmasineawnht e∆rei(cid:96)n K3, a(nmdosdu8p)p,o(cid:96)se=E3(.FS)u[ptopross]e∼=EZ/F/2i√(cid:96)snaZn. eBllyipTtihcecourrevme − · ∈ { } ∈ Z O ⊂ 2.1, ∆ = (2(cid:15)(cid:96)a)2 (cid:96) for (cid:15) 0, 1 and a ≥0. If (cid:98)(cid:15)n(cid:99)=O 0, then⊂K2 K F ( d) = F K byO[5, Lemma(cid:98)3.1(cid:99)5] and Proposition 2.4. By Corollary 2.8, K(cid:96) 2 K(ζ(cid:96)n) F K. Since 2 ramiﬁes in K2 K n O O an·d K(cid:96) −2 K(ζ|(cid:96)n) is unQramiﬁed ·away· f−rom· (cid:96),|theseQﬁelds are linearly disjoint over K K. Thus 3 h ((cid:96) 1)(cid:96)δ [F K : ], and 3 h (cid:96) 1 (cid:96)δ [F : ]. If (cid:15) = 1, then the ring class ﬁeld of K of K K 2 conductor 2 is contained in KO = K(j(E)) = F K (see exercise 9.19 of Cox [12]). Since this ring class (cid:98)n(cid:99)O O · √ · − · | Q ﬁeld and K(cid:96) 2 K(ζ(cid:96)n) are linearly disjoin(cid:4)t o(cid:5)ver K K, Oit follows that 3 hQ( −(cid:96)) (cid:96) 21 (cid:96)δ(cid:4) [(cid:5)F : ∼]. Suppose (cid:96) = 3, and suppose E has CM by an order of discriminant ∆ in K and E(F )[tors] = Z · Z /F ≥ | /2 3n . As in the lemma statement, we assume n 2. We ﬁrst consider the case where 2 ∆, − · · ∈ Z ≥ | Q i.e., ∆ = (2 3a)2 3 for a ≥ . If a n , then 3δ+1 [F : ] by equation (3). If a < n , then 0 2 2 we must consider several sub-cases: • ≥ O a 1: Since 6 divides the conductor of , the ring class ﬁeld of conductor 6, KO(cid:48), is ⊂ contained in KO = K(j(E)) F K (see exercise 9.19 of Cox [12]). The prime 2 ramiﬁes in (cid:98) (cid:99) n O KO(cid:48),|so K3Q2 K(ζ3n) and KO(cid:48) are linearly disjoint over K. Since [KO(cid:48) : K] = 3, we have 3δ+1 [F : ]. • ≥ · a = 0, n 3: The proof of Theorem 2.6 shows that a rational point of order 2 3n forces − O ⊂ ≥ − O ⊂ − | Q K3α 1 K≥(ζ3n) F K, wher|e α Qn. Thus K3n 1 K(ζ3n) F K, which means 32n 3 [F : ]. Since n 3, we have 3δ+1 [F : ]. • a = 0, n = 2: Let P be the point of order 18 in E(F )[tors], where E is an elliptic curve with − · CM by an order of discriminant ∆ = 22 3. Note j(E) = 243353. Work of [18] implies E has an equation of the form − − − y2 + (1 c)xy by = x3 bx2 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Proof.

text: This

text: follows

text: from

text: Corollary

text: Let

text: T

text: be

text: a

text: group

text: which

text: appears

text: as

text: the

text: torsion

text: subgroup

text: of

text: a

text: CM

text: elliptic

text: curve

text: deﬁned

text: over

text: Q

text: a

text: number

text: ﬁeld

text: of

text: odd

text: degree.

text: We

text: will

text: identify

text: an

text: od∼d

text: positive

text: integer

text: dT

text: such

text: that

text: dT

text: [F

text: whenever

text: E/F

text: is

text: a

text: CM

text: elliptic

text: curve

text: with

text: E(F

text: )[tors]

text: T

text: and

text: F

text: is

text: of

text: odd

text: deg∼ree.

