text: 2 / Nuclear Physics A 00 (2020) 1–9 ∼ T > Tc 165MeV [1, 2]. Therefore, a chiral-symmetric QGP at high temperature is also a fundamental prediction of QCD. With such symmetry of light quark sector, the currents in QGP can be independently examined for right-handed (RH) and left-handed (LH) quarks. The chiral currents(cid:126)J can then be combined − R/L into the familiar vector current (cid:126)J = (cid:126)J + (cid:126)J and axial current (cid:126)J = (cid:126)J (cid:126)J . Recent studies on the transport R L 5 R L properties of QGP in responses to ex(cid:32)terna(cid:33)l EM(cid:32) ﬁelds have(cid:33)f(cid:32)ound(cid:33)the generalized “Ohm’s Table” for QGP: (cid:126)J = σ σ5 E(cid:126) . (1) (cid:126)J5 σχe σs B(cid:126) In addition to the familiar Ohmic conducting eﬀect, three interesting new eﬀects are identiﬁed in the above table: the Chiral Magnetic Eﬀect (CME) [3, 4, 5], the Chiral Separation Eﬀect (CSE) [6, 7], and the Chiral Electric Separation Eﬀect (CESE) [8, 9]. For detailed discussions on these anomalous transport eﬀects and for more complete reference lists, we refer the readers to recent reviews in e.g. [10, 11, 12]. In this contribution, we focus on the Chiral Magnetic Eﬀect and related phenomena in the quark-gluon plasma. The CME predicts the generation of a vector current (cid:126)J in response to external magnetic ﬁeld B(cid:126): (cid:126)J = σ5B(cid:126) (2) One may immediately realize that normally the above transport process is forbidden by symmetry con- P CP P CP siderations: (cid:126)J is -odd and -even, while B(cid:126) is -even and -odd. Indeed the CME could occur only in P CP a - and -odd environment where the CME conductivity σ5 (being a pseudo-scalar quantity) is nonzero. More speciﬁcally σ5 = CAµ5 with CA a constant. The µ5 is a chiral chemical potential that quantiﬁes the presence of chirality imbalance i.e. the diﬀerence between the numbers of RH and LH chiral fermions in the system. For a chiral QGP with nonzero µ5 (cid:44) 0, a CME current is generated along the direction of B(cid:126). Intuitively the CME may be understood in the following way. The magnetic ﬁeld leads to a spin polariza- tion (i.e. “magnetization”) eﬀect, with quarks’ spins preferably aligned along the B(cid:126) ﬁeld direction, which (cid:104) (cid:105) ∝ implies (cid:126)s (Qe)B(cid:126). Quarks with speciﬁc chirality have their momentum (cid:126)p direction correlated with spin || || − (cid:126)s orientation: (cid:126)p (cid:126)s for RH quarks, while (cid:126)p ( (cid:126)s) for LH ones. In the presence of chirality imbalance, i.e. (cid:104) (cid:105) ∝ (cid:104) (cid:105) µ5 (cid:44) 0, ther(cid:104)e w(cid:105) ∝ill be a net correlation between average spin and momentum∝ (cid:104)(cid:126)p(cid:105) ∝ µ5 (cid:126)s . It is therefore evident that (cid:126)p (Qe)µ5B(cid:126), which implies a vector current of these quarks (cid:126)J (cid:126)p (Qe)µ5B(cid:126). There remains however the crucial question of how to achieve a chiral QGP with chirality imbalance in the ﬁrst place. The process of creating nonzero chirality pertains to the famous “chiral anomaly” (or often called “triangle/axial anomaly”). This anomaly implies the breaking of axial current conservation by · · quantum loop eﬀect, ∂µJ5µ = CAE(cid:126) B(cid:126). Therefore chirality imbalance can be created through nonzero E(cid:126) B(cid:126): such ﬁelds can be either QED EM ﬁelds or QCD chromo-EM ﬁelds. · In QCD the gluon conﬁgurations with globally nonzero E(cid:126) B(cid:126) are known to exist: they are the topological (cid:82) solitons such as instantons and sphalerons. These topological objects play crucial roles in our understanding of nonperturbative dynamics in QCD vacuum [13] as well as in hot QGP [14, 15, 16]. Such conﬁgurations ∼ · have their global topological winding number Q E(cid:126) B(cid:126), and when coupled with chiral fermions, can w − generate deﬁnite amount of global chirality imbalance via chiral anomaly: N N = 2Q for each ﬂa- R L w vor of light quarks. In addition there are also strong local ﬂuctuations of topological charge density (i.e. · locally nonzero chromo E(cid:126) B(cid:126)) from the initial conditions in the glasma (see e.g. [17, 18]) which would similarly translate into local ﬂuctuations of chirality imbalance. In short, the chirality imbalance is a direct manifestation of the QCD topological ﬂuctuations and could become observable through the CME current. Summarizing the introductory discussions, the CME is a remarkable phenomenon that stems from highly nontrivial interplay of QCD chiral symmetry, axial anomaly, and gluonic topology. The pertinent CME con- T ductivity σ5 = CAµ5 has distinctive features by virtue of its topological origin: it is -even and thus non dissipative, its constant C is completely ﬁxed from anomaly relations and thus takes universal value from A weak to strong coupling and encompassing various physical systems. It is therefore of fundamental impor- tance to search for the CME in experiments. The heavy ion collisions provide a unique environment where

font: VCKFCA+rtxi

text: Nuclear

text: Physics

text: A

font: VBAZUB+txsy

text: T

font: HMMYGI+rtxmi

text: Tc

text: Therefore,

text: a

text: chiral-symmetric

text: QGP

text: at

text: high

text: temperature

text: is

text: also

text: a

text: fundamental

text: prediction

text: of

text: QCD.

text: With

text: such

text: symmetry

text: of

text: light

text: quark

text: sector,

text: the

text: currents

text: in

text: QGP

text: can

text: be

text: independently

text: examined

text: for

text: right-handed

text: (RH)

text: and

text: left-handed

text: (LH)

text: quarks.

text: The

text: chiral

text: can

text: then

text: be

text: combined

text: R/L

text: into

text: the

text: familiar

text: vector

text: current

font: ZQLNMU+rtxr

text: and

text: axial

text: current

text: Recent

text: studies

text: on

text: the

text: transport

text: R

text: L

text: R

text: L

text: properties

text: of

text: QGP

text: in

text: responses

text: to

text: ﬁelds

text: generalized

text: “Ohm’s

text: Table”

text: for

text: QGP:

text: σ

text: σ5

text: σχe

text: σs

text: In

text: addition

text: to

text: the

text: familiar

text: Ohmic

text: conducting

text: eﬀect,

text: three

text: interesting

text: new

text: eﬀects

text: are

text: identiﬁed

text: in

text: the

text: above

text: table:

text: the

text: Chiral

text: Magnetic

text: Eﬀect

text: (CME)

text: the

text: Chiral

text: Separation

text: Eﬀect

text: (CSE)

text: and

text: the

text: Chiral

text: Electric

text: Separation

text: Eﬀect

text: (CESE)

text: For

text: detailed

text: discussions

text: on

text: these

text: anomalous

text: transport

text: eﬀects

text: and

text: for

text: more

text: complete

text: reference

text: lists,

text: we

text: refer

text: the

text: readers

text: to

text: recent

text: reviews

text: in

text: e.g.

text: In

text: this

text: contribution,

text: we

text: focus

text: on

text: the

text: Chiral

text: Magnetic

text: Eﬀect

text: and

text: related

text: phenomena

text: in

text: the

text: quark-gluon

text: plasma.

text: The

text: CME

text: predicts

text: the

text: generation

text: of

text: a

text: vector

text: current

text: in

text: response

text: to

text: external

text: magnetic

text: ﬁeld

text: One

text: may

text: immediately

text: realize

text: that

text: normally

text: the

text: above

text: transport

text: process

text: is

text: forbidden

text: by

text: symmetry

text: con-

text: P

text: CP

text: P

text: CP

text: siderations:

text: is

text: -odd

text: and

text: -even,

text: while

text: is

text: -even

text: and

text: -odd.

text: Indeed

text: the

text: CME

text: could

text: occur

text: only

text: in

text: P

text: CP

text: a

text: and

text: -odd

text: environment

text: where

text: the

text: CME

text: conductivity

text: σ5

text: (being

text: a

text: pseudo-scalar

text: quantity)

text: is

text: nonzero.

text: More

text: speciﬁcally

text: σ5

text: with

text: CA

text: a

text: constant.

text: The

text: µ5

text: is

text: a

text: chiral

text: chemical

text: potential

text: that

text: quantiﬁes

text: the

text: presence

text: of

text: chirality

text: imbalance

text: i.e.

text: the

text: diﬀerence

text: between

text: the

text: numbers

text: of

text: RH

text: and

text: LH

text: chiral

text: fermions

text: in

text: the

text: system.

text: For

text: a

text: chiral

text: QGP

text: with

text: nonzero

text: µ5

text: a

text: CME

text: current

text: is

text: generated

text: along

text: the

text: direction

text: of

text: Intuitively

text: the

text: CME

text: may

text: be

text: understood

text: in

text: the

text: following

text: way.

text: The

text: magnetic

text: ﬁeld

text: leads

text: to

text: a

text: spin

text: polariza-

text: tion

text: (i.e.

text: “magnetization”)

text: eﬀect,

text: with

text: quarks’

text: spins

text: preferably

text: aligned

text: along

text: the

text: ﬁeld

text: direction,

text: which

text: implies

text: Quarks

text: with

text: speciﬁc

text: chirality

text: have

text: their

text: momentum

text: direction

text: correlated

text: with

text: spin

text: orientation:

text: for

text: RH

text: quarks,

text: while

text: for

text: LH

text: ones.

text: In

text: the

text: presence

text: of

text: chirality

text: imbalance,

text: i.e.

text: µ5

text: ∝ill

text: be

text: a

text: net

text: correlation

text: between

text: average

text: spin

text: and

text: momentum∝

text: µ5

text: It

text: is

text: therefore

text: evident

text: that

text: which

text: implies

text: a

text: vector

text: current

text: of

text: these

text: quarks

text: There

text: remains

text: however

text: the

text: crucial

text: question

text: of

text: how

text: to

text: achieve

text: a

text: chiral

text: QGP

text: with

text: chirality

text: imbalance

text: in

text: the

text: ﬁrst

text: place.

text: The

text: process

text: of

text: creating

text: nonzero

text: chirality

text: pertains

text: to

text: the

text: famous

text: “chiral

text: anomaly”

text: (or

text: often

text: called

text: “triangle/axial

text: anomaly”).

text: This

text: anomaly

text: implies

text: the

text: breaking

text: of

text: axial

text: current

text: conservation

text: by

text: quantum

text: loop

text: eﬀect,

text: ∂µJ5µ

text: Therefore

text: chirality

text: imbalance

text: can

text: be

text: created

text: through

text: nonzero

text: such

text: ﬁelds

text: can

text: be

text: either

text: QED

text: EM

text: ﬁelds

text: or

text: QCD

text: chromo-EM

text: ﬁelds.

