text: to instabilities. Note that σ(t), or equivalently the local inverse temperature β(t), is not a physical observable. It is a degree of freedom we use to parameterize non-equilibrium dy- namics. Thus exponentially behavior in σ does not have to imply instabilities. For example, despite the exponentially growing behavior in β, the energy H is always constant. In other words, the exponential behavior (3.3) carries no energy. Whether the exponential behavior shows up in other observables is a much trickier issue. In the context of Schwarzian for SYK and AdS , the SL(2, R) symmetry is a global gauge symmetry, which ensures that the 2 behavior (3.3) does not lead to any instability in any physical observables. Here we face similar issues; the exponentially growing behavior (3.3) should not lead to any instabilities in physical observables, but should lead to the exponential behavior of OTOCs (1.3) or com- mutators like (1.1). These are nontrivial requirements, which we will discuss in detail in Sec. . B. Point-wise shift symmetry We now generalize the above discussion to systems with spatial dependence. The general- ization is not unique. We ﬁrst discuss a simplest possibility and then consider more general cases. L As a simplest generalization of (3.2), we require the Lagrangian to be invariant hydro under an arbitrary spatial dependent shift symmetry → u(t, xi) u(t, xi) + a(xi) . (3.10) The physical interpretation of (3.10) is simple: each ﬂuid element has a separate shift sym- metry. For such a Lagrangian it then follows that β = β + a(xi)β eλt (3.11) 0 0 is a solution to equation of motion for arbitrary a(xi). Note that this statement holds at full nonlinear level. Invariance of (2.24) under (3.10) implies the invariance of H and G i under (3.11), and thus the exponentially increasing behavior is invisible to both the energy density and energy ﬂux. 21

text: to

text: instabilities.

text: Note

text: that

text: σ(t),

text: or

text: equivalently

text: the

text: local

text: inverse

text: temperature

text: β(t),

text: is

text: not

text: a

text: physical

text: observable.

text: It

text: is

text: a

text: degree

text: of

text: freedom

text: we

text: use

text: to

text: parameterize

text: non-equilibrium

text: dy-

text: namics.

text: Thus

text: exponentially

text: behavior

text: in

text: σ

text: does

text: not

text: have

text: to

text: imply

text: instabilities.

text: For

text: example,

text: despite

text: the

text: exponentially

text: growing

text: behavior

text: in

text: β,

text: the

text: energy

text: H

text: is

text: always

text: constant.

text: In

text: other

text: words,

text: the

text: exponential

text: behavior

text: carries

text: no

text: energy.

text: Whether

text: the

text: exponential

text: behavior

text: shows

text: up

text: in

text: other

text: observables

text: is

text: a

text: much

text: trickier

text: issue.

text: In

text: the

text: context

text: of

text: Schwarzian

text: for

text: SYK

text: and

text: AdS

text: the

text: R)

text: symmetry

text: is

text: a

text: global

text: gauge

text: symmetry,

text: which

text: ensures

text: that

text: the

text: behavior

text: does

text: not

text: lead

text: to

text: any

text: instability

text: in

text: any

text: physical

text: observables.

text: Here

text: we

text: face

text: similar

text: issues;

text: the

text: exponentially

text: growing

text: behavior

text: should

text: not

text: lead

text: to

text: any

text: instabilities

text: in

text: physical

text: observables,

text: but

text: should

text: lead

text: to

text: the

text: exponential

text: behavior

text: of

text: OTOCs

text: or

text: com-

text: mutators

text: like

text: These

text: are

text: nontrivial

text: requirements,

text: which

text: we

text: will

text: discuss

text: in

text: detail

text: in

text: Sec.

text: B.

text: Point-wise

text: shift

text: symmetry

text: We

text: now

text: generalize

text: the

text: above

text: discussion

text: to

text: systems

text: with

text: spatial

text: dependence.

text: The

text: general-

text: ization

text: is

text: not

text: unique.

text: We

text: ﬁrst

text: discuss

text: a

text: simplest

text: possibility

text: and

text: then

text: consider

text: more

text: general

text: cases.

text: L

text: As

text: a

text: simplest

text: generalization

text: of

text: we

text: require

text: the

text: Lagrangian

text: to

text: be

text: invariant

text: hydro

text: under

text: an

text: arbitrary

text: spatial

text: dependent

text: shift

text: symmetry

text: u(t,

text: xi)

text: u(t,

text: xi)

text: a(xi)

text: The

text: physical

text: interpretation

text: of

text: is

text: simple:

text: each

text: ﬂuid

text: element

text: has

text: a

text: separate

text: shift

text: sym-

text: metry.

text: For

text: such

text: a

text: Lagrangian

text: it

text: then

text: follows

text: that

text: β

text: β

text: a(xi)β

text: eλt

text: is

text: a

text: solution

text: to

text: equation

text: of

text: motion

text: for

text: arbitrary

text: a(xi).

text: Note

text: that

text: this

text: statement

text: holds

text: at

text: full

text: nonlinear

text: level.

text: Invariance

text: of

text: under

text: implies

text: the

text: invariance

text: of

text: H

text: and

text: G

text: i

text: under

text: and

text: thus

text: the

text: exponentially

text: increasing

text: behavior

text: is

text: invisible

text: to

text: both

text: the

text: energy

text: density

text: and

text: energy

text: ﬂux.


text: where − − (cid:15)˜(t ; λ ) = c (λ )eλ1t3, (cid:15)˜(t ; λ ) = c (λ )eλ1t4, c (λ ) = cL(1)[g e λ1t1]+cL(1)[g e λ1t2] . 3 1 12 1 4 1 12 1 12 1 t1 V t2 V (6.24) After imposing (6.12) we ﬁnd that there will be an exponentially growing contribution in (6.23) unless c (λ) = 0. However such a condition is precisely satisﬁed by the cou- 12 plings (6.13) and so the exponential modes again do not appear in the TOC. Since the shift symmetry of the coupling ensures the exponential modes do not appear in the TOC then they will appear instead from (6.11) in the OTOC. We then ﬁnd the contribution of the exponential mode to the OTOC is given by − − ic · · · H 1 = L(1)L(1)[g (t )g (t )eλt42] + . (6.25) 4 g g t2 t4 V 12 W 34 V W In the case of the SYK model in the infra-red limit we know the exact couplings and (6.13) is satisﬁed as a result of the SL(2, R) symmetry of generic correlation functions. To compare with the results of [15], let us consider the full G (not just the exponential parts) for the 4 example of (3.9) at maximal chaos. Using the explicit expression (C8) for G obtained in + Appendix , we ﬁnd that (in the equations below we have set a = 1, β = 2π and λ = 1) 2 0 max (cid:20)(cid:18) − (cid:19) (cid:18) (cid:19)(cid:21) − ∆ ∆ − t 2πi − t V W 12 34 G 1 = 2 + 2 + (6.26) 4 2π tanh t12 tanh t34 2 2 which agrees fully with Lorenzian continuation of (4.30) of [15].16 Similarly, for OTOC we ﬁnd − − − −i∆V ∆W e12(t3+t4 t1 t2) · · · H 1 = + (6.27) 4 sinh t12 sinh t34 2 2 which matches the analytic continuation of (4.32) of [15]. The above discussion can be readily generalized to higher dimensions to include spatial dependence in each operator insertion, e.g. H is replaced by 4 − 16 The additional 2πi term compared with (4.30) of [15] is due to the fact we are considering a slightly diﬀerent ordering. Here we consider V WWV while there V V WW is computed: the two orderings can be → related by taking t t + iβ . 1 1 0 37 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

label: paragraph

text: λ

text: c

text: (λ

text: λ

text: c

text: (λ

text: c

text: (λ

text: e

text: e

text: V

text: V

text: After

text: imposing

text: we

text: ﬁnd

text: that

text: there

text: will

text: be

text: an

text: exponentially

text: growing

text: contribution

text: in

text: unless

text: c

text: (λ)

text: However

text: such

text: a

text: condition

text: is

text: precisely

text: satisﬁed

text: by

text: the

text: cou-

text: plings

text: and

text: so

text: the

text: exponential

text: modes

text: again

text: do

text: not

text: appear

text: in

text: the

text: TOC.

text: Since

text: the

text: shift

text: symmetry

text: of

text: the

text: coupling

text: ensures

text: the

text: exponential

text: modes

text: do

text: not

text: appear

text: in

text: the

text: TOC

text: then

text: they

text: will

text: appear

text: instead

text: from

text: in

text: the

text: OTOC.

text: We

text: then

text: ﬁnd

text: the

text: contribution

text: of

text: the

text: exponential

text: mode

text: to

text: the

text: OTOC

text: is

text: given

text: by

text: ic

text: H

text: (t

text: )g

text: (t

text: g

text: g

text: V

text: W

text: V

text: W

text: In

text: the

text: case

text: of

text: the

text: SYK

text: model

text: in

text: the

text: infra-red

text: limit

text: we

text: know

text: the

text: exact

text: couplings

text: and

text: is

text: satisﬁed

text: as

text: a

text: result

text: of

text: the

text: R)

text: symmetry

text: of

text: generic

text: correlation

text: functions.

text: To

text: compare

text: with

text: the

text: results

text: of

text: let

text: us

text: consider

text: the

text: full

text: G

text: (not

text: just

text: the

text: exponential

text: parts)

text: for

text: the

text: example

text: of

text: at

text: maximal

text: chaos.