text: Once

text: we

text: exhibit

text: a

text: number

text: ﬁeld

text: F

text: of

text: degree

text: d

text: and

text: a

text: CM

text: elliptic

text: curve

text: E

text: with

text: E(F

text: )[tors]

text: T

text: the

text: Strong

text: Odd

text: T

text: /F

text: Degree

text: Theorem

text: will

text: follow

text: by

text: a

text: result

text: of

text: Theorem

text: Let

text: A

text: be

text: an

text: abelian

text: variety

text: over

text: a

text: number

text: ﬁeld,

text: and

text: let

text: d

text: There

text: are

text: /F

text: inﬁnitely

text: many

text: L/F

text: such

text: that

text: [L

text: F

text: d

text: and

text: A(L)[tors]

text: A(F

text: )[tors].

text: Proof.

text: See

text: Theorem

text: of

text: We

text: isolate

text: the

text: more

text: involved

text: case

text: in

text: the

text: following

text: lemma.

text: Here,

text: δ

text: is

text: as

text: deﬁned

text: in

text: the

text: statement

text: of

text: the

text: Strong

text: Odd

text: Degree

text: Theorem.

text: Lemma

text: Let

text: F

text: be

text: a

text: number

text: ﬁeld

text: of

text: odd

text: degree.

text: If

text: E

text: is

text: a

text: CM

text: elliptic

text: curve

text: with

text: E(F

text: )[tors]

text: Z

text: Z

text: ·/F

text: Q

text: for

text: (mod

text: where

text: n

text: if

text: then

text: hQ

text: [F

text: wPritohofC.

text: MWebywiallnﬁorrsdt

text: ecronOsiodferditshcericmasineawnht

text: eBllyipTtihcecourrevme

text: Z

text: O

text: for

text: and

text: a

text: If

text: K

text: F

text: d)

text: F

text: K

text: and

text: Proposition

text: By

text: Corollary

text: F

text: K.

text: Since

text: ramiﬁes

text: in

text: K

text: n

text: O

text: O

text: an·d

text: −2

text: is

text: unQramiﬁed

text: ·away·

text: f−rom·

text: are

text: linearly

text: disjoint

text: over

text: K

text: K.

text: Thus

text: h

text: [F

text: K

text: and

text: h

text: [F

text: If

text: then

text: the

text: ring

text: class

text: ﬁeld

text: of

text: K

text: of

text: K

text: K

text: conductor

text: is

text: contained

text: in

text: KO

text: K(j(E))

text: F

text: K

text: (see

text: exercise

text: of

text: Cox

text: Since

text: this

text: ring

text: class

text: O

text: Q

text: ﬁeld

text: and

text: are

text: linearly

text: K

text: K,

text: Oit

text: follows

text: that

text: hQ(

text: Suppose

text: and

text: suppose

text: E

text: has

text: CM

text: by

text: an

text: order

text: of

text: discriminant

text: in

text: K

text: and

text: E(F

text: )[tors]

text: Z

text: Z

text: /F

text: As

text: in

text: the

text: lemma

text: statement,

text: we

text: assume

text: n

text: We

text: ﬁrst

text: consider

text: the

text: case

text: where

text: Z

text: Q

text: i.e.,

text: for

text: a

text: If

text: a

text: n

text: then

text: [F

text: by

text: equation

text: If

text: a

text: n

text: then

text: we

text: must

text: consider

text: several

text: sub-cases:

text: O

text: a

text: Since

text: divides

text: the

text: conductor

text: of

text: the

text: ring

text: class

text: ﬁeld

text: of

text: conductor

text: is

text: contained

text: in

text: KO

text: K(j(E))

text: F

text: K

text: (see

text: exercise

text: of

text: Cox

text: The

text: prime

text: ramiﬁes

text: in

text: n

text: O

text: and

text: are

text: linearly

text: disjoint

text: over

text: K.

text: Since

text: K]

text: we

text: have

text: [F

text: a

text: n

text: The

text: proof

text: of

text: Theorem

text: shows

text: that

text: a

text: rational

text: point

text: of

text: order

text: forces

text: O

text: O

text: Q

text: F

text: K,

text: wher|e

text: α

text: Qn.