text: In

text: QCD

text: the

text: gluon

text: conﬁgurations

text: with

text: globally

text: nonzero

text: are

text: known

text: to

text: exist:

text: they

text: are

text: the

text: topological

text: solitons

text: such

text: as

text: instantons

text: and

text: sphalerons.

text: These

text: topological

text: objects

text: play

text: crucial

text: roles

text: in

text: our

text: understanding

text: of

text: nonperturbative

text: dynamics

text: in

text: QCD

text: vacuum

text: as

text: well

text: as

text: in

text: hot

text: QGP

text: Such

text: conﬁgurations

text: have

text: their

text: global

text: topological

text: winding

text: number

text: Q

text: and

text: when

text: coupled

text: with

text: chiral

text: fermions,

text: can

text: w

text: generate

text: deﬁnite

text: amount

text: of

text: global

text: chirality

text: imbalance

text: via

text: chiral

text: anomaly:

text: N

text: N

text: for

text: each

text: ﬂa-

text: R

text: L

text: w

text: vor

text: of

text: light

text: quarks.

text: In

text: addition

text: there

text: are

text: also

text: strong

text: local

text: ﬂuctuations

text: of

text: topological

text: charge

text: density

text: (i.e.

text: locally

text: nonzero

text: chromo

text: from

text: the

text: initial

text: conditions

text: in

text: the

text: glasma

text: (see

text: e.g.

text: which

text: would

text: similarly

text: translate

text: into

text: local

text: ﬂuctuations

text: of

text: chirality

text: imbalance.

text: In

text: short,

text: the

text: chirality

text: imbalance

text: is

text: a

text: direct

text: manifestation

text: of

text: the

text: QCD

text: topological

text: ﬂuctuations

text: and

text: could

text: become

text: observable

text: through

text: the

text: CME

text: current.

text: Summarizing

text: the

text: introductory

text: discussions,

text: the

text: CME

text: is

text: a

text: remarkable

text: phenomenon

text: that

text: stems

text: from

text: highly

text: nontrivial

text: interplay

text: of

text: QCD

text: chiral

text: symmetry,

text: axial

text: anomaly,

text: and

text: gluonic

text: topology.

text: The

text: pertinent

text: CME

text: con-

text: T

text: ductivity

text: σ5

text: has

text: distinctive

text: features

text: by

text: virtue

text: of

text: its

text: topological

text: origin:

text: it

text: is

text: -even

text: and

text: thus

text: non

text: dissipative,

text: its

text: constant

text: C

text: is

text: completely

text: ﬁxed

text: from

text: anomaly

text: relations

text: and

text: thus

text: takes

text: universal

text: value

text: from

text: A

text: weak

text: to

text: strong

text: coupling

text: and

text: encompassing

text: various

text: physical

text: systems.

text: It

text: is

text: therefore

text: of

text: fundamental

text: impor-

text: tance

text: to

text: search

text: for

text: the

text: CME

text: in

text: experiments.

text: The

text: heavy

text: ion

text: collisions

text: provide

text: a

text: unique

text: environment

text: where


text: x = x ) radial advection of the perturbation vertical layer system and we identiﬁed that there is a critical bg (cid:48) vorticity: u ∂ ζ. Both eﬀects act nearly opposite to one value of the Brunt-V¨ais¨al¨a frequency N for which the x cr another, but the vertical tilting mechanism appears to location of the critical layer goes from being found out- dominate in the jet proﬁles we have tested here in this side the shear layer (gβ > N2 ) to appearing inside the cr study. shear layer (0 < gβ < N2 ). The discussion in the Among other things, this results in an asymmetric jet- previous section concerning ctrhe self-reproductive spatial like structure which grows in amplitude until it either spreading of jet creation/destruction is the expected out- saturates or the jet itself gets destroyed because of roll-up come of the former of the two conditions. What might due to the RWI. While the spawned jet grows it can, on happen when the critical layer appears inside the shear its own accord, induce new jet-like structures at new crit- layer? ical points associated with the spawned jet and the linear Consider the nonlinear vorticity forcing arising from analysis done here largely supports the picture sketched the conditions shown in Figure 10b in which the criti- in M13 of jets begetting jets. As a “parent” structure, cal layer is clearly inside the shear layer and the shear a jet-like proﬁle can give rise to new “child” jets only layer itself is stable against the destructive RWI. We show while the parent jet maintains its structural integrity. in Figure 19 the corresponding mean vertical vorticity This can be inferred from the linear stability of the asym- forcing proﬁle arising under these parameter conditions metric je|t±ex| amined in Section 6.3: as the amplitude of gβ < Nc2r. The imprint of the primary≈sh±ear layer is evi- the jet ωa is increased fr|om±| zero ﬁrst only Z-modes dent in the vicinity of the locations x 0.14 while the are unstable. However, as ωa increases eventually the nonline≈ar±forcing at the critical layer is also clearly visible most unstable mode is the RWI which induces transfor- near x 0.045. The resulting aggregate forcing proﬁle mative/destructive roll up of the parent jet. It is only qualitatively resembles forcing proﬁles we have discussed during the low amplitude range of jet amplitudes, prior thus far for when the critical layers appear outside. We to the RWI becoming important, will a given jet gener- observe that unlike the examples of the vorticity-step and ate into existence a new jet(s) at its associated buoyant jet proﬁles, the critical layer jets are primarily driven by critical layer(s). the radial advection of the perturbation vertical vorticity − In this way, the parent jet must stay as such long ( u∂ ζ) while the vertical tilting of radial perturbation x enough for child jets to grow into maturity so they too vorticity (ζ ∂ w) is relatively weak by comparison but x x can generate the next generation of jets, and so on per- acts with the same sense as the radial advection of the petuating the self-replication process. Critical to this perturbation vertical vorticity. is the time spent growing each jet - as it must grow The implications are interesting: These results suggest slow enough so that the Z-mode instability (which is rel- that under those conditions in which the Brunt-V¨ais¨al¨a atively slow by our current account) may manifest itself frequency is small, that jets will grow inside the shear and start spawning the next generation of jets. If a jet layer and, as per our earlier insights, once the jet ampli- grows too fast then the Z-mode process has no time to tude grows large enough it will self-destruct while gen- birth the new generation and the whole process shuts oﬀ. erating new jets associated with its critical layers. If the If the jet grows quite slowly it could, in a general sense, whole process remains conﬁned to inside the shear layer, give rise to a pattern state like that reported in M13. then there are at least two possible outcomes: In the Conditions lying somewhere between these two extremes ﬁrst of these, the parent shear layer structure remains might give rise to a turbulent ﬂow state that could be stable (also as suggested by linear analysis) while the jet either decaying or self-sustaining - perhaps something in inside grows and once having reached a suﬃcient ampli- the ﬂavor of a “chaotic propeller” identiﬁed as being op- tude it will self-destruct having spawned child jets along erative in subcritical transitions in both plane-Couette the way (presumably contained inside the shear layer as and rotating-plane-Couette ﬂows (e.g., see discussion in well). The second possibility is that while the jet inside Rincon et al. 2007). grows, the parent shear layer nonlinearly destructs as Whether or not this process can be self-sustaining and well. If an external agent is responsible for the creation lead to a turbulent state under a wide umbrella of con- and maintenence of the parent shear layer in the ﬁrst ditions appropriate to Dead Zones remains to be deter- place – e.g, by either direct thermally driven relaxation mined. It would seem, however given our considerations, (Les & Lin, 2015, Lobo Gomes et al., 2015) or by the VSI that its self-sustainability depends centrally on what way (Richard et al. 2016) – then either of the two scenarios the original disturbances are structured since it requires envisioned could give rise to a process of unsteady jet cre- there to be some relatively large amplitude disturbance ation and destruction entirely contained inside the shear to start the whole thing oﬀ. A recent announcement by layer, where the only diﬀerence in the outcomes of the Marcus et al. (2015) on this process in a shearing sheet two possibilities is the degree of the unsteady activity – disk model with vertically varying gravity and stratiﬁca- possibly turbulent. Global numerical calculations of the tion seems to indicate that it can give rise to a strongly several processes mentioned (thermally driven relaxation turbulent state. or the VSI), when suﬃciently resolved, can test whether Draft version May 28, 2020 8.4. Spatially localized instability or not this dynamical scenario indeed manifests itself. Preprint typeset using LATEX style emulateapj v. 01/23/15 In Section we examined the properties of the shear CRITICAL LAYERS AND PROTOPLANETARY DISK TURBULENCE

text: x

label: paragraph

text: x

text: radial

text: advection

text: of

text: the

text: perturbation

text: vertical

text: layer

text: system

text: and

text: we

text: identiﬁed

text: that

text: there

text: is

text: a

text: critical

text: bg

text: vorticity:

text: u

text: ζ.

text: Both

text: eﬀects

text: act

text: nearly

text: opposite

text: to

text: one

text: value

text: of

text: the

text: Brunt-V¨ais¨al¨a

text: frequency

text: N

text: for

text: which

text: the

text: x

text: cr

text: another,

text: but

text: the

text: vertical

text: tilting

text: mechanism

text: appears

text: to

text: location

text: of

text: the

text: critical

text: layer

text: goes

text: from

text: being

text: found

text: out-

text: dominate

text: in

text: the

text: jet

text: proﬁles

text: we

text: have

text: tested

text: here

text: in

text: this

text: side

text: the

text: shear

text: layer

text: (gβ

text: to

text: appearing

text: inside

text: the

text: cr

text: study.

text: shear

text: layer

text: gβ

text: The

text: discussion

text: in

text: the

text: Among

text: other

text: things,

text: this

text: results

text: in

text: an

text: asymmetric

text: jet-

text: previous

text: section

text: concerning

text: ctrhe

text: self-reproductive

text: spatial

text: like

text: structure

text: which

text: grows

text: in

text: amplitude

text: until

text: it

text: either

text: spreading

text: of

text: jet

text: creation/destruction

text: is

text: the

text: expected

text: out-

text: saturates

text: or

text: the

text: jet

text: itself

text: gets

text: destroyed

text: because

text: of

text: roll-up

text: come

text: of

text: the

text: former

text: of

text: the

text: two

text: conditions.

text: What

text: might

text: due

text: to

text: the

text: RWI.

text: While

text: the

text: spawned

text: jet

text: grows

text: it

text: can,

text: on

text: happen

text: when

text: the

text: critical

text: layer

text: appears

text: inside

text: the

text: shear

text: its

text: own

text: accord,

text: induce

text: new

text: jet-like

text: structures

text: at

text: new

text: crit-

text: layer?

text: ical

text: points

text: associated

text: with

text: the

text: spawned

text: jet

text: and

text: the

text: linear

text: Consider

text: the

text: nonlinear

text: vorticity

text: forcing

text: arising

text: from

text: analysis

text: done

text: here

text: largely

text: supports

text: the

text: picture

text: sketched

text: the

text: conditions

text: shown

text: in

text: Figure

text: in

text: which

text: the

text: criti-

text: in

text: of

text: jets

text: begetting

text: jets.