text: Using

text: the

text: explicit

text: expression

text: for

text: G

text: obtained

text: in

text: Appendix

text: we

text: ﬁnd

text: that

text: (in

text: the

text: equations

text: below

text: we

text: have

text: set

text: a

text: β

text: 2π

text: and

text: λ

text: max

text: t

text: t

text: V

text: W

text: G

text: 2π

text: tanh

text: tanh

text: which

text: agrees

text: fully

text: with

text: Lorenzian

text: continuation

text: of

text: of

text: Similarly,

text: for

text: OTOC

text: we

text: ﬁnd

text: −i∆V

text: ∆W

text: H

text: sinh

text: sinh

text: which

text: matches

text: the

text: analytic

text: continuation

text: of

text: of

text: The

text: above

text: discussion

text: can

text: be

text: readily

text: generalized

text: to

text: higher

text: dimensions

text: to

text: include

text: spatial

text: dependence

text: in

text: each

text: operator

text: insertion,

text: e.g.

text: H

text: is

text: replaced

text: by

text: The

text: additional

text: term

text: compared

text: with

text: of

text: is

text: due

text: to

text: the

text: fact

text: we

text: are

text: considering

text: a

text: slightly

text: diﬀerent

text: ordering.

text: Here

text: we

text: consider

text: V

text: WWV

text: while

text: there

text: V

text: V

text: WW

text: is

text: computed:

text: the

text: two

text: orderings

text: can

text: be

text: related

text: by

text: taking

text: t

text: t

text: iβ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:104)P (cid:105) H (t , x , t , x , t , x , t , x ) = V (t , x )V (t , x )W (t , x )W (t , x ) 4 1 1 2 2 3 3 4 4 1 1 1 2 2 2 1 3 3 3 4 4 (cid:104) (cid:105) = W (t , x )V (t , x )W (t , x )V (t , x ) . (6.28) 4 4 2 2 3 3 1 1 For example, for point-wise shift symmetry case the discussion is completely parallel to (cid:28) (cid:28) the above, and one ﬁnds that for t , t t , x , x x 12 34 42 12 34 42 − ∼ −| | H 1 eλ(t42 x42 /vB) (6.29) 4 while the exponentials again cancel for G provided that the eﬀective vertex analogous to 4 → (6.7) respects the shift symmetry (cid:15) (cid:15) + c(x)eλt. i i VII. DISCUSSIONS AND CONCLUSIONS Let us ﬁrst summarize our main ﬁndings: with a shift symmetry, the hydrodynamic theory has a mode which grows exponentially in time, and exhibits ballistic spreading with a butterﬂy velocity v . As a result such behavior appears in OTOCs, leading to (1.5), (1.7)– B (1.8), while the shift symmetry prevents correlation functions of the energy density and ﬂux, and TOCs of generic operators from having such behavior. The theory also predicts that correlation functions of energy density and ﬂux exhibit the phenomenon of pole skipping. This phenomenon provides a simpler way than OTOCs to extract the Lyapunov exponent λ and butterﬂy velocity v . It is of interest to study this pole-skipping phenomenon more B generally in holographic systems. This direct connection between chaos and the energy density two point function also provides a new perspective on observations that in many microscopic examples the chaos diﬀusion constants D , D are comparable to the energy c 0 diﬀusion constant D [18, 19, 21, 25–29]. E While in this paper we considered a theory with only energy conservation we expect the discussion can be straightforwardly generalized to systems with full energy-momentum conservation, with other conserved quantities, or with additional light modes, since the chaos mode is associated with energy conservation. For example, as mentioned earlier, strong 38

label: equation

text: H

text: (t

text: x

text: t

text: x

text: t

text: x

text: t

text: x

text: V

text: (t

text: x

text: )V

text: (t

text: x

text: )W

text: (t

text: x

text: )W

text: (t

text: x

text: W

text: (t

text: x

text: )V

text: (t

text: x

text: )W

text: (t

text: x

text: )V

text: (t

text: x

text: For

text: example,

text: for

text: point-wise

text: shift

text: symmetry

text: case

text: the

text: discussion

text: is

text: completely

text: parallel

text: to

text: the

text: above,

text: and

text: one

text: ﬁnds

text: that

text: for

text: t

text: t

text: t

text: x

text: x

text: x

text: H

text: /vB)

text: while

text: the

text: exponentials

text: again

text: cancel

text: for

text: G

text: provided

text: that

text: the

text: eﬀective

text: vertex

text: analogous

text: to

text: respects

text: the

text: shift

text: symmetry

text: c(x)eλt.

text: i

text: i

text: VII.

text: DISCUSSIONS

text: AND

text: CONCLUSIONS

text: Let

text: us

text: ﬁrst

text: summarize

text: our

text: main

text: ﬁndings:

text: with

text: a

text: shift

text: symmetry,

text: the

text: hydrodynamic

text: theory

text: has

text: a

text: mode

text: which

text: grows

text: exponentially

text: in

text: time,

text: and

text: exhibits

text: ballistic

text: spreading

text: with

text: a

text: butterﬂy

text: velocity

text: v

text: As

text: a

text: result

text: such

text: behavior

text: appears

text: in

text: OTOCs,

text: leading

text: to

text: B

text: while

text: the

text: shift

text: symmetry

text: prevents

text: correlation

text: functions

text: of

text: the

text: energy

text: density

text: and

text: ﬂux,

text: and

text: TOCs

text: of

text: generic

text: operators

text: from

text: having

text: such

text: behavior.

text: The

text: theory

text: also

text: predicts

text: that

text: correlation

text: functions

text: of

text: energy

text: density

text: and

text: ﬂux

text: exhibit

text: the

text: phenomenon

text: of

text: pole

text: skipping.

text: This

text: phenomenon

text: provides

text: a

text: simpler

text: way

text: than

text: OTOCs

text: to

text: extract

text: the

text: Lyapunov

text: exponent

text: λ

text: and

text: butterﬂy

text: velocity

text: v

text: It

text: is

text: of

text: interest

text: to

text: study

text: this

text: pole-skipping

text: phenomenon

text: more

text: B

text: generally

text: in

text: holographic

text: systems.

text: This

text: direct

text: connection

text: between

text: chaos

text: and

text: the

text: energy

text: density

text: two

text: point

text: function

text: also

text: provides

text: a

text: new

text: perspective

text: on

text: observations

text: that

text: in

text: many

text: microscopic

text: examples

text: the

text: chaos

text: diﬀusion

text: constants

text: D

text: D

text: are

text: comparable

text: to

text: the

text: energy

text: c

text: diﬀusion

text: constant

text: D

text: E

text: While

text: in

text: this

text: paper

text: we

text: considered

text: a

text: theory

text: with

text: only

text: energy

text: conservation

text: we

text: expect

text: the

text: discussion

text: can

text: be

text: straightforwardly

text: generalized

text: to

text: systems

text: with

text: full

text: energy-momentum

text: conservation,

text: with

text: other

text: conserved

text: quantities,

text: or

text: with

text: additional

text: light

text: modes,

text: since

text: the

text: chaos

text: mode

text: is

text: associated

text: with

text: energy

text: conservation.

text: For

text: example,

text: as

text: mentioned

text: earlier,

text: strong


text: FIG. 2. We propose that operator growth of V (t) can be described in terms of a bare operator Vˆ (t) surrounded by an expanding hydrodynamic cloud. eralize to C(t, (cid:126)x) ≡ −(cid:10)[V (t, (cid:126)x), W (0)]2(cid:11) ∼ N1 eλ(t−v|(cid:126)xB| ) (1.7) where v in (1.7) is often referred to as the butterﬂy velocity which characterizes the scram- B bling/expansion of an operator in space. Equation (1.7) is observed in a SYK chain and essentially all holographic systems [8, 9, 18]. In other systems diﬀerent forms of spatial propagation are often seen instead [19–21]. For instance in [20, 21] chaos is described by a diﬀusive spreading around the exponential growth2 C(t, (cid:126)x) ∼ N1 eλt−D|(cid:126)x0|2t . (1.8) ∼ We will be interested in those systems for which λ 1, so that the chaotic behavior is (cid:126) intrinsically quantum, i.e. does not have a straightforward semi-classical limit. The behaviors (1.5)–(1.8) have been found in many model systems, often through com- plicated model speciﬁc calculations. A uniﬁed understanding of how they emerge across diﬀerent systems is still lacking. It is the purpose of this paper to propose such an de- scription using “quantum hydrodynamics” whose precise meaning will be elaborated more below. N → ∞ We propose that to leading order in the limit , the scrambling of a generic few- body operator in a chaotic system allows a coarse-grained description in which the growth 2 In [9, 18], the behavior below has been observed as transient behavior to (1.7). 5 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: FIG.

text: We

label: caption

text: propose

text: that

text: operator

text: growth

text: of

text: V

text: (t)

text: can

text: be

text: described

text: in

text: terms

text: of

text: a

text: bare

text: operator

text: Vˆ

text: (t)

text: surrounded

text: by

text: an

text: expanding

text: hydrodynamic

text: cloud.

text: eralize

text: to

text: C(t,

text: (t,

text: W

text: where

text: v

text: in

text: is

text: often

text: referred

text: to

text: as

text: the

text: butterﬂy

text: velocity

text: which

text: characterizes

text: the

text: scram-

text: B

text: bling/expansion

text: of

text: an

text: operator

text: in

text: space.