text: Thus

text: F

text: K,

text: which

text: means

text: [F

text: Since

text: n

text: we

text: have

text: [F

text: a

text: n

text: Let

text: P

text: be

text: the

text: point

text: of

text: order

text: in

text: E(F

text: )[tors],

text: where

text: E

text: is

text: an

text: elliptic

text: curve

text: with

text: CM

text: by

text: an

text: order

text: of

text: discriminant

text: Note

text: j(E)

text: Work

text: of

text: implies

text: E

text: has

text: an

text: equation

text: of

text: the

text: form

text: c)xy

text: by

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∈ for some b, c F and P = (0∈, 0Q(cid:40)). We let j(b, c) denote the j-invariant of E. As in [9], we may obtain a polynomial f [b, c] that vanishes when (0, 0) has order 18. A computation 18 shows that if f (b, c) = 0 18 , j(b, c) = 243353 | Q Q then 9 [ (b, c) : ] (see the research website of the ﬁrst author for the Magma scripts used). | Q Hence 9 [F : ], as desired. Next, suppose 2 (cid:45) ∆, i.e., ∆ = −(3a)2 ·3 for a ∈ Z≥ . If a ≥ 1, then by Propos√ition 2.2, there exists → (cid:48) (cid:48) 0 Q − an F -rational isogeny ι : E E , where E has CM by the order in K = ( 3) of conductor 3. 3 − Since the kernel of this isogeny has size 3a 1, a point of order 2 in E(F ) induces a point of order 2 (cid:48) (cid:48) − in E (F ). But E is a quadratic twist of the elliptic curve E : y2 = x3 480x + 4048, and points 0 of order 2 are invariant under quadratic twists. Thus E (F ) contains a point of order 2, and F − 0 (cid:98)n(cid:99)O contains a root α of x3 480x + 4080. S|ince 2Qramiﬁes in K(α), the ﬁelds K(α) and K3 2 K(ζ3n) are linearly disjoint over K; hence 3δ+1 [F : ]. ≥ − O ⊂ | Q If a = 0 and n 3, K3n 1 K(ζ3n) F|K and 3δ+1 [F : ] as above. Fi|nally, Qif a = 0 and n = 2, then j(E) = 0. As in the case where 2 ∆, a computation shows that 9 [F : ]. Again, see the (cid:3) research website of the ﬁrst author for the Magma scripts used. (cid:4) (cid:5) ≡ ∈ Z Theorem 2.12 (Strong Odd Degree Theorem). Let (cid:96) 3 (mod 4) and n +. Deﬁne δ as  follows: − (cid:4)3n(cid:5) 1, (cid:96) > 3, 2 δ = 0, (cid:96) = 3 and n = 1, − ≥ 3n 2, (cid:96) = 3 and n 2. 2 Then: {•} Z Z Z Z Z Z ⊕ Z Z (1) For any odd positive integer d, the groups , /2 , /4 , and /2 /2 appear as the torsion subgroup of a CM elliptic curve deﬁned over a number ﬁeld of degree d. Z Z (2) /(cid:96)n appears as the torsion subgroup of a CM elliptic curve deﬁned over a number ﬁeld of ≡ √ · − · odd degree d if and only if (cid:96) 3 (mod 8) and d is a multiple of hQ − (cid:96) 1 (cid:96)δ. Z Z ( (cid:96)) 2 (3) /2(cid:96)n appears as the torsion subgroup of a CM elliptic curve deﬁned over a number ﬁeld of odd degree d if and only if one of the following holds: ≡ ≥ · √ · − · a. (cid:96) 3 (mod 8), where n 2 if (cid:96) = 3, and d is a multiple of 3 hQ − (cid:96) 1 (cid:96)δ, or ( (cid:96)) 2 b. (cid:96) = 3 and n = 1 and d is any odd positive integer, or ≡ √ · − · c. (cid:96) 7 (mod 8) and d is a multiple of hQ − (cid:96) 1 (cid:96)δ. ( (cid:96)) 2 {•} Z Z Z Z Z Z ⊕ Z Z Proof. The groups , /2 , /4 , and /2 /2 appear as torsion subgroups of CM elliptic Q curves deﬁned over by work of Olson [26], so part (1) is an immediate co∼nsequence of Theorem Z Z ≡ 2.10. For part (2), suppose E is a CM elliptic curve with E(F )[tors] = /(cid:96)n . Then (cid:96) 3 /F Q √ · − · (mod 8) by the Odd Degree Theorem, and [F : ] is a multiple of hQ − (cid:96) 1 (cid:96)δ by Theorem 2.6. ≡ ( (cid:96)) 2 √ · − · Conversely, if for each (cid:96) 3 (mod 8) the∼reZexisZts a number ﬁeld F of degree hQ( −(cid:96)) (cid:96) 21 (cid:96)δ and a CM elliptic curve E with E(F )[tors] = /(cid:96)n , part (2) will follow from Theorem 2.10. ≡/F (cid:54) First suppose (cid:96) 3 (mod 8), (cid:96) = 3. By Theorem 2.6 there exists a CM elliptic curve E deﬁned √ · − · (cid:107) over a number ﬁeld F √of degree hQ − (cid:96) 1 (cid:96)δ with (cid:96)n E(F )[tors]. By the Odd Degree Theorem, Q − ( (cid:96)) 2 ∼ Z Z Z Z ∼ Z Z E has CM by K = ( (cid:96)) and E(F )[tors] = /(cid:96)n or /2(cid:96)n . But if E(F )[tors] = ∼/2(cid:96)n , then · · − · | Q Z Z 3 h (cid:96) 1 (cid:96)δ [F : ] by Lemma 2.11, which is a contradiction. Thus E(F )[tors] = /(cid:96)n , as K 2 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: for