text: As

text: a

text: “parent”

text: structure,

text: cal

text: layer

text: is

text: clearly

text: inside

text: the

text: shear

text: layer

text: and

text: the

text: shear

text: a

text: jet-like

text: proﬁle

text: can

text: give

text: rise

text: to

text: new

text: “child”

text: jets

text: only

text: layer

text: itself

text: is

text: stable

text: against

text: the

text: destructive

text: RWI.

text: We

text: show

text: while

text: the

text: parent

text: jet

text: maintains

text: its

text: structural

text: integrity.

text: in

text: Figure

text: the

text: corresponding

text: mean

text: vertical

text: vorticity

text: This

text: can

text: be

text: inferred

text: from

text: the

text: linear

text: stability

text: of

text: the

text: asym-

text: forcing

text: proﬁle

text: arising

text: under

text: these

text: parameter

text: conditions

text: metric

text: je|t±ex|

text: amined

text: in

text: Section

text: as

text: the

text: amplitude

text: of

text: gβ

text: The

text: imprint

text: of

text: the

text: primary≈sh±ear

text: layer

text: is

text: evi-

text: the

text: jet

text: ωa

text: is

text: increased

text: fr|om±|

text: zero

text: ﬁrst

text: only

text: Z-modes

text: dent

text: in

text: the

text: vicinity

text: of

text: the

text: locations

text: x

text: while

text: the

text: are

text: unstable.

text: However,

text: as

text: ωa

text: increases

text: eventually

text: the

text: nonline≈ar±forcing

text: at

text: the

text: critical

text: layer

text: is

text: also

text: clearly

text: visible

text: most

text: unstable

text: mode

text: is

text: the

text: RWI

text: which

text: induces

text: transfor-

text: near

text: x

text: The

text: resulting

text: aggregate

text: forcing

text: proﬁle

text: mative/destructive

text: roll

text: up

text: of

text: the

text: parent

text: jet.

text: It

text: is

text: only

text: qualitatively

text: resembles

text: forcing

text: proﬁles

text: we

text: have

text: discussed

text: during

text: the

text: low

text: amplitude

text: range

text: of

text: jet

text: amplitudes,

text: prior

text: thus

text: far

text: for

text: when

text: the

text: critical

text: layers

text: appear

text: outside.

text: We

text: to

text: the

text: RWI

text: becoming

text: important,

text: will

text: a

text: given

text: jet

text: gener-

text: observe

text: that

text: unlike

text: the

text: examples

text: of

text: the

text: vorticity-step

text: and

text: ate

text: into

text: existence

text: a

text: new

text: jet(s)

text: at

text: its

text: associated

text: buoyant

text: jet

text: proﬁles,

text: the

text: critical

text: layer

text: jets

text: are

text: primarily

text: driven

text: by

text: critical

text: layer(s).

text: the

text: radial

text: advection

text: of

text: the

text: perturbation

text: vertical

text: vorticity

text: In

text: this

text: way,

text: the

text: parent

text: jet

text: must

text: stay

text: as

text: such

text: long

text: u∂

text: ζ)

text: while

text: the

text: vertical

text: tilting

text: of

text: radial

text: perturbation

text: x

text: enough

text: for

text: child

text: jets

text: to

text: grow

text: into

text: maturity

text: so

text: they

text: too

text: vorticity

text: (ζ

text: w)

text: is

text: relatively

text: weak

text: by

text: comparison

text: but

text: x

text: x

text: can

text: generate

text: the

text: next

text: generation

text: of

text: jets,

text: and

text: so

text: on

text: per-

text: acts

text: with

text: the

text: same

text: sense

text: as

text: the

text: radial

text: advection

text: of

text: the

text: petuating

text: the

text: self-replication

text: process.

text: Critical

text: to

text: this

text: perturbation

text: vertical

text: vorticity.

text: is

text: the

text: time

text: spent

text: growing

text: each

text: jet

text: as

text: it

text: must

text: grow

text: The

text: implications

text: are

text: interesting:

text: These

text: results

text: suggest

text: slow

text: enough

text: so

text: that

text: the

text: Z-mode

text: instability

text: (which

text: is

text: rel-

text: that

text: under

text: those

text: conditions

text: in

text: which

text: the

text: Brunt-V¨ais¨al¨a

text: atively

text: slow

text: by

text: our

text: current

text: account)

text: may

text: manifest

text: itself

text: frequency

text: is

text: small,

text: that

text: jets

text: will

text: grow

text: inside

text: the

text: shear

text: and

text: start

text: spawning

text: the

text: next

text: generation

text: of

text: jets.

text: If

text: a

text: jet

text: layer

text: and,

text: as

text: per

text: our

text: earlier

text: insights,

text: once

text: the

text: jet

text: ampli-

text: grows

text: too

text: fast

text: then

text: the

text: Z-mode

text: process

text: has

text: no

text: time

text: to

text: tude

text: grows

text: large

text: enough

text: it

text: will

text: self-destruct

text: while

text: gen-

text: birth

text: the

text: new

text: generation

text: and

text: the

text: whole

text: process

text: shuts

text: oﬀ.

text: erating

text: new

text: jets

text: associated

text: with

text: its

text: critical

text: layers.

text: If

text: the

text: If

text: the

text: jet

text: grows

text: quite

text: slowly

text: it

text: could,

text: in

text: a

text: general

text: sense,

text: whole

text: process

text: remains

text: conﬁned

text: to

text: inside

text: the

text: shear

text: layer,

text: give

text: rise

text: to

text: a

text: pattern

text: state

text: like

text: that

text: reported

text: in

text: then

text: there

text: are

text: at

text: least

text: two

text: possible

text: outcomes:

text: In

text: the

text: Conditions

text: lying

text: somewhere

text: between

text: these

text: two

text: extremes

text: ﬁrst

text: of

text: these,

text: the

text: parent

text: shear

text: layer

text: structure

text: remains

text: might

text: give

text: rise

text: to

text: a

text: turbulent

text: ﬂow

text: state

text: that

text: could

text: be

text: stable

text: (also

text: as

text: suggested

text: by

text: linear

text: analysis)

text: while

text: the

text: jet

text: either

text: decaying

text: or

text: self-sustaining

text: perhaps

text: something

text: in

text: inside

text: grows

text: and

text: once

text: having

text: reached

text: a

text: suﬃcient

text: ampli-

text: the

text: ﬂavor

text: of

text: a

text: “chaotic

text: propeller”

text: identiﬁed

text: as

text: being

text: op-

text: tude

text: it

text: will

text: self-destruct

text: having

text: spawned

text: child

text: jets

text: along

text: erative

text: in

text: subcritical

text: transitions

text: in

text: both

text: plane-Couette

text: the

text: way

text: (presumably

text: contained

text: inside

text: the

text: shear

text: layer

text: as

text: and

text: rotating-plane-Couette

text: ﬂows

text: (e.g.,

text: see

text: discussion

text: in

text: well).

text: The

text: second

text: possibility

text: is

text: that

text: while

text: the

text: jet

text: inside

text: Rincon

text: et

text: al.

text: grows,

text: the

text: parent

text: shear

text: layer

text: nonlinearly

text: destructs

text: as

text: Whether

text: or

text: not

text: this

text: process

text: can

text: be

text: self-sustaining

text: and

text: well.

text: If

text: an

text: external

text: agent

text: is

text: responsible

text: for

text: the

text: creation

text: lead

text: to

text: a

text: turbulent

text: state

text: under

text: a

text: wide

text: umbrella

text: of

text: con-

text: and

text: maintenence

text: of

text: the

text: parent

text: shear

text: layer

text: in

text: the

text: ﬁrst

text: ditions

text: appropriate

text: to

text: Dead

text: Zones

text: remains

text: to

text: be

text: deter-

text: place

text: e.g,

text: by

text: either

text: direct

text: thermally

text: driven

text: relaxation

text: mined.

text: It

text: would

text: seem,

text: however

text: given

text: our

text: considerations,

text: (Les

text: Lin,

text: Lobo

text: Gomes

text: et

text: al.,

text: or

text: by

text: the

text: VSI

text: that

text: its

text: self-sustainability

text: depends

text: centrally

text: on

text: what

text: way

text: (Richard

text: et

text: al.

text: then

text: either

text: of

text: the

text: two

text: scenarios

text: the

text: original

text: disturbances

text: are

text: structured

text: since

text: it

text: requires

text: envisioned

text: could

text: give

text: rise

text: to

text: a

text: process

text: of

text: unsteady

text: jet

text: cre-

text: there

text: to

text: be

text: some

text: relatively

text: large

text: amplitude

text: disturbance

text: ation

text: and

text: destruction

text: entirely

text: contained

text: inside

text: the

text: shear

text: to

text: start

text: the

text: whole

text: thing

text: oﬀ.

text: A

text: recent

text: announcement

text: by

text: layer,

text: where

text: the

text: only

text: diﬀerence

text: in

text: the

text: outcomes

text: of

text: the

text: Marcus

text: et

text: al.

text: on

text: this

text: process

text: in

text: a

text: shearing

text: sheet

text: two

text: possibilities

text: is

text: the

text: degree

text: of

text: the

text: unsteady

text: activity

text: disk

text: model

text: with

text: vertically

text: varying

text: gravity

text: and

text: stratiﬁca-

text: possibly

text: turbulent.

text: Global

text: numerical

text: calculations

text: of

text: the

text: tion

text: seems

text: to

text: indicate

text: that

text: it

text: can

text: give

text: rise

text: to

text: a

text: strongly

text: several

text: processes

text: mentioned

text: (thermally

text: driven

text: relaxation

text: turbulent

text: state.

text: or

text: the

text: VSI),

text: when

text: suﬃciently

text: resolved,

text: can

text: test

text: whether

text: Draft

text: version

text: May

text: Spatially

text: localized

text: instability

text: or

text: not

text: this

text: dynamical

text: scenario

text: indeed

text: manifests

text: itself.

text: Preprint

text: typeset

text: using

text: LATEX

text: style

text: emulateapj

text: v.