text: Equation

text: is

text: observed

text: in

text: a

text: SYK

text: chain

text: and

text: essentially

text: all

text: holographic

text: systems

text: In

text: other

text: systems

text: diﬀerent

text: forms

text: of

text: spatial

text: propagation

text: are

text: often

text: seen

text: instead

text: For

text: instance

text: in

text: chaos

text: is

text: described

text: by

text: a

text: diﬀusive

text: spreading

text: around

text: the

text: exponential

text: C(t,

text: We

text: will

text: be

text: interested

text: in

text: those

text: systems

text: for

text: which

text: λ

text: so

text: that

text: the

text: chaotic

text: behavior

text: is

text: intrinsically

text: quantum,

text: i.e.

text: does

text: not

text: have

text: a

text: straightforward

text: semi-classical

text: limit.

text: The

text: behaviors

text: have

text: been

text: found

text: in

text: many

text: model

text: systems,

text: often

text: through

text: com-

text: plicated

text: model

text: speciﬁc

text: calculations.

text: A

text: uniﬁed

text: understanding

text: of

text: how

text: they

text: emerge

text: across

text: diﬀerent

text: systems

text: is

text: still

text: lacking.

text: It

text: is

text: the

text: purpose

text: of

text: this

text: paper

text: to

text: propose

text: such

text: an

text: de-

text: scription

text: using

text: “quantum

text: hydrodynamics”

text: whose

text: precise

text: meaning

text: will

text: be

text: elaborated

text: more

text: below.

text: N

text: We

text: propose

text: that

text: to

text: leading

text: order

text: in

text: the

text: limit

text: the

text: scrambling

text: of

text: a

text: generic

text: few-

text: body

text: operator

text: in

text: a

text: chaotic

text: system

text: allows

text: a

text: coarse-grained

text: description

text: in

text: which

text: the

text: growth

text: In

text: the

text: behavior

text: below

text: has

text: been

text: observed

text: as

text: transient

text: behavior

text: to

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: and (cid:20) − (cid:21) − 1 t2 π2 − − G (t) = t sinh t + a + b cosh t + iπt iπ sinh t . (C8) + 2π 2 where a, b are some undetermined integration constants. Note that (C8) precisely agrees with the Lorentzian analytic continuation of (4.28) of [15]. [1] J. Deutsch, “Quantum statistical mechanics in a closed system,” Physical Review A 43, no. 4 (1991): 2046. [2] Srednicki, “Chaos and quantum thermalization,” Physical Review E 50, no. 2 (1994): 888. [3] M. Srednicki, “The approach to thermal equilibrium in quantized chaotic systems,” Journal of Physics A: Mathematical and General 32.7 (1999): 1163. [4] M. Rigol, V. Dunjko, and M. Olshanii, “Thermalization and its mechanism for generic isolated quantum systems,” Nature, (2008): 452(7189), 854-858, [arXiv:0708.1324]. [5] L. D’Alessio, Y. Kafri, A. Polkovnikov, M. Rigol, “From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics,” Adv. Phys. 65, 239 (2016), [arXiv:1509.06411]. [6] A. Dymarsky and H. Liu, “Canonical Universality,” arXiv:1702.07722 [cond-mat.stat-mech]. [7] S. H. Shenker and D. Stanford, “Black holes and the butterﬂy eﬀect,” JHEP 1403, 067 (2014) [arXiv:1306.0622 [hep-th]]. [8] D. A. Roberts, D. Stanford and L. Susskind, “Localized shocks,” JHEP 1503, 051 (2015) [arXiv:1409.8180 [hep-th]]. [9] S. H. Shenker and D. Stanford, “Stringy eﬀects in scrambling,” JHEP 1505, 132 (2015) [arXiv:1412.6087 [hep-th]]. [10] J. Maldacena, S. H. Shenker and D. Stanford, “A bound on chaos,” JHEP 1608, 106 (2016) [arXiv:1503.01409 [hep-th]]. [11] A. Kitaev, Talks at KITP, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev/, http://online.kitp.ucsb.edu/online/entangled15/kitaev2/. 45 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: and

text: π2

text: G

text: (t)

text: t

text: sinh

text: t

text: a

text: b

text: cosh

text: t

text: iπt

text: iπ

text: sinh

text: t

text: 2π

text: where

text: a,

text: b

text: are

text: some

text: undetermined

text: integration

text: constants.

text: Note

text: that

text: precisely

text: agrees

text: with

text: the

text: Lorentzian

text: analytic

text: continuation

text: of

text: of

label: reference

text: J.

text: Deutsch,

text: “Quantum

text: statistical

text: mechanics

text: in

text: a

text: closed

text: system,”

text: Physical

text: Review

text: A

text: no.

text: Srednicki,

text: “Chaos

text: and

text: quantum

text: thermalization,”

text: Physical

text: Review

text: E

text: no.

text: M.

text: Srednicki,

text: “The

text: approach

text: to

text: thermal

text: equilibrium

text: in

text: quantized

text: chaotic

text: systems,”

text: Journal

text: of

text: Physics

text: A:

text: Mathematical

text: and

text: General

text: M.

text: Rigol,

text: V.

text: Dunjko,

text: and

text: M.

text: Olshanii,

text: “Thermalization

text: and

text: its

text: mechanism

text: for

text: generic

text: isolated

text: quantum

text: systems,”

text: Nature,

text: L.

text: D’Alessio,

text: Y.

text: Kafri,

text: A.

text: Polkovnikov,

text: M.

text: Rigol,

text: “From

text: Quantum

text: Chaos

text: and

text: Eigenstate

text: Thermalization

text: to

text: Statistical

text: Mechanics

text: and

text: Thermodynamics,”

text: Adv.

text: Phys.

text: A.

text: Dymarsky

text: and

text: H.

text: Liu,

text: “Canonical

text: Universality,”

text: [cond-mat.stat-mech].

text: S.

text: H.

text: Shenker

text: and

text: D.

text: Stanford,

text: “Black

text: holes

text: and

text: the

text: butterﬂy

text: eﬀect,”

text: JHEP

text: [hep-th]].

text: D.

text: A.

text: Roberts,

text: D.

text: Stanford

text: and

text: L.

text: Susskind,

text: “Localized

text: shocks,”

text: JHEP

text: [hep-th]].

text: S.

text: H.

text: Shenker

text: and

text: D.

text: Stanford,

text: “Stringy

text: eﬀects

text: in

text: scrambling,”

text: JHEP

text: [hep-th]].

text: J.

text: Maldacena,

text: S.

text: H.

text: Shenker

text: and

text: D.

text: Stanford,

text: “A

text: bound

text: on

text: chaos,”

text: JHEP

text: [hep-th]].

text: A.

text: Kitaev,

text: Talks

text: at

text: KITP,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Prepared for submission to JHEP Numerical analysis of the unintegrated double gluon distribution Edgar Eliasa Krzysztof Golec-Biernatb,c Anna M. Sta´stoa aDepartment of Physics, The Pennsylvania State University, University Park, PA 16802, U.S.A. bInstitute of Nuclear Physics, Polish Academy of Sciences, 31-342 Cracow, Poland cFaculty of Mathematics and Natural Sciences, University of Rzesz´ow, 35-959 Rzesz´ow, Poland Abstract: We present detailed numerical analysis of the unintegrated double gluon distribution which includes the dependence on the transverse momenta of partons. The unintegrated double gluon distribution was obtained following the Kimber-Martin-Ryskin method as a convolution of the perturbative gluon splitting function with the collinear integrated double gluon distribution and the Sudakov form factors. We analyze the de- pendence on the transverse momenta, longitudinal momentum fractions and hard scales. We ﬁnd that the unintegrated gluon distribution factorizes into a product of two single unintegrated gluon distributions in the region of small values of x, provided the splitting contribution is included and the momentum sum rule is satisﬁed. Keywords: Quantum Chromodynamics, parton distributions, transverse momentum de- pendence, double parton scattering, evolution equations

text: Prepared

text: for

text: submission

text: to

text: JHEP

text: Numerical

label: title

text: analysis

text: of

text: the

text: unintegrated

text: double

text: gluon

text: distribution

text: Edgar

label: author

text: Eliasa

text: Krzysztof

text: Golec-Biernatb,c

text: Anna

text: M.

text: Sta´stoa

text: aDepartment

text: of

text: Physics,

text: The

text: Pennsylvania

text: State

text: University,

text: University

text: Park,

text: PA

text: U.S.A.

text: bInstitute

text: of

text: Nuclear

text: Physics,

text: Polish

text: Academy

text: of

text: Sciences,

text: Cracow,

text: Poland

text: cFaculty

text: of

text: Mathematics

text: and

text: Natural

text: Sciences,

text: University

text: of

text: Rzesz´ow,

text: Rzesz´ow,

text: Poland

text: Abstract:

text: We

label: abstract

text: present

text: detailed

text: numerical

text: analysis

text: of

text: the

text: unintegrated

text: double

text: gluon

text: distribution

text: which

text: includes

text: the

text: dependence

text: on

text: the

text: transverse

text: momenta

text: of

text: partons.

text: The

text: unintegrated

text: double

text: gluon

text: distribution

text: was

text: obtained

text: following

text: the

text: Kimber-Martin-Ryskin

text: method

text: as

text: a

text: convolution

text: of

text: the

text: perturbative

text: gluon

text: splitting

text: function

text: with

text: the

text: collinear

text: integrated

text: double

text: gluon

text: distribution

text: and

text: the

text: Sudakov

text: form

text: factors.

text: We

text: analyze

text: the

text: de-

text: pendence

text: on

text: the

text: transverse

text: momenta,

text: longitudinal

text: momentum

text: fractions

text: and

text: hard

text: scales.

