Galois theory of periods, and the Andr e-Oort conjecture Yves Andr - - PDF document

Galois theory of periods, and the Andr´ e-Oort conjecture Yves Andr´ e, CNRS, Univ. Paris 6

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Outline of Galois theory of periods

ş

∆ ω,

∆ and ω “algebraic” (ω diff. form on an algebraic variety X defined

• ver some number field k, ∆ Ă XpRq defined

by algebraic inequations {k), Transcendence of periods? Algebraic relations between them (period relations)? Leibniz (1691, letters to Huygens): specula- tion about transcendence of π and some other (1-dim) periods. Inquiry about “accidental” cases when they are algebraic: “nothing hap- pens without a reason”...

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General conjectures: Grothendieck (’66): any period relation is of motivic origin. X smooth {k, H˚pXpCq, Qq b H˚

dRpXq Ñ C

expressed by period matrix ΩX. If X proper, Z alg. subvariety dim. r of Xm , ω P H2r

dRpXmq Ă HdRpXqbm ❀ ş Z ω P p2πiqrk

conjecturally, period relations always come in this way.

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Kontsevich (’98) (-Zagier): any period relation comes from the basic rules for ş : linearity, product, algebraic change of variable

ş

∆ f˚ω “ ş f˚∆ ω, Stokes ş ∆ dω “ ş

B∆ ω.

When made precise, these two conjectures can be proven to be equivalent. Remark. Functional analog

• f

periods: Q ❀ Cptq. Ayoub (2015) proved analogs

• f

Grothendieck’s and Kontsevich’s conjec- tures in this case.

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Motives: categorification of the Grothendieck ring of varieties K0pV arkq. ❀ abelian b-category MMpkq. 3 unconditional, compatible theories:

• A. (pure motives, ie.

motives attached to projective smooth varieties), Nori, Ayoub; cf. Bourbaki nov. 2015).

• eg. X ❀ motive of X ❀ xXyb – RepQ GX

GX Ă GLpHpXpCqq, Qq motivic Galois group of X.

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Example. X

“

abelian variety

{C.

H1pXpCqq, Qq b C “ Ω1pXq ‘ Ω1pXq cλµ :“ λ ¨ idΩ1pXq ` µ ¨ idΩ1pXq. Fact (A. ’96): GX is isomorphic to the Mumford-Tate group

• f

X, ie. the smallest algebraic Q-subgroup H Ă GLpH1pXpCqq, Qq such that @λ, µ P Cˆ, cλµ P HpCq. eg. X “ non CM elliptic curve: GX “ GL2, X “ CM elliptic curve by K: GXpQq “ Kˆ.

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xXyb

HB,HDR

Ñ

V ecQ pk “ Qq ❀ ΠX period torsor Period pairing ô canonical point in ΠXpCq: Spec C ̟X

Ñ ΠX.

Grothendieck’s period conjecture: PCX: ̟X is a generic point. Equivalently: ΠX is connected, and TrDegQ QrΩXs “ dim GX. If so, one can develop a bit of Galois theory of periods: GXpQq would act on QrΩXs ❀ Conju- gates of periods... Examples: X

“

P1 : GXpQq

“

Qˆ, QrΩXs “ Qr2πis (PCX: Lindemann), X “ CM elliptic curve by K: GXpQq “ Kˆ, QrΩXs “ Qrω1, η1s (PCX: Chudnovsky).

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What if k Ă C is no longer algebraic over Q? generalized PCX: TrDegQ krΩXs ě dim GX (A. ’97). Should hold for any motive; in the case

• f

1-motives

rZn Ñ

Gn

ms,

this amounts to Schanuel’s conjecture: if x1, . . . , xn

P

C are Q-linearly independent, TrDegQ Qrx1, . . . , xn, ex1, . . . , exns ě n.

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Interlude: motivic Galois groups and specializations problems. X Ñ S family of projective smooth varieties s P S ❀ Xs ❀ GXs Variation of GXs with s? e.g. non-constant family of elliptic curves: GXs “ GL2 if Xs non CM. General result: Outside a countable union Sex

• f algebraic subvarieties Sn Ĺ S, GXs is (lo-

cally) constant. If the family is defined over ¯ Q, Sexp¯ Qq ‰ Sp¯ Qq. (A. ’96) Application: H2pXsqGXs “ NSXs. Thus: NSXs is constant outside Sex; if the family is defined over ¯ Q, there exists s P Sp¯ Qq such that NS specializes isomorphically at s.