text: In

text: Section

text: we

text: examined

text: the

text: properties

text: of

text: the

text: shear

text: CRITICAL

text: LAYERS

text: AND

text: PROTOPLANETARY

text: DISK

text: TURBULENCE


text: Critical layers and disk turbulence 7 −1 (a) Single Jump (Vortex Defect) −1.45 (b) Double Jump (Shear Layer) −1 (c) Triple Jump (Asymmetric Jet) ∼ 2∆ −1.2 −1.5 −1.2 −1.55 dv −1.4 ∼ ωa −1.4 ∼ ∆+ ∼ ωa+ 0 −1.6 dx −1.6 ∼ −ωa −1.6 −1.65 ∼ ωa ∼ ∆− −1.8 −1.7 −1.8 ∼ ωa− −2 −1 0 1 −1.75 −2 −1 0 1 −1 0 1 x − Figure 1. The total shear proﬁle (dv /dx = 3Ω /2 + dV /dx) of the two barotropic velocity ﬁelds detailed in Section : (a) The “vortex 0 0 0 step”, (b) the “shear layer”, and (c) the “asymmetric jet”. The shear is scaled in units −of Ω0 while the radial length x is scaled to the box scale L. Note that all shear ﬁelds shown include the background Keplerian state (= 3Ω /2) which is indicated for reference by the 0 dashed horizontal line in panel (a). The deviation from Keplerian shear proﬁle shown in panel (b) is anti-cyclonic while in panel (c) it is − − − anti-cyclonic in the region ∆ < x < 0 provided ω < 0 and it is cyclonic for 0 < x < ∆+ provided ω+ > 0. a a ##LTFigure##

text: Critical

text: layers

text: and

text: disk

text: turbulence

text: −1

font: default

text: (a)

text: Single

text: Jump

text: (Vortex

text: Defect)

text: (b)

text: Double

text: Jump

text: (Shear

text: Layer)

text: −1

text: (c)

text: Triple

text: Jump

text: (Asymmetric

text: Jet)

text: 2∆

text: dv

text: ωa

text: ωa+

text: dx

text: −ωa

text: ωa

text: ωa−

text: −2

text: −1

text: −2

text: −1

text: −1

text: x

text: Figure

text: The

text: total

text: shear

text: proﬁle

text: (dv

text: /dx

text: 3Ω

text: dV

text: /dx)

text: of

text: the

text: two

text: barotropic

text: velocity

text: ﬁelds

text: detailed

text: in

text: Section

text: (a)

text: The

text: “vortex

text: step”,

text: (b)

text: the

text: “shear

text: layer”,

text: and

text: (c)

text: the

text: “asymmetric

text: jet”.

text: The

text: shear

text: is

text: scaled

text: in

text: units

text: −of

text: Ω0

text: while

text: the

text: radial

text: length

text: x

text: is

text: scaled

text: to

text: the

text: box

text: scale

text: L.

text: Note

text: that

text: all

text: shear

text: ﬁelds

text: shown

text: include

text: the

text: background

text: Keplerian

text: state

text: 3Ω

text: which

text: is

text: indicated

text: for

text: reference

text: by

text: the

text: dashed

text: horizontal

text: line

text: in

text: panel

text: (a).

text: The

text: deviation

text: from

text: Keplerian

text: shear

text: proﬁle

text: shown

text: in

text: panel

text: (b)

text: is

text: anti-cyclonic

text: while

text: in

text: panel

text: (c)

text: it

text: is

text: anti-cyclonic

text: in

text: the

text: region

text: x

text: provided

text: ω

text: and

text: it

text: is

text: cyclonic

text: for

text: x

text: provided

text: ω+

text: a

text: a

text: ##LTFigure##


text: NCB = 6000 NCB = 9000 1000 1000 800 800 600 600 400 400 200 200 0 0 10001200140016001800200022002400 10001200140016001800200022002400 m @GeVD m @GeVD k±± k±± ±± FIG. 1: The ﬁgures represent the allowed region between the mass of k and the trilinear term (cid:46) (cid:46) of µ to satisfy 5.3 GeV Γ 45 GeV coming from the experimental resolution and the best S S ﬁt value of ATLAS respectively, where each of colored region (yellow for N = 6000 and blue for CB N = 9000) is allowed only and the upper line corresponds to Γ = 45 GeV and the lower line CB S corresponds to Γ = 5.3 GeV. S Then the resulting neutrino mass has to be multiplied by N , and can be estimated as CB ∗ O − ∗ ≈ 16N µg (fm )2 × (cid:46) (103 104) µg f2[GeV]2 CB (cid:96) m (loop factor) ν (4π)4 M2 16π4 M2 max max ∗ ≈ O − µg f2[GeV]2 (1 10) , (I.7) M2 max ≡ ≈ O where M Max[m ±±, m ±], we have used m = m (1) [GeV], and loop factor max k h (cid:96) τ is order 1. It suggests that the neutrino mass scale is determined by the trilinear coupling µ and the Yukawa couplings, and N that almost compensates the two loop suppression CB eﬀect. Therefore the two loop neutrino mass scale is equivalent to the tree level scale. Applying this fact, we will discuss our radiative neutrino model at the four loop level in the next section, which could be equivalent to a typical two loop radiative model. Then we will conclude and discuss in Sec. III. 4 DVeG@ SΜ DVeG@ SΜ ##LTFigure## ##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## ##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## ##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## ##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## ##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## ##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## ##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## ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: NCB

text: NCB

text: m

text: @GeVD

text: m

text: @GeVD

text: k±±

text: k±±

label: caption

text: FIG.

text: The

text: ﬁgures

text: represent

text: the

text: allowed

text: region

text: between

text: the

text: mass

text: of

text: k

text: and

text: the

text: trilinear

text: term

text: of

text: µ

text: to

text: satisfy

text: GeV

text: Γ

text: GeV

text: coming

text: from

text: the

text: experimental

text: resolution

text: and

text: the

text: best

text: S

text: S

text: ﬁt

text: value

text: of

text: ATLAS

text: respectively,

text: where

text: each

text: of

text: colored

text: region

text: (yellow

text: for

text: N

text: and

text: blue

text: for

text: CB

text: N

text: is

text: allowed

text: only

text: and

text: the

text: upper

text: line

text: corresponds

text: to

text: Γ

text: GeV

text: and

text: the

text: lower

text: line

text: CB

text: S

text: corresponds

text: to

text: Γ

text: GeV.

text: S

text: Then

text: the

text: resulting

text: neutrino

text: mass

text: has

text: to

text: be

text: multiplied

text: by

text: N

text: and

text: can

text: be

text: estimated

text: as

text: CB

text: O

text: µg

text: (fm

text: µg

text: CB

text: m

text: (loop

text: factor)

text: ν

text: max

text: max

text: O

text: µg

text: max

text: O

text: where

text: M

text: Max[m

text: m

text: we

text: have

text: used

text: m

text: m

text: [GeV],

text: and

text: loop

text: factor

text: max

text: k

text: h

text: τ

text: is

text: order

text: It

text: suggests

text: that

text: the

text: neutrino

text: mass

text: scale

text: is

text: determined

text: by

text: the

text: trilinear

text: coupling

text: µ

text: and

text: the

text: Yukawa

text: couplings,

text: and

text: N

text: that

text: almost

text: compensates

text: the

text: two

text: loop

text: suppression

text: CB

text: eﬀect.

text: Therefore

text: the

text: two

text: loop

text: neutrino

text: mass

text: scale

text: is

text: equivalent

text: to

text: the

text: tree

text: level

text: scale.

text: Applying

text: this

text: fact,

text: we

text: will

text: discuss

text: our

text: radiative

text: neutrino

text: model

text: at

text: the

text: four

text: loop

text: level

text: in

text: the

text: next

text: section,

text: which

text: could

text: be

text: equivalent

text: to

text: a

text: typical

text: two

text: loop

text: radiative

text: model.

text: Then

text: we

text: will

text: conclude

text: and

text: discuss

text: in

text: Sec.

text: III.

text: DVeG@

text: SΜ

text: DVeG@

text: SΜ

text: ##LTFigure##

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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: as shown in Figure 2, and we can respectively estimate the order of masses as follows: m = mI + mII + mIII + mIV , (II.3) ν ν ν ν ν ∗ ≈ [N µm M fg h]2 mI CB (cid:96) E G (x , x , x , x ), (II.4) ν (4π)8M4 M I (cid:96) E h k max N ∗ ≈ N2 M [µm M fg h]2 − mi CB R (cid:96) E G (x , x , x , x , x ), (i = II IV ), (II.5) ν (4π)8M6 i (cid:96) E h k N max where the left-top side of ﬁgure corresponds to mI , the right-top side of ﬁgure corresponds ν to mII, the left-bottom side of ﬁgure corresponds to mIII, the right-bottom side of ﬁgure ν ν ≡ corresponds to mIV , and we deﬁne x (m /M )2 and M = Max[M , m , m , M ]. ν i i max max E h k N G consists of two pairs of the Zee-Babu like two-loop function. Obviously mI can be greater I ν − ≈ than mII IV under the condition G G , since the ratio is given by ν I i (cid:18) (cid:19) mI ≈ M 2 (cid:29) −ν max 1. (II.6) mII IV M ν N Hence we can approximate the neutrino masses as ∗ ≈ [N µm M fg h]2 m = mI CB (cid:96) E , (II.7) ν ν (4π)8M4 M max N O (cid:46) (cid:46) where we take G = (1), and m should be 0.001 eV m 0.1 eV from the neutrino I ν ν oscillation data [141]. Muon anomalous magnetic moment: − The muon anomalous magnetic moment (muon g 2) has been measured at Brookhaven National Laboratory that suggests there is a discrepancy between the experimental data ≡ − and the prediction in the SM. The diﬀerence ∆a aexp aSM is respectively calculated in µ µ µ Ref. [142] and Ref. [143] as ± × − ± × − ∆a = (29.0 9.0) 10 10, ∆a = (33.5 8.2) 10 10. (II.8) µ µ 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: as

text: shown

text: in

text: Figure

text: and

text: we

text: can

text: respectively

text: estimate

text: the

text: order

text: of

text: masses

text: as

text: follows:

text: m

label: equation

text: mI

text: mII

text: mIII

text: mIV

text: ν

text: ν

text: ν

text: ν

text: ν

text: [N

text: µm

text: M

text: fg

text: mI

text: CB

text: E

text: G

text: (x

text: x

text: x

text: x

text: ν

text: M

text: I

text: E

text: h

text: k

text: max

text: N

text: M

text: [µm

text: M

text: fg

text: mi

text: CB

text: R

text: E

text: G

text: (x

text: x

text: x

text: x

text: x

text: (i

text: II

text: IV

text: ν

text: i

text: E

text: h

text: k

text: N

text: max

text: where

text: the

text: left-top

text: side

text: of

text: ﬁgure

text: corresponds

text: to

text: mI

text: the

text: right-top

text: side

text: of

text: ﬁgure

text: corresponds

text: ν

text: to

text: mII,

text: the

text: left-bottom

text: side

text: of

text: ﬁgure

text: corresponds

text: to

text: mIII,

text: the

text: right-bottom

text: side

text: of

text: ﬁgure

text: ν

text: ν

text: corresponds

text: to

text: mIV

text: and

text: we

text: deﬁne

text: x

text: (m

text: /M

text: and

text: M

text: Max[M

text: m

text: m

text: M

text: ν

text: i

text: i

text: max

text: max

text: E

text: h

text: k

text: N

text: G

text: consists

text: of

text: two

text: pairs

text: of

text: the

text: Zee-Babu

text: like

text: two-loop

text: function.

text: Obviously

text: mI

text: can

text: be

text: greater

text: I

text: ν

text: than

text: mII

text: IV

text: under

text: the

text: condition

text: G

text: G

text: since

text: the

text: ratio

text: is

text: given

text: by

text: ν

text: I

text: i

text: mI

text: M

text: −ν

text: max

text: mII

text: IV

text: M

text: ν

text: N

text: Hence

text: we

text: can

text: approximate

text: the

text: neutrino

text: masses

text: as

text: [N

text: µm

text: M

text: fg

text: m

text: mI

text: CB

text: E

text: ν

text: ν

text: M

text: max

text: N

text: O

text: where

text: we

text: take

text: G

text: and

text: m

text: should

text: be

text: eV

text: m

text: eV

text: from

text: the

text: neutrino

text: I

text: ν

text: ν

text: oscillation

text: data

text: Muon

text: anomalous

text: magnetic

text: moment:

text: The

text: muon

text: anomalous

text: magnetic

text: moment

text: (muon

text: g

text: has

text: been

text: measured

text: at

text: Brookhaven

text: National

text: Laboratory

text: that

text: suggests

text: there

text: is

text: a

text: discrepancy

text: between

text: the

text: experimental

text: data

text: and

text: the

text: prediction

text: in

text: the

text: SM.