text: We

text: ﬁnd

text: that

text: the

text: unintegrated

text: gluon

text: distribution

text: factorizes

text: into

text: a

text: product

text: of

text: two

text: single

text: unintegrated

text: gluon

text: distributions

text: in

text: the

text: region

text: of

text: small

text: values

text: of

text: x,

text: provided

text: the

text: splitting

text: contribution

text: is

text: included

text: and

text: the

text: momentum

text: sum

text: rule

text: is

text: satisﬁed.

text: Keywords:

text: Quantum

text: Chromodynamics,

text: parton

text: distributions,

text: transverse

text: momentum

text: de-

text: pendence,

text: double

text: parton

text: scattering,

text: evolution

text: equations


text: SO, Q2=10 GeV2, x =0.1, x =0.1 SO, Q2=100 GeV2, x =0.1, x =0.1 1 2 1 2 1.2 1 H H 0.9 1 NH NH 0.8 H+NH H+NH 0.8 PRODUCT 0.7 PRODUCT 0.6 0.6 0.5 0.4 0.4 0.3 0.2 0.2 0.1 0 0 1 10 1 10 102 k2 k2 T T SO, Q2=10 GeV2, x =0.4, x =0.001 SO, Q2=100 GeV2, x =0.4, x =0.001 1 2 1 2 0.5 0.7 H H 0.45 NH 0.6 NH 0.4 H+NH H+NH 0.35 PRODUCT 0.5 PRODUCT 0.3 0.4 0.25 0.3 0.2 0.15 0.2 0.1 0.1 0.05 0 0 1 10 1 10 102 k2 k2 T T Figure 10. As in Figure 9 but for large values of x or x when factorization does not hold. 1 2 Q2=100 GeV2, k2=10 GeV2 T - 0.5 1 - 1 1.2 0.8 - 1.5 1 0.8 - 2 0.6 0.6 - 2.5 0.4 0.4 - 3 0.2 0- 4 - 3.5 - 3 - lo2g.510-(x22)- 1.5 - 1 - 0.5 - 0.-51- 1l.o-5g21(-0x2).1-53- 3.5- 4 - 3- .54- 4 - 3.5 - 3 - 2.5 - 2 - 1.5 - 1 l-o0g.150(x1) 00.2 Figure 11. The ratio (5.2) as a function of the longitudinal momentum fractions (x , x ) for ﬁxed 1 2 values of transverse momenta, k2⊥ = k2⊥ = 10 GeV2, and Q2 = 100 GeV2, in the strong ordering 1 2 case. hard scale Q = Q = Q for ﬁxed values of (x , x ). We indeed observe that the sum of 1 2 1 2 the homogenous and non-homogenous terms nicely factorizes into a product of two single – 17 – f x x f x x gg 2 1 gg 2 1 oitar f x x f x x gg 2 1 gg 2 1 ) x( gol 2 01 ##LTFigure## ##LTFigure## ##LTFigure## ##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## 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text: obeys the momentum sum rule (3.1), see [64] for details. Note that, the initial double gluon distribution (3.3) is completely determined by the parameters of the single gluon distributions, Nk, αk, βk, known from the global ﬁts [65]. It is also worth emphasizing g g g that the distribution (3.3) is not a product of two single gluon distributions, even for small values of parton momentum fractions. The momentum sum rule is preserved by evolution equations for the double parton ≡ distribution [16, 17]. Thus, the rule (3.1) is independent of the hard scale Q Q = Q at 1 2 which the double gluon distribution is deﬁned. The distribution (3.3) will be used in the forthcoming discussion as the initial condition for the evolution equations at some initial scale Q and we shall refer to it as the GBLS3 input. For the comparison, we also use the 0 distribution proposed in [22] (referred to subsequently as the GS input), − − (1 x x )2 D (x , x , Q , Q ) = D (x , Q ) D (x , Q ) − 1 −2 . (3.4) gg 1 2 0 0 g 1 0 g 1 0 (1 x )2 (1 x )2 1 2 This form is approximately factorizable into product of two single PDFs for small val- ues of momentum fractions and includes the correlating factor which guarantees smooth → vanishing of the distribution when (x + x ) 1. This ansatz does not satisfy the momen- 1 2 tum sum rule (3.1), although we checked that the violation is relatively small for almost all values of x, that is below 5% percent for x < 0.5, and only becomes signiﬁcant, up to 30% level, for higher values of x. 4 Unintegrated double gluon distribution In this section we present the construction of the double gluon distribution which is unintegrated over the transverse momenta. The framework discussed in the current work, proposed recently in [54], allows to construct the unintegrated double parton distri- butions which additionally depend on parton transverse momenta, k ⊥ and k ⊥. In the 1 2 following analysis, we are only interested in the unintegrated double gluon distribution, f (x , x , k ⊥, k ⊥, Q , Q ), though similar investigations can be performed for the other gg 1 2 1 2 1 2 distributions1. In the proposed construction [54], which follows the original proposal of Kimber- Martin-Ryskin for the single PDFs [55, 56], the dependence on the transverse momenta is obtained from unfolding the last step in the DGLAP-like evolution equations (2.2) re- duced to the gluonic sector. The integrated double gluon distribution is the sum of two contributions described in the previous section and illustrated in Fig. 1, D = D(h)(x , x , Q , Q ) + D(nh)(x , x , Q , Q ) . (4.1) gg gg 1 2 1 2 gg 1 2 1 2 Similarly, the unintegrated double gluon distribution can also be written as a sum of the homogenous and non-homogeneous contributions, f = f(h)(x , x , k ⊥, k ⊥, Q , Q ) + f(nh)(x , x , k ⊥, k ⊥, Q , Q ) . (4.2) gg gg 1 2 1 2 1 2 gg 1 2 1 2 1 2 1 In this case though one would need to specify the appropriate initial conditions for the integrated DPDFs which satisfy the momentum and quark sum rules with known single PDFs, which is a much more diﬃcult task. – 6 – ##LTLine## ##LTLine##

text: obeys

text: the

text: momentum

text: sum

text: rule

text: see

text: for

text: details.

text: Note

text: that,

text: the

text: initial

text: double

text: gluon

text: distribution

text: is

text: completely

text: determined

text: by

text: the

text: parameters

text: of

text: the

text: single

text: gluon

text: distributions,

text: Nk,

text: αk,

text: βk,

text: known

text: from

text: the

text: global

text: ﬁts

text: It

text: is

text: also

text: worth

text: emphasizing

text: g

text: g

text: g

text: that

text: the

text: distribution

text: is

text: not

text: a

text: product

text: of

text: two

text: single

text: gluon

text: distributions,

text: even

text: for

text: small

text: values

text: of

text: parton

text: momentum

text: fractions.

text: The

text: momentum

text: sum

text: rule

text: is

text: preserved

text: by

text: evolution

text: equations

text: for

text: the

text: double

text: parton

text: distribution

text: Thus,

text: the

text: rule

text: is

text: independent

text: of

text: the

text: hard

text: scale

text: Q

text: Q

text: Q

text: at

text: which

text: the

text: double

text: gluon

text: distribution

text: is

text: deﬁned.

text: The

text: distribution

text: will

text: be

text: used

text: in

text: the

text: forthcoming

text: discussion

text: as

text: the

text: initial

text: condition

text: for

text: the

text: evolution

text: equations

text: at

text: some

text: initial

text: scale

text: Q

text: and

text: we

text: shall

text: refer

text: to

text: it

text: as

text: the

text: input.

text: For

text: the

text: comparison,

text: we

text: also

text: use

text: the

text: distribution

text: proposed

text: in

text: (referred

text: to

text: subsequently

text: as

text: the

text: GS

text: input),

text: x

text: x

text: D

text: (x

text: x

text: Q

text: Q

text: D

text: (x

text: Q

text: D

text: (x

text: Q

text: −2

text: gg

text: g

text: g

text: x

text: x

text: This

text: form

text: is

text: approximately

text: factorizable

text: into

text: product

text: of

text: two

text: single

text: PDFs

text: for

text: small

text: val-

text: ues

text: of

text: momentum

text: fractions

text: and

text: includes

text: the

text: correlating

text: factor

text: which

text: guarantees

text: smooth

text: vanishing

text: of

text: the

text: distribution

text: when

text: (x

text: x

text: This

text: ansatz

text: does

text: not

text: satisfy

text: the

text: momen-

text: tum

text: sum

text: rule

text: although

text: we

text: checked

text: that

text: the

text: violation

text: is

text: relatively

text: small

text: for

text: almost

text: all

text: values

text: of

text: x,

text: that

text: is

text: below

text: percent

text: for

text: x

text: and

text: only

text: becomes

text: signiﬁcant,

text: up

text: to

text: level,

text: for

text: higher

text: values

text: of

text: x.

text: Unintegrated

text: double

text: gluon

text: distribution

text: In

text: this

text: section

text: we

text: present

text: the

text: construction

text: of

text: the

text: double

text: gluon

text: distribution

text: which

text: is

text: unintegrated

text: over

text: the

text: transverse

text: momenta.