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(Similarly, if G is the motivic Galois of some complex abelian variety, there is an abelian va- riety A defined over ¯ Q with GA “ G.) Hint: Xs defined over K ❀ ρXs : GK

Ñ

GXspQℓq Ă GLpHetpXs, ¯

K, Qℓqq.

Conjecturally, Im ρXs Zariski dense; thus if η ❀ s is a specialization and GXs is smaller than GXη, then Im ρXs is smaller than Im ρXη. But this can be proved unconditionally. Conclude by Hilbert irreducibility argument (Serre’s “in- finite” variant). Refinement (Cadoret - Tamagawa): when S is a curve defined over a number field, the set of points of Sex of bounded degree is finite.

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Similar situation with periods instead of Galois representations: Conjecturally (PC), Im (Spec C

̟Xs

Ñ

ΠXs) Zariski-dense. Thus if η ❀ s is a specializa- tion and GXs is smaller than GXη, then Im ̟Xs is smaller than Im ̟Xη. But this can be proved unconditionally (one of the threads which led me to the AO conjecture...)

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Outline of the AO conjecture. Geometry of Ag, the algebraic variety which parametrizes principally polarized abelian vari- eties of dimension g (e.g. A1 “ j-line). Special subvarieties of Ag: subvarieties which parametrize PPAV with “extra symmetries”. PPAV with maximal symmetry (complex mul- tiplication) are parametrized by special points. AO conjecture: special subvarieties of Ag are characterized by the density of their special points.

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Remarks.

• Ag and its special subvarieties

share a common geometric nature: they are Shimura varieties g “ 1 : C H

Ó ℘ Ó j

E – C{pZω1 ` Zω2q A1 – H{SL2pZq

• “extra symmetries” ?

... Prescribed en- domorphisms on A, or more generally, pre- scribed Hodge cycles on powers of A; looks transcendental, but is an algebraic condition: amounts to prescribe algebraic cycles on prod- uct of powers of A and some compact abelian pencils (A. ’96). The AO conjecture (for Ag) is now a theo- rem (2015), after two decades of collaborative efforts putting together many different areas. Some key contributors: A. Yafaev, E. Ullmo,

• B. Klingler, J. Pila, J. Tsimerman...

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Connections between AO and PC. 1. Early circle of ideas which gave rise to the AO conjecture. G ´ fct PC AO G-functions Ø periods Variational approach to PC (for abelian peri-

• ds) through G-functions?
• Example. Eλ : y2 “ xpx ´ 1qpx ´ λq,

ω1pλq „ πFpλq, η1 „ πF 1pλq, F “ Fp1

2, 1 2, 1; λ).

Diophantine theory of special values of G- functions F, F 1 ❀ new proof of PC for CM elliptic curves (A. (’96)).

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For λ singular modulus (ie special point), FpλqpF 1pλq ` αFpλqq “ β{π, α, β P ¯

• Q. One can-

not eliminate π... other solutions of the HGE are useless (log singularity at 0). But for AV

• f dim. g ą 1 instead parametrized by a curve

in Ag (instead of λ-line), one may get enough G-functions and relations between their special values. Existence of lots of special points on the curve would allow to apply G-function theory effi-

• ciently. But analogy with Manin-Mumford ren-

ders the existence of 8ly many special points unlikely in the non-modular case! This was one source of my formulation of AO (’89) (Oort’s later but independent formula- tion came from another source: CM liftings, Coleman conjecture...). AO bounds the hope for an application of G-

• fct. theory to PC; nevertheless, more intricate

alternative connections between AO and PC exist.

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• 2. Curves in products of modular curves

Case of C Ă A1 ˆ A1 Ă A2 (A. (’93) - first, and only, unconditional case of AO, until Pila (2011)): AOA1ˆA1: if C contains 8ly many pairs of sin- gular moduli pj, j1q, C is either a vertical or hor- izontal line or some X0pNq. i) pjn, j1

nq singular moduli on C, pDn, D1 nq (dis-

criminants of quadratic orders). Class field theory ❀ for n ąą 0, Qp

?

Dnq “ Qp

?