text: The

text: diﬀerence

text: ∆a

text: aexp

text: aSM

text: is

text: respectively

text: calculated

text: in

text: µ

text: µ

text: µ

text: Ref.

text: and

text: Ref.

text: as

text: ∆a

text: ∆a

text: µ

text: µ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 JENNIFER S. BALAKRISHNAN AND NETAN DOGRA To give some context for our results, let us begin by recalling the Chabauty- Coleman method. Let J denote the Jacobian of X, let p be a prime of good reduction, and let p be a prime above p. Let → ∗ log : J(K ) H0(X , Ω1) J p p (cid:54) ∅ be the p-adic logarithm map for the abelian variety J. Suppose that X(K) = , and for convenience, that we know one point b in X(K). If the Mordell-Weil rank of J is less than g, the method of Chabauty [9] produces a ﬁnite set of p-adic points on X, which we shall denote X(K ) . The set X(K ) is a subset of X(K ), and p 1 p 1 p further, X(K) is a subset of X(K(cid:26)) . Following (cid:90)Coleman (cid:27)[11] the set X(K ) may p 1 p 1 be interpreted as the zeroes of a p-adic path integral ∈ z X(K ) = z X(K ) : ω = 0 p 1 p b for some diﬀerential ω in H0(X , Ω1 ). By further interpreting this p-adic path integral as a p-adic power series Kanpd sXolving for its zeros, in practice, one can often recover X(K). This is known as the Chabauty-Coleman method. The Chabauty-Coleman method requires that the Mordell-Weil rank of the Ja- cobian be less than the genus, which is somewhat restrictive. As such one would like to have a reﬁnement of the Jacobian which remembers more information about the set X(K). The insight of Kim [21] is that rather than trying to generalise the Jacobian of X, it is easier its Galois cohomological avatar: the Selmer group. In [22], Kim deﬁned a family of Selmer varieties Sel(U ) giving a decreasing sequence n of subsets [2] ⊃ ⊃ X(K ) X(K ) . . . p 1 p 2 of X(K ) , which can be computed in terms of iterated p-adic path integrals. p n The sets X(K ) contain X(K), so by proving ﬁniteness of X(K ) and explicitly computing it, ponne can hope to recover X(K). Note that when Kp=nQ, conjectures of Bloch and Kato imply that X(Q ) is ﬁnite for n suﬃciently large [22]. p n However, at present the only documented example of a curve X where X(K ) has been proved to give more information than X(K ) is when K = Q and Xp ins p 1 a curve whose Jacobian is isogenous to a product of CM abelian varieties. In this case, Coates and Kim prove in [10] that for n (cid:29) 0, X(Q ) is ﬁnite. Even in this case it is not clear how to actually compute X(Q ) . p n p n In this paper, we give techniques to handle some cases beyond the scope of classical Chabauty, by computing ﬁnite sets containing X(K ) . The methods p 2 used are a generalisation of those employed to study integral points on hyperelliptic curves using p-adic heights [6], combined with new methods for relating unipotent path torsors to p-adic heights [16]. In [6], one works with a hyperelliptic curve X/Q of genus g with a model (1) y2 = f(x) = x2g+1 + a x2g + · · · + a , a ∈ Z. 2g 0 i Let T denote the set of primes of bad reduction for this model and let p be 0 a prime of good reduction. Suppose that the polynomial f does not reduce to a square in (Z/q)[x] for any prime q. Let Y = Spec(Z[x, y]/(y2 −f(x))), so that Y (Z) denotes the set of integral solutions to (1). Using p-adic heights, one can compute Y (Z):

text: JENNIFER

text: S.

text: BALAKRISHNAN

text: AND

text: NETAN

text: DOGRA

text: To

text: give

text: some

text: context

text: for

text: our

text: results,

text: let

text: us

text: begin

text: by

text: recalling

text: the

text: Chabauty-

text: Coleman

text: method.

text: Let

text: J

text: denote

text: the

text: Jacobian

text: of

text: X,

text: let

text: p

text: be

text: a

text: prime

text: of

text: good

text: reduction,

text: and

text: let

text: p

text: be

text: a

text: prime

text: above

text: p.

text: Let

text: log

text: J(K

text: J

text: p

text: p

text: be

text: the

text: p-adic

text: logarithm

text: map

text: for

text: the

text: abelian

text: variety

text: J.

text: Suppose

text: that

text: X(K)

text: and

text: for

text: convenience,

text: that

text: we

text: know

text: one

text: point

text: b

text: in

text: X(K).

text: If

text: the

text: Mordell-Weil

text: rank

text: of

text: J

text: is

text: less

text: than

text: g,

text: the

text: method

text: of

text: Chabauty

text: produces

text: a

text: ﬁnite

text: set

text: of

text: p-adic

text: points

text: on

text: X,

text: which

text: we

text: shall

text: denote

text: X(K

text: The

text: set

text: X(K

text: is

text: a

text: subset

text: of

text: X(K

text: and

text: p

text: p

text: p

text: further,

text: X(K)

text: is

text: a

text: subset

text: of

text: Following

text: the

text: set

text: X(K

text: may

text: p

text: p

text: be

text: interpreted

text: as

text: the

text: zeroes

text: of

text: a

text: p-adic

text: path

text: integral

text: z

text: X(K

text: z

text: X(K

text: ω

text: p

text: p

text: b

text: for

text: some

text: diﬀerential

text: ω

text: in

text: Ω1

text: By

text: further

text: interpreting

text: this

text: p-adic

text: path

text: integral

text: as

text: a

text: p-adic

text: power

text: series

text: Kanpd

text: sXolving

text: for

text: its

text: zeros,

text: in

text: practice,

text: one

text: can

text: often

text: recover

text: X(K).

text: This

text: is

text: known

text: as

text: the

text: Chabauty-Coleman

text: method.

text: The

text: Chabauty-Coleman

text: method

text: requires

text: that

text: the

text: Mordell-Weil

text: rank

text: of

text: the

text: Ja-

text: cobian

text: be

text: less

text: than

text: the

text: genus,

text: which

text: is

text: somewhat

text: restrictive.

text: As

text: such

text: one

text: would

text: like

text: to

text: have

text: a

text: reﬁnement

text: of

text: the

text: Jacobian

text: which

text: remembers

text: more

text: information

text: about

text: the

text: set

text: X(K).

text: The

text: insight

text: of

text: Kim

text: is

text: that

text: rather

text: than

text: trying

text: to

text: generalise

text: the

text: Jacobian

text: of

text: X,

text: it

text: is

text: easier

text: its

text: Galois

text: cohomological

text: avatar:

text: the

text: Selmer

text: group.

text: In

text: Kim

text: deﬁned

text: a

text: family

text: of

text: Selmer

text: varieties

text: Sel(U

text: giving

text: a

text: decreasing

text: sequence

text: n

text: of

text: subsets

text: X(K

text: X(K

text: p

text: p

text: of

text: X(K

text: which

text: can

text: be

text: computed

text: in

text: terms

text: of

text: iterated

text: p-adic

text: path

text: integrals.

text: p

text: n

text: The

text: sets

text: X(K

text: contain

text: X(K),

text: so

text: by

text: proving

text: ﬁniteness

text: of

text: X(K

text: and

text: explicitly

text: computing

text: it,

text: ponne

text: can

text: hope

text: to

text: recover

text: X(K).

text: Note

text: that

text: when

text: Kp=nQ,

text: conjectures

text: of

text: Bloch

text: and

text: Kato

text: imply

text: that

text: X(Q

text: is

text: ﬁnite

text: for

text: n

text: suﬃciently

text: large

text: p

text: n

text: However,

text: at

text: present

text: the

text: only

text: documented

text: example

text: of

text: a

text: curve

text: X

text: where

text: X(K

text: has

text: been

text: proved

text: to

text: give

text: more

text: information

text: than

text: X(K

text: is

text: when

text: K

text: Q

text: and

text: Xp

text: ins

text: p

text: a

text: curve

text: whose

text: Jacobian

text: is

text: isogenous

text: to

text: a

text: product

text: of

text: CM

text: abelian

text: varieties.

text: In

text: this

text: case,

text: Coates

text: and

text: Kim

text: prove

text: in

text: that

text: for

text: n

text: X(Q

text: is

text: ﬁnite.

text: Even

text: in

text: this

text: case

text: it

text: is

text: not

text: clear

text: how

text: to

text: actually

text: compute

text: X(Q

text: p

text: n

text: p

text: n

text: In

text: this

text: paper,

text: we

text: give

text: techniques

text: to

text: handle

text: some

text: cases

text: beyond

text: the

text: scope

text: of

text: classical

text: Chabauty,

text: by

text: computing

text: ﬁnite

text: sets

text: containing

text: X(K

text: The

text: methods

text: p

text: used

text: are

text: a

text: generalisation

text: of

text: those

text: employed

text: to

text: study

text: integral

text: points

text: on

text: hyperelliptic

text: curves

text: using

text: p-adic

text: heights

text: combined

text: with

text: new

text: methods

text: for

text: relating

text: unipotent

text: path

text: torsors

text: to

text: p-adic

text: heights

text: In

text: one

text: works

text: with

text: a

text: hyperelliptic

text: curve

text: X/Q

text: of

text: genus

text: g

text: with

text: a

text: model

text: f(x)

text: a

text: a

text: a

text: Z.

text: i

text: Let

text: T

text: denote

text: the

text: set

text: of

text: primes

text: of

text: bad

text: reduction

text: for

text: this

text: model

text: and

text: let

text: p

text: be

text: a

text: prime

text: of

text: good

text: reduction.

text: Suppose

text: that

text: the

text: polynomial

text: f

text: does

text: not

text: reduce

text: to

text: a

text: square

text: in

text: (Z/q)[x]

text: for

text: any

text: prime

text: q.