text: The

text: framework

text: discussed

text: in

text: the

text: current

text: work,

text: proposed

text: recently

text: in

text: allows

text: to

text: construct

text: the

text: unintegrated

text: double

text: parton

text: distri-

text: butions

text: which

text: additionally

text: depend

text: on

text: parton

text: transverse

text: momenta,

text: k

text: and

text: k

text: In

text: the

text: following

text: analysis,

text: we

text: are

text: only

text: interested

text: in

text: the

text: unintegrated

text: double

text: gluon

text: distribution,

text: f

text: (x

text: x

text: k

text: k

text: Q

text: Q

text: though

text: similar

text: investigations

text: can

text: be

text: performed

text: for

text: the

text: other

text: gg

text: In

text: the

text: proposed

text: construction

text: which

text: follows

text: the

text: original

text: proposal

text: of

text: Kimber-

text: Martin-Ryskin

text: for

text: the

text: single

text: PDFs

text: the

text: dependence

text: on

text: the

text: transverse

text: momenta

text: is

text: obtained

text: from

text: unfolding

text: the

text: last

text: step

text: in

text: the

text: DGLAP-like

text: evolution

text: equations

text: re-

text: duced

text: to

text: the

text: gluonic

text: sector.

text: The

text: integrated

text: double

text: gluon

text: distribution

text: is

text: the

text: sum

text: of

text: two

text: contributions

text: described

text: in

text: the

text: previous

text: section

text: and

text: illustrated

text: in

text: Fig.

text: D

text: D(h)(x

text: x

text: Q

text: Q

text: D(nh)(x

text: x

text: Q

text: Q

text: gg

text: gg

text: gg

text: Similarly,

text: the

text: unintegrated

text: double

text: gluon

text: distribution

text: can

text: also

text: be

text: written

text: as

text: a

text: sum

text: of

text: the

text: homogenous

text: and

text: non-homogeneous

text: contributions,

text: f

text: f(h)(x

text: x

text: k

text: k

text: Q

text: Q

text: f(nh)(x

text: x

text: k

text: k

text: Q

text: Q

text: gg

text: gg

text: gg

text: In

text: this

text: case

text: though

text: one

text: would

text: need

text: to

text: specify

text: the

text: appropriate

text: initial

text: conditions

text: for

text: the

text: integrated

text: DPDFs

text: which

text: satisfy

text: the

text: momentum

text: and

text: quark

text: sum

text: rules

text: with

text: known

text: single

text: PDFs,

text: which

text: is

text: a

text: much

text: more

text: diﬃcult

text: task.

text: ##LTLine##

text: ##LTLine##


text: 16 ROBERT AUFFARTH AND GIANCARLO LUCCHINI ARTECHE an isomorphism. By inspection, we see that F = E [2] and we can thus σ,B σ,B apply Proposition 3.4, which tells us that elements in ∆ are invariant by every transposition, hence of diagonal type. Assume now that ∆ contains an element t¯ of hyperplanar type. Then, since − ∈ ∆ is G-stable, we have that, for ρ = (ζ , 1, ζ 1, 1, . . . , 1) H, 1 m m − − ∈ (1 ρ )(t¯) = ((1 ζ )t, 0, . . . , 0) ∆. 1 m But, by construction, there are no elements of the form (x, 0, . . . , 0) in ∆. We deduce then that t is ζ -invariant. The assertion on the torsion of its m coordinates follows immediately. ∈ ∈ Assume ﬁnally that m = 6 and let (t , . . . , t ) ∆. Deﬁne σ = (1 i) ⊂ − ∈ ⊂1 n i S G and ρ = (ζ 1, ζ , 1, . . . , 1) H G. Then n 2 6 6 − − ∈ [(1 ρ )(1 ρ )σ ](t¯) = (t , 0, . . . , 0) ∆, 2 1 i i which implies as above that t = 0 and thus ∆ = 0. (cid:94)¨ i (cid:111) Let us study now pseudoreﬂections in ∆ G. Deﬁne the elements − ∈ ⊂ ρ := (ζ , ζ 1, 1, . . . , 1) H G; m m ∈ ⊂ σ := (1 2) S G; n ∈ ⊂ τ := (ζp , 1, . . . , 1) H G. m Then there are two types of pseudoreﬂections in G: ≤ (I) conjugates of ρaσ for 0 a < p; (II) conjugates of powers of τ (these do not exist if m = p); and the corresponding elliptic curves in B are respectively: { − | ∈ } E = (x, ζa x, 0, . . . , 0) x E ; ρaσ m { | ∈ } E = (x, 0, 0, . . . , 0) x E . τ Note now that elements of the form (x, 0, . . . , 0) are not in ∆ by construction → of the isogeny π : B A. Using Lemmas 3.5 and 4.6, we see then that (cid:111) pseudoreﬂections in ∆ G that are not in G must be of the form ∈ (cid:111) ≤ (III) conjugates of (t¯, ρaσ) ∆ G for 0 a < p; − ∈ where t¯ = (t, t, 0, . . . , 0) ∆ and t is ζ -invariant. m With these considerations, we can restrict further on the structure of ∆. For instance, diagonal elements in ∆ are bound to bring problems since they do ∈ not belong to any elliptic curve E for a pseudoreﬂection υ G. Thus, they υ (cid:111) cannot bring up new pseudoreﬂections in ∆ G unless they are generated by hyperplanar elements. This is explained by the following proposition. ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ROBERT

text: AUFFARTH

text: AND

text: GIANCARLO

text: LUCCHINI

text: ARTECHE

text: an

text: isomorphism.

text: By

text: inspection,

text: we

text: see

text: that

text: F

text: E

text: and

text: we

text: can

text: thus

text: σ,B

text: σ,B

text: apply

text: Proposition

text: which

text: tells

text: us

text: that

text: elements

text: in

text: are

text: invariant

text: by

text: every

text: transposition,

text: hence

text: of

text: diagonal

text: type.

text: Assume

text: now

text: that

text: contains

text: an

text: element

text: t¯

text: of

text: hyperplanar

text: type.

text: Then,

text: since

text: is

text: G-stable,

text: we

text: have

text: that,

text: for

text: ρ

text: (ζ

text: ζ

text: H,

text: m

text: m

text: ρ

text: )(t¯)

text: ζ

text: )t,

text: m

text: But,

text: by

text: construction,

text: there

text: are

text: no

text: elements

text: of

text: the

text: form

text: (x,

text: in

text: We

text: deduce

text: then

text: that

text: t

text: is

text: ζ

text: -invariant.

text: The

text: assertion

text: on

text: the

text: torsion

text: of

text: its

text: m

text: coordinates

text: follows

text: immediately.

text: Assume

text: ﬁnally

text: that

text: m

text: and

text: let

text: (t

text: t

text: Deﬁne

text: σ

text: i)

text: ⊂1

text: n

text: i

text: S

text: G

text: and

text: ρ

text: (ζ

text: ζ

text: H

text: G.

text: Then

text: n

text: ρ

text: ρ

text: )σ

text: ](t¯)

text: (t

text: i

text: i

text: which

text: implies

text: as

text: above

text: that

text: t

text: and

text: thus

text: i

text: Let

text: us

text: study

text: now

text: pseudoreﬂections

text: in

text: G.

text: Deﬁne

text: the

text: elements

text: ρ

text: (ζ

text: ζ

text: H

text: G;

text: m

text: m

text: σ

text: S

text: G;

text: n

text: τ

text: (ζp

text: H

text: G.

text: m

text: Then

text: there

text: are

text: two

text: types

text: of

text: pseudoreﬂections

text: in

text: G:

text: (I)

text: conjugates

text: of

text: ρaσ

text: for

text: a

text: p;

text: (II)

text: conjugates

text: of

text: powers

text: of

text: τ

text: (these

text: do

text: not

text: exist

text: if

text: m

text: p);

text: and

text: the

text: corresponding

text: elliptic

text: curves

text: in

text: B

text: are

text: respectively:

text: E

text: (x,

text: ζa

text: x,

text: x

text: E

text: ρaσ

text: m

text: E

text: (x,

text: x

text: E

text: τ

text: Note

text: now

text: that

text: elements

text: of

text: the

text: form

text: (x,

text: are

text: not

text: in

text: by

text: construction

text: of

text: the

text: isogeny

text: π

text: B

text: A.

text: Using

text: Lemmas

text: and

text: we

text: see

text: then

text: that

text: pseudoreﬂections

text: in

text: G

text: that

text: are

text: not

text: in

text: G

text: must

text: be

text: of

text: the

text: form

text: (III)

text: conjugates

text: of

text: (t¯,

text: ρaσ)

text: G

text: for

text: a

text: p;

text: where

text: t¯

text: (t,

text: t,

text: and

text: t

text: is

text: ζ

text: -invariant.

text: m

text: With

text: these

text: considerations,

text: we

text: can

text: restrict

text: further

text: on

text: the

text: structure

text: of

text: For

text: instance,

text: diagonal

text: elements

text: in

text: are

text: bound

text: to

text: bring

text: problems

text: since

text: they

text: do

text: not

text: belong

text: to

text: any

text: elliptic

text: curve

text: E

text: for

text: a

text: pseudoreﬂection

text: υ

text: G.

text: Thus,

text: they

text: υ

text: cannot

text: bring

text: up

text: new

text: pseudoreﬂections

text: in

text: G

text: unless

text: they

text: are

text: generated

text: by

text: hyperplanar

text: elements.