D1

nq and

D1

n{Dn takes finitely many values.

ii) Linear forms in elliptic periods ❀ if 8ly many special points on C, a branch of C goes to p8, 8q: if pjn “ jpτnq, j1

nq Ñ p8, j1 “

jpτ1qq, then τ1 “ ω1

1{ω1 2 is well-approximated

by quadratic numbers τn; contradicts Masser’s lower bound for |ω1

1 ´ τn ω1 2|.]

iii) analysis of Puiseux expansions.

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• 3. Hypergeometric values

a, b, c P Q, Rpcq ą Rpbq ą 0, n “ denpa, b, cq, Fpa, b, c; λq “ ř paqmpbqm

pcqm m! λm

“

ş1

0 xb´1p1´xqc´b´1p1´λxq´adx

Bpb,c´bq

satisfies HG diff. equation, monodromy = Schwarz triangle group ∆. numerator = period of Jnew

n,a,b,c,λ

yn “ xnpb´1qp1 ´ xqnpc´b´1qp1 ´ λxq´na denominator Bpb, c ´ bq = period of simple CM quotient Fb,c of Fermat jacobian.

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Question (J. Wolfart): for which pa, b, cq are there 8ly many λ P ¯ Q with Fpa, b, c; λq P ¯ Q? Answer (W¨ ustholz-Wolfart-Cohen-Edixhoven - Yafaev): iff ∆ finite or arithmetic. [“if” due to Wolfart. “Only if”: 3 steps: i) W¨ ustholz (special case of PC): ¯ Q-linear rela- tions between periods of abelian periods come from endomorphisms ❀ pλ, Fpa, b, c; λq P ¯ Qq ñ Jnew

n,a,b,c,λ „ Fb,c,

ii) for P1zt0, 1, 8u φ

Ñ Ag : λ ÞÑ Jnew

n,a,b,c,λ,

Impφq special iff ∆ finite or arithmetic. iii) AO ❀ Jnew

n,a,b,c,λ has CM for 8ly many λ’s iff

Impφq special.]

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• 4. Bialgebraicity

Hg

Ă H_

g

plagrangian grassmannianq

j Ó τ “ Ω1Ω´1

2

ÞÑ jpτq

Ag Both H_

g and Ag are algebraic varieties/Q, but

j is transcendental. Bialgebraic characterization of special subva- rieties (W¨ ustholz-Cohen-Shiga-Wolfart-Ullmo- Yafaev): S Ă Ag is special iff both S and a branch of j´1pSq Ă H_

g

are algebraic and de- fined over ¯ Q. CSW: case of dim 0: τ, jpτq P ¯ Q ô jpτq is a special point (Schneider).

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Remark. H_

g and Ag are transcendentally re-

lated by j, but are also algebraically related by the relative period torsor: Πg

ρ

Ñ

H_

g

Ó

Ag Πg is a Sp2g-torsor on Ag, with coordinates corresponding to

ˆ

Ω1 N1 Ω2 N2

˙

, and ρ is the Sp2g-equivariant surjective map

ˆ

Ω1 N1 Ω2 N2

˙ ÞÑ

Ω1Ω´1

2 .

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• 5. Minimal special subvarieties

Given a PPAV A of dim. g, ie a point jA P Ag, there is a (unique) minimal special subvariety SA containing jA. Question (Wolfart): if A is defined over ¯ Q, what is the dimension of SA? Answer: PCA ñ dim SA “ TrDeg.Q Qpτq, for any τ P H such that jpτq “ jAq. via an analysis of (a reduction of) the relative period torsor Πg. CSW is the case ‘0= 0” of this equality.

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An afterthought. One breakthrough in the proof of AO was the introduction of o-minimal methods (Pila- Zannier). In such “counting arguments”, an exceptional (semi-)algebraic set is left out. To handle it, one needs some functional transcen- dence results, which are functional analogs of the generalized PC. Very recently, the Pila-Zannier method led to very powerful functional transcendence results (“Ax-Schanuel”).

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Question: is there any way to use the Pila- Zannier method in transcendental Number Theory? For instance, Schneider’s problem: prove that “τ, jpτq P ¯ Q ô jpτq is a special point” using only the j-function (no ℘!) is still open... can “counting arguments” ` a la Pila-Wilkie help in this (and other similar) context(s)?

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