text: Let

text: Y

text: Spec(Z[x,

text: −f(x))),

text: so

text: that

text: Y

text: (Z)

text: denotes

text: the

text: set

text: of

text: integral

text: solutions

text: to

text: Using

text: p-adic

text: heights,

text: one

text: can

text: compute

text: Y

text: (Z):


text: 16 JENNIFER S. BALAKRISHNAN AND NETAN DOGRA construction with some kind of linearity properties analogous to those of the global height pairing. For this reason, in this section we deﬁne a way to embed Sel(U) into H1 (G , U(Q , V, Q (1)) via twisting. We then apply Nekovář’s construction, giving s(tviaTcompopsition)p local functions Sel(U) → Q . Note that if Q (1) is re- p − p placed by a diﬀerent Galois representation W of motivic weight 2 arising in U[2], one may mimic Nekovář’s construction with the cohomology class χ replaced by ∗ a cohomology class in H1(G , W (1)) which is nontrivial and noncrystalline at p T (assuming one can prove such a class exists). This will be pursued in future work. To construct a mixed extension associated to an element of H1(G, U), we deﬁne a G-representation with an equivariant U-module structure, which will be denoted A(b), and then send a U-torsor P to the twist of A(b) by P. A(b) will be deﬁned to be a certain ﬁnite-dimensional quotient of the universal Q enveloping algebra of πe´t, p(X, b). By the theory of Malcev completion, this has a 1 very concrete description, which we now recall. Let I denote the kernel of (cid:88) (cid:88) Q → Q [πe´t(X, x)] p 1 p (cid:55)→ λ γ λ . γ γ Since πe´t(X, x) is isomorphic to the proﬁnite completion of a free group F 1 2g on 2g generators modulo one relation lying in [F , F ], we have that A∞(x) is 2g 2g isomorphic to the completion of an algebra obtained by quotienting a free associative Q -algebra R on 2g generators by a 2-sided ideal generated by an element e ∈ I2. p 2g Deﬁnition 8. Let A (b) = Q [πe´t(X, x)]/In+1. n p 1 A (b) is equipped with the structure of a Galois-equivariant πe´t(X, x)-module, n 1 since it is a quotient of the group algebra by a Galois stable ideal. Hence for any Galois-equivariant πe´t(X, x) torsor P we can twist A (b) by P to get a Galois rep- 1 n resentation A (b)(P). When P = πe´t(X; x, y), A (b)(P) may be identiﬁed with the Galois-equivarniant A (b)-module A1 (b, z) obtainned by tensoring Q [πe´t(X; x, y)], thought of as a Q [πne´t(X, x)]-modunle, with A (b). For this reasonpwe1 refer to it p 1 n A (b, z) as a path module. It follows from the theory of Malcev completion that the n action of πe´t(X, b) on A (b) factors through the homomorphism 1 n → πe´t(X, b) U (b). 1 n Furthermore, A (b) is a quotient of the enveloping algebra of U (b), and a faithful representation onf U (b). More generally we can view the Q -vectnor space generated n p by the torsor of paths from b to z, denoted Q [πe´t(X; b, z)], as a G-equivariant free rank 1 module over Q [πe´t(X, b)]. Hence we pma1y make the following deﬁnition. p 1 Deﬁnition 9. Let A (b, z) be the G-equivariant free rank 1 A (b) module n n Q × [πe´t(X; b, z)] Q A (b). p 1 p[π1e´t(X;b,z)] n Note that A (b, z) is naturally equipped with a G-stable ﬁltration n ⊃ ⊃ ⊃ A (b, z) IA (b, z) . . . In+1A (b, z) = 0. n n n ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: JENNIFER

text: S.

text: BALAKRISHNAN

text: AND

text: NETAN

text: DOGRA

text: construction

text: with

text: some

text: kind

text: of

text: linearity

text: properties

text: analogous

text: to

text: those

text: of

text: the

text: global

text: height

text: pairing.

text: For

text: this

text: reason,

text: in

text: this

text: section

text: we

text: deﬁne

text: a

text: way

text: to

text: embed

text: Sel(U)

text: into

text: (G

text: U(Q

text: V,

text: Q

text: via

text: twisting.

text: We

text: then

text: apply

text: Nekovář’s

text: construction,

text: giving

text: s(tviaTcompopsition)p

text: local

text: functions

text: Sel(U)

text: Q

text: Note

text: that

text: if

text: Q

text: is

text: re-

text: p

text: p

text: placed

text: by

text: a

text: diﬀerent

text: Galois

text: representation

text: W

text: of

text: motivic

text: weight

text: arising

text: in

text: one

text: may

text: mimic

text: Nekovář’s

text: construction

text: with

text: the

text: cohomology

text: class

text: χ

text: replaced

text: by

text: a

text: cohomology

text: class

text: in

text: W

text: which

text: is

text: nontrivial

text: and

text: noncrystalline

text: at

text: p

text: T

text: (assuming

text: one

text: can

text: prove

text: such

text: a

text: class

text: exists).

text: This

text: will

text: be

text: pursued

text: in

text: future

text: work.

text: To

text: construct

text: a

text: mixed

text: extension

text: associated

text: to

text: an

text: element

text: of

text: U),

text: we

text: deﬁne

text: a

text: G-representation

text: with

text: an

text: equivariant

text: U-module

text: structure,

text: which

text: will

text: be

text: denoted

text: A(b),

text: and

text: then

text: send

text: a

text: U-torsor

text: P

text: to

text: the

text: twist

text: of

text: A(b)

text: by

text: P.

text: A(b)

text: will

text: be

text: deﬁned

text: to

text: be

text: a

text: certain

text: ﬁnite-dimensional

text: quotient

text: of

text: the

text: universal

text: Q

text: enveloping

text: algebra

text: of

text: πe´t,

text: p(X,

text: b).

text: By

text: the

text: theory

text: of

text: Malcev

text: completion,

text: this

text: has

text: a

text: very

text: concrete

text: description,

text: which

text: we

text: now

text: recall.

text: Let

text: I

text: denote

text: the

text: kernel

text: of

text: Q

text: Q

text: [πe´t(X,

text: x)]

text: p

text: p

text: λ

text: γ

text: λ

text: γ

text: γ

text: Since

text: πe´t(X,

text: x)

text: is

text: isomorphic

text: to

text: the

text: proﬁnite

text: completion

text: of

text: a

text: free

text: group

text: F

text: on

text: generators

text: modulo

text: one

text: relation

text: lying

text: in

text: [F

text: F

text: we

text: have

text: that

text: A∞(x)

text: is

text: isomorphic

text: to

text: the

text: completion

text: of

text: an

text: algebra

text: obtained

text: by

text: quotienting

text: a

text: free

text: associative

text: Q

text: -algebra

text: R

text: on

text: generators

text: by

text: a

text: ideal

text: generated

text: by

text: an

text: element

text: e

text: p

text: Deﬁnition

text: Let

text: A

text: (b)

text: Q

text: [πe´t(X,

text: n

text: p

text: A

text: (b)

text: is

text: equipped

text: with

text: the

text: structure

text: of

text: a

text: Galois-equivariant

text: πe´t(X,

text: x)-module,

text: n

text: since

text: it

text: is

text: a

text: quotient

text: of

text: the

text: group

text: algebra

text: by

text: a

text: Galois

text: stable

text: ideal.

text: Hence

text: for

text: any

text: Galois-equivariant

text: πe´t(X,

text: x)

text: torsor

text: P

text: we

text: can

text: twist

text: A

text: (b)

text: by

text: P

text: to

text: get

text: a

text: Galois

text: rep-

text: n

text: resentation

text: A

text: (b)(P).

text: When

text: P

text: πe´t(X;

text: x,

text: y),

text: A

text: (b)(P)

text: may

text: be

text: identiﬁed

text: with

text: the

text: Galois-equivarniant

text: A

text: (b)-module

text: (b,

text: z)

text: obtainned

text: by

text: tensoring

text: Q

text: [πe´t(X;

text: x,

text: y)],

text: thought

text: of

text: as

text: a

text: Q

text: [πne´t(X,

text: x)]-modunle,

text: with

text: A

text: (b).

text: For

text: this

text: refer

text: to

text: it

text: p

text: n

text: A

text: (b,

text: z)

text: as

text: a

text: path

text: module.

text: It

text: follows

text: from

text: the

text: theory

text: of

text: Malcev

text: completion

text: that

text: the

text: n

text: action

text: of

text: πe´t(X,

text: b)

text: on

text: A

text: (b)

text: factors

text: through

text: the

text: homomorphism

text: n

text: πe´t(X,

text: b)

text: U

text: (b).

text: n

text: Furthermore,

text: A

text: (b)

text: is

text: a

text: quotient

text: of

text: the

text: enveloping

text: algebra

text: of

text: U

text: (b),

text: and

text: a

text: faithful

text: representation

text: onf

text: U

text: (b).

text: More

text: generally

text: we

text: can

text: view

text: the

text: Q

text: -vectnor

text: space

text: generated

text: n

text: p

text: by

text: the

text: torsor

text: of

text: paths

text: from

text: b

text: to

text: z,

text: denoted

text: Q

text: [πe´t(X;

text: b,

text: z)],

text: as

text: a

text: G-equivariant

text: free

text: rank

text: module

text: over

text: Q

text: [πe´t(X,

text: b)].

text: Hence

text: we

text: make

text: the

text: following

text: deﬁnition.

text: p

text: Deﬁnition

text: Let

text: A

text: (b,

text: z)

text: be

text: the

text: G-equivariant

text: free

text: rank

text: A

text: (b)

text: module

text: n

text: n

text: Q

text: [πe´t(X;

text: b,

text: z)]

text: Q

text: A

text: (b).

text: p

text: n

text: Note

text: that

text: A

text: (b,

text: z)

text: is

text: naturally

text: equipped

text: with

text: a

text: G-stable

text: ﬁltration

text: n

text: A

text: (b,

text: z)

text: IA

text: (b,

text: z)

text: (b,

text: z)

text: n

text: n

text: n

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: QUADRATIC CHABAUTY AND RATIONAL POINTS I: p-ADIC HEIGHTS 3 Theorem 1 (“Quadratic Chabauty(cid:88)”, [6]). Let Ω ⊂ Q be the explicitly computable, p ﬁnite set of values taken by the sum of the local heights − − ∞ (cid:81) h (z ), v v ∈ v T0 for (z ) in ∈ Y (Z ). Suppose that the Mordell-Weil rank of J is g. Then there is a syvmmetrivc Tb0ilinearv map ∗ × ∗ → Q B : H0(XQ , Ω1) H0(XQ , Ω1) p p p such that Y (Z) ⊂ Y (Z ) is contained inside the ﬁnite set of solutions to p − ∞ − ∞ − ∞ ∈ h (z ) + B(log (z ), log (z )) Ω. p J J In the present work, we give a generalisation of this theorem which also allows us to study rational points in some special cases where the Mordell-Weil rank is not less than the genus. To state our results more precisely, we ﬁx some notation. Let K be Q or an imaginary quadratic ﬁeld, and let X/K be a smooth projective curve of genus g > 1 with a K-rational point b. Let T be the set of primes of bad reduction for X, let p be a prime of Q such that {v|0p} ∩ T is empty, and let ∪ { | } 0 T = T v p . We show that X(K ) is ﬁnite whenever the rank of J(K) minus 0 p 2 − the rank of the Néron-Severi group of J, denoted NS(J), is less than g 1 (see Lemma 3). In particular, the main example we consider is the situation when the rank of the Jacobian of X is g and the rank of NS(J) is greater than 1 (for a result applying when the rank is greater than the genus, see Proposition 2). We further assume that the map ⊗ Q −(cid:39)→ ⊗ Q J(K) Z J(K ) Z × p p p p is an isomorphism. Let X := X K. By our assumptions on the Néron-Severi × K ∧ group, there is a cycle Z in X X whose image in 2H1 (X) under e´t Q − → × Q → Q ⊗ Q → ∧ Q ( 1) H2 (X X, ) H1 (X, ) H1 (X, ) 2H1 (X, ) p e´t p e´t p e´t p e´t p is nonzero (where the maps are, from left to right, the cycle class, the Kunneth projector, and the antisymmetric projection), and such that the intersection number − × − × of Z with ∆ X P P X is zero, where P and P are any points on X. 1 2 − 1 2 For distinct points b and z in X intersecting ∆ 1Z properly, we associate a cycle X D(b, z) to the triple (b, z, Z) (see Deﬁnition 11). Theorem 2. Let X/K be a smooth projective curve of genus g > 1. Let b, z, D(b, z) (cid:48) − − | | be as above. Let X := X i 1 Z . (i): For each prime v (cid:45) p, h∆(z, D(b, z)) takes only ﬁnitely many values for z in X(cid:48)(Q ). If v is a prime of potvential good reduction, then h (z, D(b, z)) is identically v v zero. (ii): Suppose the rank of J(K) i(cid:88)s g and the rank of NS(J) is greater than 1. Let Ω ⊂ Q be the ﬁnite set of values taken by the sum of local heights p − (cid:81) h (z , D(b, z )) v v v (cid:45) v p (cid:48) for (z ) in (cid:45) X (K ). Then there is a symmetric bilinear map v v p v ∗ × ∗ → Q B : H0(X , Ω1) H0(X , Ω1) Kp Kp p ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: QUADRATIC