text: This

text: is

text: explained

text: by

text: the

text: following

text: proposition.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: SMOOTH QUOTIENTS OF ABELIAN VARIETIES BY FINITE GROUPS 17 ∈ Proposition 4.7 (∆ is not diagonal). Assume that there exists s E such ∈ − (cid:54)∈ that (s, . . . , s) ∆ but (s, s, 0, . . . , 0) ∆. Then A/G is not smooth. In particular, we see that ∆ has to contain at least one hyperplanar element. (cid:39) (cid:111) Proof. Since A/G B/(∆ G), we will work with this last quotient using the Chevalley-Shephard-Todd Theorem. ∈ Let s¯ ∆ denote the diagonal element in the statement of the Proposition. We will prove ﬁrst that an element of the form (s¯, υ) cannot be generated by (cid:111) pseudoreﬂections in ∆ G. Indeed, the only pseudoreﬂections that are not in G are those of type (III), so that if (s¯, υ) was generated by pseudoreﬂections, we should be able to write (cid:88) (cid:96) (2) s¯ = υ (t¯ ), i i i=1 − ∈ ∈ with t¯ = (t , t , 0 . . . , 0) ∆ a hyperplanar element and υ G. In i i i i particular, s¯ would be contained in the sub-G-module of ∆ generated by the t¯ . But since t is ζ -invariant, the only way in which G acts on the t¯ is i i m i by permuting their coordinates. Thus, by looking at the ﬁrst coordinate in equation (2), we get that s is a linear combination of the t , which implies − i immediately that (s, s, 0, . . . , 0) is a linear combination of the t¯ and hence i is in ∆, contradicting our hypothesis. ∈ Having proved this, it suﬃces then to exhibit an element x¯ B such that (cid:111) its stabilizer in ∆ G has an element of the form (s¯, υ). In other words, we ∈ ∈ need υ G and x¯ B such that υ(x¯)+s¯ = x¯, and this is a direct consequence of Lemma 3.8. (cid:94)¨ Denote by E the subgroup of ζ -invariant elements of E. Then E is equal (cid:39) Z 0Z m Z Z 0 to E[2] ( /2 )2 if m = 2, isomorphic to /3 if m = 3 and isomorphic to Z Z /2 if m = 4. Now that we know that diagonal elements in ∆ only appear if generated by h(cid:40)yperplanar elements, Lemma 4(cid:41).6 tells us that ∆ is contained (cid:88) in En = BH, and more precisely in the “hyperplane” 0 ∈ | n ⊂ ⊂ (x , . . . , x ) En x = 0 En B. 1 n 0 i 0 i=1 But for m = 3, 4 the mere presence of a hyperplanar element implies by G- stability that ∆ is actually the whole “hyperplane”. For m = 2, one hyper- planar element does not suﬃce to generate the whole hyperplanar subgroup of ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: SMOOTH

text: QUOTIENTS

text: OF

text: ABELIAN

text: VARIETIES

text: BY

text: FINITE

text: GROUPS

text: Proposition

text: is

text: not

text: diagonal).

text: Assume

text: that

text: there

text: exists

text: s

text: E

text: such

text: that

text: (s,

text: s)

text: but

text: (s,

text: s,

text: Then

text: A/G

text: is

text: not

text: smooth.

text: In

text: particular,

text: we

text: see

text: that

text: has

text: to

text: contain

text: at

text: least

text: one

text: hyperplanar

text: element.

text: Proof.

text: Since

text: A/G

text: B/(∆

text: G),

text: we

text: will

text: work

text: with

text: this

text: last

text: quotient

text: using

text: the

text: Chevalley-Shephard-Todd

text: Theorem.

text: Let

text: s¯

text: denote

text: the

text: diagonal

text: element

text: in

text: the

text: statement

text: of

text: the

text: Proposition.

text: We

text: will

text: prove

text: ﬁrst

text: that

text: an

text: element

text: of

text: the

text: form

text: (s¯,

text: υ)

text: cannot

text: be

text: generated

text: by

text: pseudoreﬂections

text: in

text: G.

text: Indeed,

text: the

text: only

text: pseudoreﬂections

text: that

text: are

text: not

text: in

text: G

text: are

text: those

text: of

text: type

text: (III),

text: so

text: that

text: if

text: (s¯,

text: υ)

text: was

text: generated

text: by

text: pseudoreﬂections,

text: we

text: should

text: be

text: able

text: to

text: write

text: s¯

text: υ

text: (t¯

text: i

text: i

text: with

text: t¯

text: (t

text: t

text: a

text: hyperplanar

text: element

text: and

text: υ

text: G.

text: In

text: i

text: i

text: i

text: i

text: particular,

text: s¯

text: would

text: be

text: contained

text: in

text: the

text: sub-G-module

text: of

text: generated

text: by

text: the

text: t¯

text: But

text: since

text: t

text: is

text: ζ

text: -invariant,

text: the

text: only

text: way

text: in

text: which

text: G

text: acts

text: on

text: the

text: t¯

text: is

text: i

text: i

text: m

text: i

text: by

text: permuting

text: their

text: coordinates.

text: Thus,

text: by

text: looking

text: at

text: the

text: ﬁrst

text: coordinate

text: in

text: equation

text: we

text: get

text: that

text: s

text: is

text: a

text: linear

text: combination

text: of

text: the

text: t

text: which

text: implies

text: i

text: immediately

text: that

text: (s,

text: s,

text: is

text: a

text: linear

text: combination

text: of

text: the

text: t¯

text: and

text: hence

text: i

text: is

text: in

text: contradicting

text: our

text: hypothesis.

text: Having

text: proved

text: this,

text: it

text: suﬃces

text: then

text: to

text: exhibit

text: an

text: element

text: x¯

text: B

text: such

text: that

text: its

text: stabilizer

text: in

text: G

text: has

text: an

text: element

text: of

text: the

text: form

text: (s¯,

text: υ).

text: In

text: other

text: words,

text: we

text: need

text: υ

text: G

text: and

text: x¯

text: B

text: such

text: that

text: υ(x¯)+s¯

text: x¯,

text: and

text: this

text: is

text: a

text: direct

text: consequence

text: of

text: Lemma

text: Denote

text: by

text: E

text: the

text: subgroup

text: of

text: ζ

text: -invariant

text: elements

text: of

text: E.

text: Then

text: E

text: is

text: equal

text: Z

text: m

text: Z

text: Z

text: to

text: if

text: m

text: isomorphic

text: to

text: if

text: m

text: and

text: isomorphic

text: to

text: Z

text: Z

text: if

text: m

text: Now

text: that

text: we

text: know

text: that

text: diagonal

text: elements

text: in

text: only

text: appear

text: if

text: generated

text: by

text: elements,

text: Lemma

text: tells

text: us

text: that

text: is

text: contained

text: in

text: En

text: BH,

text: and

text: more

text: precisely

text: in

text: the

text: “hyperplane”

text: n

text: (x

text: x

text: En

text: x

text: En

text: B.

text: n

text: i

text: But

text: for

text: m

text: the

text: mere

text: presence

text: of

text: a

text: hyperplanar

text: element

text: implies

text: by

text: G-

text: stability

text: that

text: is

text: actually

text: the

text: whole

text: “hyperplane”.

text: For

text: m

text: one

text: hyper-

text: planar

text: element

text: does

text: not

text: suﬃce

text: to

text: generate

text: the

text: whole

text: hyperplanar

text: subgroup

text: of

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 24 ROBERT AUFFARTH AND GIANCARLO LUCCHINI ARTECHE its stabilizer is not generated by pseudoreﬂections. The result follows then from the Chevalley-Shephard-Todd theorem. | | We start with the groups G such that det(S) = 1, so that we only need to verify Popov’s explicit lattices. For these, Table 1 gives: • The group G (by giving its number in Shephard-Todd’s list). • (cid:111) Popov’s name for the group Λ0 G. • Z { } A rational linear combination v of the -basis e , . . . , e of Λ = Λ0. • 0 1 2n The order of the stabilizer S = Stab (x ) of the image x of v in the C 0 G 0 0 0 abelian variety A = n/Λ. • The order of the subgroup P of S that is generated by pseudoreﬂec- 0 0 tions. § Z We refer to [Pop82, 2.6, pp. 37–44] for the explicit -basis. In each case, the ﬁrst n elements of the basis are Popov’s e , . . . , e and the (n + i)-th element ∈ C 1 n is τ e for some explicit τ . i i i (cid:111) ∈ ⊗ Q | | | | #G Λ0 G v Λ0 Z S P 0 0 0 5 [K ] (1, 1, 1, 0) 3 1 5 3 3 3 8 [K ] (1, 1, 1, 0) 3 1 8 3 3 3 12 [K ] (0, 0, 0, 1) 16 8 12 − − 2 − 24 [K ] (1, 1, 1, 1, 1, 1) 4 1 24 4 4 4 2 4 4 26 [K ] (1, 1, 1, 0, 1, 1) 36 18 26 1 2 2−2 2 2 26 [K ] (0, 0, 1, 0, 0, 1) 72 24 26 2 3 3 28 [F ]α (1, 1, 1, 1, 1, 0, 0, 0) 12 6 4 1 2 2 2 2 2 28 [F ]β (0, 1, 1, 0, 0, 0, 1, 0) 16 8 4 2 2 2 2 28 [F ]γ (0, 1, 0, 0, 0, 0, 1, 0) 16 8 4 3 2 2 29 [K ] (1, 0, 0, 0, 1, 0, 0, 0) 768 384 29 2 2 31 [K ] (1, 1, 0, 0, 0, 0, 0, 0) 384 192 31 2 2 32 [K ] (1, 0, 0, 0, 0, 0, 0, 0) 1296 648 32 2 − 34 [K ] (1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0) 155520 51840 34 3 3 37 [E ]α (1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1) 103680 51840 8 2 2 2 2 2 2 2 2 2 2 Table 1. Examples of non-smooth points in A/G for sporadic | | groups G such that det(S) = 1. ##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: ROBERT

text: AUFFARTH

text: AND

text: GIANCARLO

text: LUCCHINI

text: ARTECHE

text: its

text: stabilizer

text: is

text: not

text: generated

text: by

text: pseudoreﬂections.