text: some

text: b,

text: c

text: F

text: and

text: P

text: We

text: let

text: j(b,

text: c)

text: denote

text: the

text: j-invariant

text: of

text: E.

text: As

text: in

text: we

text: may

text: obtain

text: a

text: polynomial

text: f

text: [b,

text: c]

text: that

text: vanishes

text: when

text: has

text: order

text: A

text: computation

text: shows

text: that

text: if

text: f

text: (b,

text: c)

text: j(b,

text: c)

text: Q

text: Q

text: then

text: (b,

text: c)

text: (see

text: the

text: research

text: website

text: of

text: the

text: ﬁrst

text: author

text: for

text: the

text: Magma

text: scripts

text: used).

text: Q

text: Hence

text: [F

text: as

text: desired.

text: Next,

text: suppose

text: i.e.,

text: ·3

text: for

text: a

text: Z≥

text: If

text: a

text: then

text: by

text: Propos√ition

text: there

text: exists

text: Q

text: an

text: F

text: -rational

text: isogeny

text: ι

text: E

text: E

text: where

text: E

text: has

text: CM

text: by

text: the

text: order

text: in

text: K

text: of

text: conductor

text: Since

text: the

text: kernel

text: of

text: this

text: isogeny

text: has

text: size

text: a

text: point

text: of

text: order

text: in

text: E(F

text: induces

text: a

text: point

text: of

text: order

text: in

text: E

text: (F

text: But

text: E

text: is

text: a

text: quadratic

text: twist

text: of

text: the

text: elliptic

text: curve

text: E

text: and

text: points

text: of

text: order

text: are

text: invariant

text: under

text: quadratic

text: twists.

text: Thus

text: E

text: (F

text: contains

text: a

text: point

text: of

text: order

text: and

text: F

text: contains

text: a

text: root

text: α

text: of

text: S|ince

text: in

text: K(α),

text: the

text: ﬁelds

text: K(α)

text: and

text: are

text: linearly

text: disjoint

text: over

text: K;

text: hence

text: [F

text: O

text: Q

text: If

text: a

text: and

text: n

text: F|K

text: and

text: [F

text: as

text: above.