text: CHABAUTY

text: AND

text: RATIONAL

text: POINTS

text: I:

text: p-ADIC

text: HEIGHTS

text: Theorem

text: (“Quadratic

text: Let

text: Ω

text: Q

text: be

text: the

text: explicitly

text: computable,

text: p

text: ﬁnite

text: set

text: of

text: values

text: taken

text: by

text: the

text: sum

text: of

text: the

text: local

text: heights

text: h

text: (z

text: v

text: v

text: v

text: for

text: (z

text: in

text: Y

text: (Z

text: Suppose

text: that

text: the

text: Mordell-Weil

text: rank

text: of

text: J

text: is

text: g.

text: Then

text: there

text: is

text: a

text: syvmmetrivc

text: map

text: Q

text: B

text: p

text: p

text: p

text: such

text: that

text: Y

text: (Z)

text: Y

text: (Z

text: is

text: contained

text: inside

text: the

text: ﬁnite

text: set

text: of

text: solutions

text: to

text: p

text: h

text: (z

text: B(log

text: (z

text: log

text: (z

text: Ω.

text: p

text: J

text: J

text: In

text: the

text: present

text: work,

text: we

text: give

text: a

text: generalisation

text: of

text: this

text: theorem

text: which

text: also

text: allows

text: us

text: to

text: study

text: rational

text: points

text: in

text: some

text: special

text: cases

text: where

text: the

text: Mordell-Weil

text: rank

text: is

text: not

text: less

text: than

text: the

text: genus.

text: To

text: state

text: our

text: results

text: more

text: precisely,

text: we

text: ﬁx

text: some

text: notation.

text: Let

text: K

text: be

text: Q

text: or

text: an

text: imaginary

text: quadratic

text: ﬁeld,

text: and

text: let

text: X/K

text: be

text: a

text: smooth

text: projective

text: curve

text: of

text: genus

text: g

text: with

text: a

text: K-rational

text: point

text: b.

text: Let

text: T

text: be

text: the

text: set

text: of

text: primes

text: of

text: bad

text: reduction

text: for

text: X,

text: let

text: p

text: be

text: a

text: prime

text: of

text: Q

text: such

text: that

text: T

text: is

text: empty,

text: and

text: let

text: T

text: T

text: v

text: p

text: We

text: show

text: that

text: X(K

text: is

text: ﬁnite

text: whenever

text: the

text: rank

text: of

text: J(K)

text: minus

text: p

text: the

text: rank

text: of

text: the

text: Néron-Severi

text: group

text: of

text: J,

text: denoted

text: NS(J),

text: is

text: less

text: than

text: g

text: (see

text: Lemma

text: In

text: particular,

text: the

text: main

text: example

text: we

text: consider

text: is

text: the

text: situation

text: when

text: the

text: rank

text: of

text: the

text: Jacobian

text: of

text: X

text: is

text: g

text: and

text: the

text: rank

text: of

text: NS(J)

text: is

text: greater

text: than

text: (for

text: a

text: result

text: applying

text: when

text: the

text: rank

text: is

text: greater

text: than

text: the

text: genus,

text: see

text: Proposition

text: We

text: further

text: assume

text: that

text: the

text: map

text: Q

text: Q

text: J(K)

text: Z

text: J(K

text: Z

text: p

text: p

text: p

text: p

text: is

text: an

text: isomorphism.

text: Let

text: X

text: X

text: K.

text: By

text: our

text: assumptions

text: on

text: the

text: Néron-Severi

text: K

text: group,

text: there

text: is

text: a

text: cycle

text: Z

text: in

text: X

text: X

text: whose

text: image

text: in

text: (X)

text: under

text: e´t

text: Q

text: Q

text: Q

text: Q

text: Q

text: (X

text: X,

text: (X,

text: (X,

text: (X,

text: p

text: e´t

text: p

text: e´t

text: p

text: e´t

text: p

text: e´t

text: p

text: is

text: nonzero

text: (where

text: the

text: maps

text: are,

text: from

text: left

text: to

text: right,

text: the

text: cycle

text: class,

text: the

text: Kunneth

text: projector,

text: and

text: the

text: antisymmetric

text: projection),

text: and

text: such

text: that

text: the

text: intersection

text: number

text: of

text: Z

text: with

text: X

text: P

text: P

text: X

text: is

text: zero,

text: where

text: P

text: and

text: P

text: are

text: any

text: points

text: on

text: X.

text: For

text: distinct

text: points

text: b

text: and

text: z

text: in

text: X

text: intersecting

text: properly,

text: we

text: associate

text: a

text: cycle

text: X

text: D(b,

text: z)

text: to

text: the

text: triple

text: (b,

text: z,

text: Z)

text: (see

text: Deﬁnition

text: Theorem

text: Let

text: X/K

text: be

text: a

text: smooth

text: projective

text: curve

text: of

text: genus

text: g

text: Let

text: b,

text: z,

text: D(b,

text: z)

text: be

text: as

text: above.

text: Let

text: X

text: X

text: i

text: Z

text: (i):

text: For

text: each

text: prime

text: v

text: p,

text: h∆(z,

text: D(b,

text: z))

text: takes

text: only

text: ﬁnitely

text: many

text: values

text: for

text: z

text: in

text: If

text: v

text: is

text: a

text: prime

text: of

text: potvential

text: good

text: reduction,

text: then

text: h

text: (z,

text: D(b,

text: z))

text: is

text: identically

text: v

text: v

text: zero.

text: (ii):

text: Suppose

text: the

text: rank

text: of

text: J(K)

text: g

text: and

text: the

text: rank

text: of

text: NS(J)

text: is

text: greater

text: than

text: Let

text: Ω

text: Q

text: be

text: the

text: ﬁnite

text: set

text: of

text: values

text: taken

text: by

text: the

text: sum

text: of

text: local

text: heights

text: p

text: h

text: (z

text: D(b,

text: z

text: v

text: v

text: v

text: v

text: p

text: for

text: (z

text: in

text: X

text: (K

text: Then

text: there

text: is

text: a

text: symmetric

text: bilinear

text: map

text: v

text: v

text: p

text: v

text: Q

text: B

text: Kp

text: Kp

text: p

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 24 JENNIFER S. BALAKRISHNAN AND NETAN DOGRA together with the isomorphisms (cid:39) (cid:39) (cid:39) H1(Y ; b) H1(Y ; z) H1(Y ) H1(X). Hence we get a short exact sequence → ∪ → × ∪ ∪ → ⊗ → 0 H1(Y ; b z) H2(Y Y ; X X ∆) H1(X) H1(X) 0. 1 2 × ∪ ∪ The dual of A (X)(b, z) is isomorphic to H2(X X; X X ∆), which sits in 2 1 2 an exact sequence 0 → H1(X; b∪z) → H2(X×X; X ∪X ∪∆) → H2(X×X) (i∗1−,i→∗2,i∗∆) H2(X)⊕3 → 0. 1 2 The natural map × ∪ ∪ → × ∪ ∪ H2(X X; X X ∆) H2(Y Y ; X X ∆) 1 2 1 2 is injective and gives a short exact sequence → × ∪ ∪ → × ∪ ∪ → → 0 H2(X X; X X ∆) H2(Y Y ; X X ∆) H2(X) 0. 1 2 1 2 × × Via the above, if Z is a cycle in X X whose image in H2(Y Y ) is nonzero then Z × ∪ ∪ deﬁnes a subobject of H2(Y Y ; X X ∆). If the intersection number of Z with − − 1 2 × ∪ ∪ ∆ X X is zero then this subobject is in the image of H2(X X; X X ∆) 1 2 × 1 2 (speciﬁcally, its image in H2(X X) is the Kunneth projector of Z). Via Beilinson’s cohomological characterisation of the mixed extension A (Y )(b, z), 2 characterising the local heights of A(b, z) in terms of the height pairings on algebraic − | | | | cycles is reduced to relating subquotients of H1(Y Z ; Z ) to subquotients of × | | 1 2 H2(Y Y ; W ), where the divisors W arise from Beilinson’s theorem. Let Z be a divisor of S intersecting X , X and ∆ properly. We somewhat 1 2 abusively denote the composite map Q (−1) −cl→Z H2(S) → H1(X)⊗2 −(cid:39)→ H2(S; X ∪ X ) p 1 2 by cl , where the last map is the isomorphism induced by Z ∪ → H2(S; X X ) H2(S). 1 2 ∈ ∗ − ∗ − ∗ Deﬁnition 11. Deﬁne D(b, z) Div0(X) to be the cycle i Z i Z i Z+(Z.X + − ∆ 1 2 1 Z.X Z.∆)x. 2 Let E = E (b, z) be the mixed extension with graded pieces obtained by pulling Z Z∪ ∪ back H2(S; X X ∆)(1) by the image of Z: 1 2 Q E Z p cl Z ∪ ∪ ⊗ H2(S; X X ∆)(1) H1(Y ) H1(Y )(1). 1 2 As in §5.1, if cl is nonzero it deﬁnes a surjection A[2](Y ) → Q (1) and hence a Z p quotient A(b, z) of A (Y )(b, z). The representation A(b, z) is the dual of E . If the 2 − − Z intersection number of Z with ∆ X X is zero then A(b, z) in fact comes from 1 2 a quotient of A (X)(b, z). The following theorem says that the mixed extension 2 ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: JENNIFER

text: S.