text: The

text: result

text: follows

text: then

text: from

text: the

text: Chevalley-Shephard-Todd

text: theorem.

text: We

text: start

text: with

text: the

text: groups

text: G

text: such

text: that

text: det(S)

text: so

text: that

text: we

text: only

text: need

text: to

text: verify

text: Popov’s

text: explicit

text: lattices.

text: For

text: these,

text: Table

text: gives:

label: list

text: The

text: group

text: G

text: (by

text: giving

text: its

text: number

text: in

text: Shephard-Todd’s

text: list).

text: Popov’s

text: name

text: for

text: the

text: group

text: Λ0

text: G.

text: Z

text: A

text: rational

text: linear

text: combination

text: v

text: of

text: the

text: -basis

text: e

text: e

text: of

text: Λ

text: The

text: order

text: of

text: the

text: stabilizer

text: S

text: Stab

text: (x

text: of

text: the

text: image

text: x

text: of

text: v

text: in

text: the

text: C

text: G

text: abelian

text: variety

text: A

text: n/Λ.

text: The

text: order

text: of

text: the

text: subgroup

text: P

text: of

text: S

text: that

text: is

text: generated

text: by

text: pseudoreﬂec-

text: tions.

text: Z

text: We

text: refer

text: to

text: pp.

text: for

text: the

text: explicit

text: -basis.

text: In

text: each

text: case,

text: the

text: ﬁrst

text: n

text: elements

text: of

text: the

text: basis

text: are

text: Popov’s

text: e

text: e

text: and

text: the

text: (n

text: i)-th

text: element

text: C

text: n

text: is

text: τ

text: e

text: for

text: some

text: explicit

text: τ

text: i

text: i

text: i

text: Q

text: #G

text: Λ0

text: G

text: v

text: Λ0

text: Z

text: S

text: P

text: [K

text: [K

text: [K

text: [K

text: [K

text: [K

text: [F

text: ]α

text: [F

text: ]β

text: [F

text: ]γ

text: [K

text: [K

text: [K

text: [K

text: [E

text: ]α

text: Table

text: Examples

text: of

text: non-smooth

text: points

text: in

text: A/G

text: for

text: sporadic

text: groups

text: G

text: such

text: that

text: det(S)

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text: SMOOTH QUOTIENTS OF ABELIAN VARIETIES BY FINITE GROUPS 27 basic SageMath algorithms (we thank once again Antonio Behn for his enor- mous help in optimizing our ﬁrst algorithms). These are given in the appendix, but the reader is welcome to use his or her own! The main idea in order to ﬁnd these elements was to check the stabilizers (and the pseudoreﬂections therein) of small torsion elements chosen via the following principle: for every element g of the matrix group G, we decomposed Z − ⊕ − ⊥ ⊥ 2n as ker(g I ) ker(g I ) , where the is taken with respect to a G- 2n 2nC − ⊥ invariant Hermitian form H on n. By restricting g to ker(g I ) , we obtain − Q2n an integer-valued matrix g˜ such that g˜ I is invertible over . The columns − − of (g˜ I) 1 that contain rational, non-integer numbers therefore correspond to ﬁxed points of g in A that do not come from the eigenspace associated to 1 of g. These were the vectors whose stabilizers we calculated and analyzed. 5. Galois embeddings In [Yos07], Yoshihara introduces the notion of a Galois embedding of a smooth projective variety. If X is a smooth projective variety of dimension → P n and D is a very ample divisor that induces an embedding X (cid:44) N, then − − the embedding is said to be Galois if there exists an (N n 1)-dimensional P ∩ ∅ linear subspace W of N such that X W = and the restriction of the linear P (cid:57)(cid:57)(cid:75) P projection π : N n to X is Galois. W In [Yos07], Yoshihara speciﬁcally studies when abelian surfaces have a Ga- lois embedding. He gives a classiﬁcation of abelian surfaces having a Galois embedding, along with their Galois groups.2 Let A be an abelian variety and let G be a ﬁnite group acting on A. We have an exact sequence → → → → 0 G G H 1 0 where G is the set of translations that lie in G and H is the quotient. More- 0 over, H acts on B = A/G as automorphisms that ﬁx the origin and the 0 quotient of B by H is isomorphic to A/G. Yoshihara proved that in the case (cid:39) of an abelian surface, B E2 for some elliptic curve E. In [Auf17], the ﬁrst author shows that if A has a Galois embedding, then A is isogenous to the self product of an elliptic curve. Our results generalize Yoshihara’s results to arbitrary dimension: ≥ Theorem 5.1. If an abelian variety A of dimension n 3 possesses a Galois (cid:39) embedding with Galois group G, then A/G En for some elliptic curve E. 0 2Unfortunately, upon ﬁnishing this article, we realized that Yoshihara’s results are in- complete, so that a full classiﬁcation in dimension 2 is still not achieved, cf. Remark 4.1. ##LTLine##

text: SMOOTH

text: QUOTIENTS

text: OF

text: ABELIAN

text: VARIETIES

text: BY

text: FINITE

text: GROUPS

text: basic

text: SageMath

text: algorithms

text: (we

text: thank

text: once

text: again

text: Antonio

text: Behn

text: for

text: his

text: enor-

text: mous

text: help

text: in

text: optimizing

text: our

text: ﬁrst

text: algorithms).

text: These

text: are

text: given

text: in

text: the

text: appendix,

text: but

text: the

text: reader

text: is

text: welcome

text: to

text: use

text: his

text: or

text: her

text: own!

text: The

text: main

text: idea

text: in

text: order

text: to

text: ﬁnd

text: these

text: elements

text: was

text: to

text: check

text: the

text: stabilizers

text: (and

text: the

text: pseudoreﬂections

text: therein)

text: of

text: small

text: torsion

text: elements

text: chosen

text: via

text: the

text: following

text: principle:

text: for

text: every

text: element

text: g

text: of

text: the

text: matrix

text: group

text: G,

text: we

text: decomposed

text: Z

text: as

text: ker(g

text: I

text: ker(g

text: I

text: where

text: the

text: is

text: taken

text: with

text: respect

text: to

text: a

text: G-

text: invariant

text: Hermitian

text: form

text: H

text: on

text: n.

text: By

text: restricting

text: g

text: to

text: ker(g

text: I

text: we

text: obtain

text: an

text: integer-valued

text: matrix

text: g˜

text: such

text: that

text: g˜

text: I

text: is

text: invertible

text: over

text: The

text: columns

text: of

text: (g˜

text: I)

text: that

text: contain

text: rational,

text: non-integer

text: numbers

text: therefore

text: correspond

text: to

text: ﬁxed

text: points

text: of

text: g

text: in

text: A

text: that

text: do

text: not

text: come

text: from

text: the

text: eigenspace

text: associated

text: to

text: of

text: g.

text: These

text: were

text: the

text: vectors

text: whose

text: stabilizers

text: we

text: calculated

text: and

text: analyzed.

text: Galois

text: embeddings

text: In

text: Yoshihara

text: introduces

text: the

text: notion

text: of

text: a

text: Galois

text: embedding

text: of

text: a

text: smooth

text: projective

text: variety.

text: If

text: X

text: is

text: a

text: smooth

text: projective

text: variety

text: of

text: dimension

text: P

text: n

text: and

text: D

text: is

text: a

text: very

text: ample

text: divisor

text: that

text: induces

text: an

text: embedding

text: X

text: N,

text: then

text: the

text: embedding

text: is

text: said

text: to

text: be

text: Galois

text: if

text: there

text: exists

text: an

text: (N

text: n

text: P

text: linear

text: subspace

text: W

text: of

text: N

text: such

text: that

text: X

text: W

text: and

text: the

text: restriction

text: of

text: the

text: linear

text: P

text: P

text: projection

text: π

text: N

text: n

text: to

text: X

text: is

text: Galois.

text: W

text: In

text: Yoshihara

text: speciﬁcally

text: studies

text: when

text: abelian

text: surfaces

text: have

text: a

text: Ga-

text: lois

text: embedding.

text: He

text: gives

text: a

text: classiﬁcation

text: of

text: abelian

text: surfaces

text: having

text: a

text: Galois

text: embedding,

text: along

text: with

text: their

text: Galois

text: Let

text: A

text: be

text: an

text: abelian

text: variety

text: and

text: let

text: G

text: be

text: a

text: ﬁnite

text: group

text: acting

text: on

text: A.

text: We

text: have

text: an

text: exact

text: sequence

text: G

text: G

text: H

text: where

text: G

text: is

text: the

text: set

text: of

text: translations

text: that

text: lie

text: in

text: G

text: and

text: H

text: is

text: the

text: quotient.

text: More-

text: over,

text: H

text: acts

text: on

text: B

text: A/G

text: as

text: automorphisms

text: that

text: ﬁx

text: the

text: origin

text: and

text: the

text: quotient

text: of

text: B

text: by

text: H

text: is

text: isomorphic

text: to

text: A/G.

text: Yoshihara

text: proved

text: that

text: in

text: the

text: case

text: of

text: an

text: abelian

text: surface,

text: B

text: for

text: some

text: elliptic

text: curve

text: E.