text: Fi|nally,

text: Qif

text: a

text: and

text: n

text: then

text: j(E)

text: As

text: in

text: the

text: case

text: where

text: a

text: computation

text: shows

text: that

text: [F

text: Again,

text: see

text: the

text: research

text: website

text: of

text: the

text: ﬁrst

text: author

text: for

text: the

text: Magma

text: scripts

text: used.

text: Z

text: Theorem

text: (Strong

text: Odd

text: Degree

text: Theorem).

text: Let

text: (mod

text: and

text: n

text: Deﬁne

text: δ

text: as

text: follows:

text: δ

text: and

text: n

text: and

text: n

text: Then:

text: Z

text: Z

text: Z

text: Z

text: Z

text: Z

text: Z

text: Z

text: For

text: any

text: odd

text: positive

text: integer

text: d,

text: the

text: groups

text: and

text: appear

text: as

text: the

text: torsion

text: subgroup

text: of

text: a

text: CM

text: elliptic

text: curve

text: deﬁned

text: over

text: a

text: number

text: ﬁeld

text: of

text: degree

text: d.

text: Z

text: Z

text: appears

text: as

text: the

text: torsion

text: subgroup

text: of

text: a

text: CM

text: elliptic

text: curve

text: deﬁned

text: over

text: a

text: number

text: ﬁeld

text: of

text: odd

text: degree

text: d

text: if

text: and

text: only

text: if

text: (mod

text: and

text: d

text: is

text: a

text: multiple

text: of

text: hQ

text: Z

text: Z

text: appears

text: as

text: the

text: torsion

text: subgroup

text: of

text: a

text: CM

text: elliptic

text: curve

text: deﬁned

text: over

text: a

text: number

text: ﬁeld

text: of

text: odd

text: degree

text: d

text: if

text: and

text: only

text: if

text: one

text: of

text: the

text: following

text: holds:

text: a.

text: (mod

text: where

text: n

text: if

text: and

text: d

text: is

text: a

text: multiple

text: of

text: hQ

text: or

text: b.

text: and

text: n

text: and

text: d

text: is

text: any

text: odd

text: positive

text: integer,

text: or

text: c.

text: (mod

text: and

text: d

text: is

text: a

text: multiple

text: of

text: hQ

text: Z

text: Z

text: Z

text: Z

text: Z

text: Z

text: Z

text: Z

text: Proof.

text: The

text: groups

text: and

text: appear

text: as

text: torsion

text: subgroups

text: of

text: CM

text: elliptic

text: Q

text: curves

text: deﬁned

text: over

text: by

text: work

text: of

text: Olson

text: so

text: part

text: is

text: an

text: immediate

text: co∼nsequence

text: of

text: Theorem

text: Z

text: Z

text: For

text: part

text: suppose

text: E

text: is

text: a

text: CM

text: elliptic

text: curve

text: with

text: E(F

text: )[tors]

text: Then

text: /F

text: Q

text: (mod

text: by

text: the

text: Odd

text: Degree

text: Theorem,

text: and

text: [F

text: is

text: a

text: multiple

text: of

text: hQ

text: by

text: Theorem

text: Conversely,

text: if

text: for

text: each

text: (mod

text: the∼reZexisZts

text: a

text: number

text: ﬁeld

text: F

text: of

text: degree

text: hQ(

text: and

text: a

text: CM

text: elliptic

text: curve

text: E

text: with

text: E(F

text: )[tors]

text: part

text: will

text: follow

text: from

text: Theorem

text: ≡/F

text: First

text: suppose

text: (mod

text: By

text: Theorem

text: there

text: exists

text: a

text: CM

text: elliptic

text: curve

text: E

text: deﬁned

text: over

text: a

text: number

text: ﬁeld

text: F

text: √of

text: degree

text: hQ

text: with

text: E(F

text: )[tors].

text: By

text: the

text: Odd

text: Degree

text: Theorem,

text: Q

text: Z

text: Z

text: Z

text: Z

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