text: BALAKRISHNAN

text: AND

text: NETAN

text: DOGRA

text: together

text: with

text: the

text: isomorphisms

text: b)

text: z)

text: Hence

text: we

text: get

text: a

text: short

text: exact

text: sequence

text: b

text: z)

text: Y

text: X

text: X

text: The

text: dual

text: of

text: A

text: (X)(b,

text: z)

text: is

text: isomorphic

text: to

text: X;

text: X

text: X

text: which

text: sits

text: in

text: an

text: exact

text: sequence

text: b∪z)

text: X

text: ∪X

text: The

text: natural

text: map

text: X;

text: X

text: X

text: Y

text: X

text: X

text: is

text: injective

text: and

text: gives

text: a

text: short

text: exact

text: sequence

text: X;

text: X

text: X

text: Y

text: X

text: X

text: Via

text: the

text: above,

text: if

text: Z

text: is

text: a

text: cycle

text: in

text: X

text: X

text: whose

text: image

text: in

text: Y

text: is

text: nonzero

text: then

text: Z

text: deﬁnes

text: a

text: subobject

text: of

text: Y

text: X

text: X

text: If

text: the

text: intersection

text: number

text: of

text: Z

text: with

text: X

text: X

text: is

text: zero

text: then

text: this

text: subobject

text: is

text: in

text: the

text: image

text: of

text: X;

text: X

text: X

text: (speciﬁcally,

text: its

text: image

text: in

text: X)

text: is

text: the

text: Kunneth

text: projector

text: of

text: Z).

text: Via

text: Beilinson’s

text: cohomological

text: characterisation

text: of

text: the

text: mixed

text: extension

text: A

text: (Y

text: )(b,

text: z),

text: characterising

text: the

text: local

text: heights

text: of

text: A(b,

text: z)

text: in

text: terms

text: of

text: the

text: height

text: pairings

text: on

text: algebraic

text: cycles

text: is

text: reduced

text: to

text: relating

text: subquotients

text: of

text: Z

text: Z

text: to

text: subquotients

text: of

text: Y

text: W

text: where

text: the

text: divisors

text: W

text: arise

text: from

text: Beilinson’s

text: theorem.

text: Let

text: Z

text: be

text: a

text: divisor

text: of

text: S

text: intersecting

text: X

text: X

text: and

text: properly.

text: We

text: somewhat

text: abusively

text: denote

text: the

text: composite

text: map

text: Q

text: −cl→Z

text: X

text: X

text: p

text: by

text: cl

text: where

text: the

text: last

text: map

text: is

text: the

text: isomorphism

text: induced

text: by

text: Z

text: X

text: X

text: Deﬁnition

text: Deﬁne

text: D(b,

text: z)

text: to

text: be

text: the

text: cycle

text: i

text: Z

text: i

text: Z

text: i

text: Z+(Z.X

text: Z.X

text: Z.∆)x.

text: Let

text: E

text: E

text: (b,

text: z)

text: be

text: the

text: mixed

text: extension

text: with

text: graded

text: pieces

text: obtained

text: by

text: pulling

text: Z

text: Z∪

text: back

text: X

text: X

text: by

text: the

text: image

text: of

text: Z:

text: Q

text: E

text: Z

text: p

text: cl

text: Z

text: X

text: X

text: As

text: in

text: if

text: cl

text: is

text: nonzero

text: it

text: deﬁnes

text: a

text: surjection

text: Q

text: and

text: hence

text: a

text: Z

text: p

text: quotient

text: A(b,

text: z)

text: of

text: A

text: (Y

text: )(b,

text: z).

text: The

text: representation

text: A(b,

text: z)

text: is

text: the

text: dual

text: of

text: E

text: If

text: the

text: Z

text: intersection

text: number

text: of

text: Z

text: with

text: X

text: X

text: is

text: zero

text: then

text: A(b,

text: z)

text: in

text: fact

text: comes

text: from

text: a

text: quotient

text: of

text: A

text: (X)(b,

text: z).

text: The

text: following

text: theorem

text: says

text: that

text: the

text: mixed

text: extension

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: ##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: 30 JENNIFER S. BALAKRISHNAN AND NETAN DOGRA Let X be a hyperelliptic curve given by an equation of the form y2 = f(x) O (cid:80) −∞ where f(x) is a monic polynomial in [x] of degree 2g +1. Let Y = X . Take { } ⊂ K × { } Z to be the cycle Γ = (z, w(z)) X X. Let z , . . . , z denote the set of K w 1 n points of X with y-coordinate zero, and let W denote the divisor z . Let b and i i z be points of Y with nonzero y-coordinate. Then ∗ i Γ = w(b) 1 w ∗ i Γ = w(z) 2 w ∗ ∞ i Γ = W + ∆ w − − − − ∞ hence D(b, z) = W w(b) w(z) (2g 1) . So the class of A(Y )(b, z) is dual − − − − − ∞ to H (z b, W w(b) w(z) (2g 1) ), by Theorem 6. The following lemma X illustrates how Theorem 1 may be deduced from Theorem 6 together with the aﬃne version of Theorem 2. Lemma 14. For any prime v, − − ∞ − − ∞ h (z b, D(b, z)) = h (z ) h (b ). v v v Proof. First, note that additivity yields − − − ∞ − − ∞ − − h (z b, D(b, z)) = h (z b, W (2g + 1) ) h (z b, 2 w(z) w(b)). v v v ∞ − − Since 2(g + 1) W = div(y), the ﬁrst term is equal to χ(y(z)) χ(y(b)). For the − ∞ − − second term, since z b and 2 w(z) w(b) are disjoint, − ∞− − 1 − ∞− − 1 ∞− − − h (z b, 2 w(z) w(b)) = h (z b, 2 w(z) w(b))+ h (2 w(z) w(b), z b). v 2 v 2 v By additivity − ∞ − − − ∞ ∞ − − ∞ ∞ − h (z b, 2 w(z) w(b)) = h (z , w(z)) + h (z , w(b)) v v v ∞ − ∞ − ∞ − ∞ − + h ( b, w(z)) + h ( b, w(b)) v v ∞ − − − and similarly for h (2 w(z) w(b), z b). Using the fact that h (D , D ) = v v 1 2 h (w(D ), w(D )), this gives v 1 2 − ∞ − − − ∞ ∞ − ∞ − ∞ − h (z b, 2 w(z) w(b)) = h (z , w(z)) + h ( b, w(b)). v v v (cid:3) The result now follows from Lemma 13. In this subsection we return to the case where X is a genus 2 curve of the form y2 = x6 + ax4 + bx2 + c, and explain how to deduce Theorem 3 from Theorem 2. Let h and h denote v (local and global, resp.) heights on X, h and h heights on E , and h and h heights on E . Recall from the intEr1o,vduction Eth1 e associated el1liptic curEv2e,vs E anEd2morphisms f 2 : X → E . Deﬁne Z ⊂ X × X to be the graph of the i i i 1 automorphism ##LTLine## ##LTLine## ##LTLine##

text: JENNIFER

text: S.

text: BALAKRISHNAN

text: AND

text: NETAN

text: DOGRA

text: Let

text: X

text: be

text: a

text: hyperelliptic

text: curve

text: given

text: by

text: an

text: equation

text: of

text: the

text: form

text: f(x)

text: O

text: where

text: f(x)

text: is

text: a

text: monic

text: polynomial

text: in

text: [x]

text: of

text: degree

text: Let

text: Y

text: X

text: Take

text: K

text: Z

text: to

text: be

text: the

text: cycle

text: Γ

text: (z,

text: w(z))

text: X

text: X.

text: Let

text: z

text: z

text: denote

text: the

text: set

text: of

text: K

text: w

text: n

text: points

text: of

text: X

text: with

text: y-coordinate

text: zero,

text: and

text: let

text: W

text: denote

text: the

text: divisor

text: z

text: Let

text: b

text: and

text: i

text: i

text: z

text: be

text: points

text: of

text: Y

text: with

text: nonzero

text: y-coordinate.

text: Then

text: i

text: Γ

text: w(b)

text: w

text: i

text: Γ

text: w(z)

text: w

text: i

text: Γ

text: W

text: w

text: hence

text: D(b,

text: z)

text: W

text: w(b)

text: w(z)

text: So

text: the

text: class

text: of

text: A(Y

text: )(b,

text: z)

text: is

text: dual

text: to

text: H

text: (z

text: b,

text: W

text: w(b)

text: w(z)

text: by

text: Theorem

text: The

text: following

text: lemma

text: X

text: illustrates

text: how

text: Theorem

text: may

text: be

text: deduced

text: from

text: Theorem

text: together

text: with

text: the

text: aﬃne

text: version

text: of

text: Theorem

text: Lemma

text: For

text: any

text: prime

text: v,

text: h

text: (z

text: b,

text: D(b,

text: z))

text: h

text: (z

text: h

text: (b

text: v

text: v

text: v

text: Proof.

text: First,

text: note

text: that

text: additivity

text: yields

text: h

text: (z

text: b,

text: D(b,

text: z))

text: h

text: (z

text: b,

text: W

text: h

text: (z

text: b,

text: w(z)

text: w(b)).

text: v

text: v

text: v

text: Since

text: W

text: div(y),

text: the

text: ﬁrst

text: term

text: is

text: equal

text: to

text: χ(y(z))

text: χ(y(b)).

text: For

text: the

text: second

text: term,

text: since

text: z

text: b

text: and

text: w(z)

text: w(b)

text: are

text: disjoint,

text: h

text: (z

text: b,

text: w(z)

text: w(b))

text: h

text: (z

text: b,

text: w(z)

text: w(b))+

text: h

text: w(z)

text: w(b),

text: z

text: b).

text: v

text: v

text: v

text: By

text: additivity

text: h

text: (z

text: b,

text: w(z)

text: w(b))

text: h

text: (z

text: w(z))

text: h

text: (z

text: w(b))

text: v

text: v

text: v

text: h

text: b,

text: w(z))

text: h

text: b,

text: w(b))

text: v

text: v

text: and

text: similarly

text: for

text: h

text: w(z)

text: w(b),

text: z

text: b).

text: Using

text: the

text: fact

text: that

text: h

text: (D

text: D

text: v

text: v

text: h

text: (w(D

text: w(D

text: this

text: gives

text: v

text: h

text: (z

text: b,

text: w(z)

text: w(b))

text: h

text: (z

text: w(z))

text: h

text: b,

text: w(b)).

text: v

text: v

text: v

text: The

text: result

text: now

text: follows

text: from

text: Lemma

text: In

text: this

text: subsection

text: we

text: return

text: to

text: the

text: case

text: where

text: X

text: is

text: a

text: genus

text: curve

text: of

text: the

text: form

text: c,

text: and

text: explain

text: how

text: to

text: deduce

text: Theorem

text: from

text: Theorem

text: Let

text: h

text: and

text: h

text: denote

text: v

text: (local

text: and

text: global,

text: resp.)

text: heights

text: on

text: X,

text: h

text: and

text: h

text: heights

text: on

text: E

text: and

text: h

text: and

text: h

text: heights

text: on

text: E

text: Recall

text: from

text: the

text: e

text: associated

text: E

text: f

text: X

text: E

text: Deﬁne

text: Z

text: X

text: X

text: to

text: be

text: the

text: graph

text: of

text: the

text: i

text: i

text: i

text: automorphism

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##