text: In

text: the

text: ﬁrst

text: author

text: shows

text: that

text: if

text: A

text: has

text: a

text: Galois

text: embedding,

text: then

text: A

text: is

text: isogenous

text: to

text: the

text: self

text: product

text: of

text: an

text: elliptic

text: curve.

text: Our

text: results

text: generalize

text: Yoshihara’s

text: results

text: to

text: arbitrary

text: dimension:

text: Theorem

text: If

text: an

text: abelian

text: variety

text: A

text: of

text: dimension

text: n

text: possesses

text: a

text: Galois

text: embedding

text: with

text: Galois

text: group

text: G,

text: then

text: A/G

text: En

text: for

text: some

text: elliptic

text: curve

text: E.

text: upon

text: ﬁnishing

text: this

text: article,

text: we

text: realized

text: that

text: Yoshihara’s

text: results

text: are

text: in-

text: complete,

text: so

text: that

text: a

text: full

text: classiﬁcation

text: in

text: dimension

text: is

text: still

text: not

text: achieved,

text: cf.

text: Remark

text: ##LTLine##


text: 32 ROBERT AUFFARTH AND GIANCARLO LUCCHINI ARTECHE Theorem 6.5. If (A, Θ) is a principally polarized abelian variety with Θ irre- ducible, and G is a group consisting of automorphisms of A that preserve the numerical class of Θ and such that dim(AG) = 0, then A/G is singular. (cid:94)¨ References [Auf17] R. Auﬀarth. A note on Galois embeddings of abelian varieties. Manuscripta Math. 154(3-4), 2017, pp. 279–284. [ALR17] R. Auﬀarth, H. Lange, A. Rojas. A criterion for an abelian variety to be non- simple. J. Pure Appl. Algebra 221(8), 2017, pp. 1906–1925. [Bro90] S. A. Broughton. Classifying ﬁnite group actions on surfaces of low genus. J. Pure Appl. Algebra 69, 1990; pp. 233–270. [BS78] I. Bernˇste˘ın, O. Sˇvarcman. Chevalley’s theorem for complex crystallographic Cox- eter groups. Funktsional. Anal. i Prilozhen 12(4), 1978, pp. 79–80. [BL04] Ch. Birkenhake, H. Lange. Complex abelian varieties. Second edition. Grundlehren der Mathematischen Wissenschaften 302. Springer-Verlag, Berlin, 2004. [GR99] V. Gonz´alez-Aguilera, R. Rodr´ıguez. Families of irreducible principally polarized abelian varieties isomorphic to a product of elliptic curves. Proc. Amer. Math. Soc. 128(3), 1999, pp. 629–636. [IL15] B. Im, M. Larsen. Rational curves on quotients of abelian varieties by ﬁnite groups. Math. Res. Lett. 22(4), 2015, pp. 1145–1157. [Kan94] E. Kani. Elliptic curves on abelian surfaces. Manuscripta Math. 84, 1994, pp. 199–223. [Kan16] E. Kani. The moduli spaces of Jacobians isomorphic to a product of two elliptic curves. Collect. Math. 67, 2016, pp. 21–54. [KL09] J. Koll´ar, M. Larsen. Quotients of Calabi-Yau varieties, in Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, Progr. Math. 270, Birkhauser Boston Inc., Boston, MA, 2009, pp. 179–211. [Kri10] I. Krichever. Characterizing Jacobians via trisecants of the Kummer variety. Annals of Math. (2) 172(1), 2010, pp. 485–516. [Par91] R. Pardini. Abelian covers of algebraic varieties. J. Reine Angew. Math. 417, 1991, pp. 191–213. [Pir89] G. Pirola. Curves on generic Kummer varieties. Duke Math. J. 59(3), 1989, pp. 701–708. [Pop82] V. Popov. Discrete complex reﬂection groups. Communications of the Mathe- matical Institute, Rijksuniversiteit Utrecht, 15. Rijksuniversiteit Utrecht, Math- ematical Institute, Utrecht, 1982, 89 pp. [ST54] G. Shephard, J. Todd. Finite unitary reﬂection groups. Canadian J. Math. 6, 1954, pp. 274–304. [TY82] S. Tokunaga, M. Yoshida. Complex crystallographic groups. I. J. Math. Soc. Japan 34(4), 1982, pp. 581–593. [Yos07] H. Yoshihara. Galois embedding of algebraic variety and its application to abelian surface. Rend. Semin. Mat. Univ. Padova 117, 2007, pp. 69-85. ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ROBERT

text: AUFFARTH

text: AND

text: GIANCARLO

text: LUCCHINI

text: ARTECHE

text: Theorem

text: If

text: (A,

text: Θ)

text: is

text: a

text: principally

text: polarized

text: abelian

text: variety

text: with

text: Θ

text: irre-

text: ducible,

text: and

text: G

text: is

text: a

text: group

text: consisting

text: of

text: automorphisms

text: of

text: A

text: that

text: preserve

text: the

text: numerical

text: class

text: of

text: Θ

text: and

text: such

text: that

text: dim(AG)

text: then

text: A/G

text: is

text: singular.

text: References

text: R.

text: Auﬀarth.

text: A

text: note

text: on

text: Galois

text: embeddings

text: of

text: abelian

text: varieties.

text: Manuscripta

text: Math.

text: pp.

text: R.

text: Auﬀarth,

text: H.

text: Lange,

text: A.

text: Rojas.

text: A

text: criterion

text: for

text: an

text: abelian

text: variety

text: to

text: be

text: non-

text: simple.

text: J.

text: Pure

text: Appl.

text: Algebra

text: pp.

text: S.

text: A.

text: Broughton.

text: Classifying

text: ﬁnite

text: group

text: actions

text: on

text: surfaces

text: of

text: low

text: genus.

text: J.

text: Pure

text: Appl.

text: Algebra

text: pp.

text: I.

text: Bernˇste˘ın,

text: O.

text: Sˇvarcman.

text: Chevalley’s

text: theorem

text: for

text: complex

text: crystallographic

text: Cox-

text: eter

text: groups.

text: Funktsional.

text: Anal.

text: i

text: Prilozhen

text: pp.

text: Ch.

text: Birkenhake,

text: H.

text: Lange.

text: Complex

text: abelian

text: varieties.

text: Second

text: edition.

text: Grundlehren

text: der

text: Mathematischen

text: Wissenschaften

text: Springer-Verlag,

text: Berlin,

text: V.

text: Gonz´alez-Aguilera,

text: R.

text: Rodr´ıguez.

text: Families

text: of

text: irreducible

text: principally

text: polarized

text: abelian

text: varieties

text: isomorphic

text: to

text: a

text: product

text: of

text: elliptic

text: curves.

text: Proc.

text: Amer.

text: Math.

text: Soc.

text: pp.

text: B.

text: Im,

text: M.

text: Larsen.

text: Rational

text: curves

text: on

text: quotients

text: of

text: abelian

text: varieties

text: by

text: ﬁnite

text: groups.

text: Math.

text: Res.

text: Lett.

text: pp.

text: E.

text: Kani.

text: Elliptic

text: curves

text: on

text: abelian

text: surfaces.

text: Manuscripta

text: Math.

text: pp.

text: E.

text: Kani.

text: The

text: moduli

text: spaces

text: of

text: Jacobians

text: isomorphic

text: to

text: a

text: product

text: of

text: two

text: elliptic

text: curves.

text: Collect.

text: Math.

text: pp.

text: J.

text: Koll´ar,

text: M.

text: Larsen.

text: Quotients

text: of

text: Calabi-Yau

text: varieties,

text: in

text: Algebra,

text: arithmetic,

text: and

text: geometry:

text: in

text: honor

text: of

text: Yu.

text: I.

text: Manin.

text: Vol.

text: II,

text: Progr.

text: Math.

text: Birkhauser

text: Boston

text: Inc.,

text: Boston,

text: MA,

text: pp.

text: I.

text: Krichever.

text: Characterizing

text: Jacobians

text: via

text: trisecants

text: of

text: the

text: Kummer

text: variety.

text: Annals

text: of

text: Math.

text: pp.

text: R.

text: Pardini.

text: Abelian

text: covers

text: of

text: algebraic

text: varieties.

text: J.

text: Reine

text: Angew.

text: Math.

text: pp.

text: G.

text: Pirola.

text: Curves

text: on

text: generic

text: Kummer

text: varieties.

text: Duke

text: Math.

text: J.

text: pp.

text: V.

text: Popov.

text: Discrete

text: complex

text: reﬂection

text: groups.

text: Communications

text: of

text: the

text: Mathe-

text: matical

text: Institute,

text: Rijksuniversiteit

text: Utrecht,

text: Rijksuniversiteit

text: Utrecht,

text: Math-

text: ematical

text: Institute,

text: Utrecht,

text: pp.

text: G.

text: Shephard,

text: J.

text: Todd.

text: Finite

text: unitary

text: reﬂection

text: groups.

text: Canadian

text: J.

text: Math.

text: pp.

text: S.

text: Tokunaga,

text: M.

text: Yoshida.

text: Complex

text: crystallographic

text: groups.

text: I.

text: J.

text: Math.

text: Soc.

text: Japan

text: pp.

text: H.

text: Yoshihara.

text: Galois

text: embedding

text: of

text: algebraic

text: variety

text: and

text: its

text: application

text: to

text: abelian

text: surface.

text: Rend.

text: Semin.

text: Mat.

text: Univ.

text: Padova

text: pp.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##