Galois theory of periods, and the Andr´ e-Oort conjecture Yves Andr´ e, CNRS, Univ. Paris 6 1
Outline of Galois theory of periods ş ∆ and ω “algebraic” ∆ ω, ( ω diff. form on an algebraic variety X defined over some number field k , ∆ Ă X p R q 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”... 2
General conjectures: Grothendieck (’66): any period relation is of motivic origin . X smooth { k , H ˚ p X p C q , Q q b H ˚ dR p X q Ñ C expressed by period matrix Ω X . If X proper, Z alg. subvariety dim. r of X m , dR p X m q Ă H dR p X q b m ❀ ş ω P H 2 r Z ω P p 2 πi q r k conjecturally, period relations always come in this way. 3
Kontsevich (’98) (-Zagier): any period relation comes from the basic rules for ş : linearity, product, algebraic change of variable ∆ f ˚ ω “ ş ş f ˚ ∆ ω , Stokes ş ∆ dω “ ş B ∆ ω. When made precise, these two conjectures can be proven to be equivalent. Remark. Functional analog of periods: Q ❀ C p t q . Ayoub (2015) proved analogs of Grothendieck’s and Kontsevich’s conjec- tures in this case. 4
Motives : categorification of the Grothendieck ring of varieties K 0 p V ar k q . ❀ abelian b -category MM p k q . 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 ❀ x X y b – Rep Q G X G X Ă GL p H p X p C qq , Q q motivic Galois group of X . 5
Example. “ abelian variety { C . X H 1 p X p C qq , Q q b C “ Ω 1 p X q ‘ Ω 1 p X q c λµ : “ λ ¨ id Ω 1 p X q ` µ ¨ id Ω 1 p X q . Fact (A. ’96): is isomorphic G X to the Mumford-Tate group of X , ie. the smallest algebraic Q -subgroup H Ă GL p H 1 p X p C qq , Q q such that @ λ, µ P C ˆ , c λµ P H p C q . eg. X “ non CM elliptic curve: G X “ GL 2 , X “ CM elliptic curve by K : G X p Q q “ K ˆ . 6
H B ,H DR x X y b Ñ V ec Q p k “ Q q ❀ Π X period torsor Period pairing ô canonical point in Π X p C q : Spec C ̟ X Ñ Π X . Grothendieck’s period conjecture: PC X : ̟ X is a generic point . Equivalently: Π X is connected, and TrDeg Q Q r Ω X s “ dim G X . If so, one can develop a bit of Galois theory of periods: G X p Q q would act on Q r Ω X s ❀ Conju- gates of periods... P 1 Q ˆ , “ G X p Q q “ Examples : : X Q r Ω X s “ Q r 2 πi s ( PC X : Lindemann), G X p Q q “ K ˆ , X “ CM elliptic curve by K : Q r Ω X s “ Q r ω 1 , η 1 s ( PC X : Chudnovsky). 7
What if k Ă C is no longer algebraic over Q ? generalized PC X : TrDeg Q k r Ω X s ě dim G X (A. ’97). Should hold for any motive; in the r Z n G n case of 1-motives Ñ m s , this amounts to Schanuel’s conjecture: if P are Q -linearly independent, x 1 , . . . , x n C TrDeg Q Q r x 1 , . . . , x n , e x 1 , . . . , e x n s ě n . 8
Interlude: motivic Galois groups and specializations problems. X Ñ S family of projective smooth varieties s P S ❀ X s ❀ G X s Variation of G X s with s ? e.g. non-constant family of elliptic curves: G X s “ GL 2 if X s non CM. General result: Outside a countable union S ex of algebraic subvarieties S n Ĺ S , G X s is (lo- cally) constant. If the family is defined over ¯ Q , S ex p ¯ Q q ‰ S p ¯ Q q . (A. ’96) Application: H 2 p X s q G Xs “ NS X s . Thus: NS X s is constant outside S ex ; if the family is defined over ¯ Q , there exists s P S p ¯ Q q such that NS specializes isomorphically at s . 9
(Similarly, if G is the motivic Galois of some complex abelian variety, there is an abelian va- riety A defined over ¯ Q with G A “ G .) Ñ Hint: X s defined over K : G K ❀ ρ X s G X s p Q ℓ q Ă GL p H et p X s, ¯ K , Q ℓ qq . Conjecturally, Im ρ X s Zariski dense; thus if η ❀ s is a specialization and G X s is smaller than G X η , then Im ρ X s is smaller than Im ρ X η . 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 S ex of bounded degree is finite. 10
Similar situation with periods instead of Galois representations: ̟ Xs Conjecturally ( PC ), Im (Spec C Ñ Π X s ) Zariski-dense. Thus if η ❀ s is a specializa- tion and G X s is smaller than G X η , then Im ̟ X s is smaller than Im ̟ X η . But this can be proved unconditionally (one of the threads which led me to the AO conjecture...) 11
Outline of the AO conjecture. Geometry of A g , the algebraic variety which parametrizes principally polarized abelian vari- eties of dimension g (e.g. A 1 “ j -line). Special subvarieties of A g : subvarieties which parametrize PPAV with “extra symmetries”. PPAV with maximal symmetry (complex mul- tiplication) are parametrized by special points . AO conjecture: special subvarieties of A g are characterized by the density of their special points . 12
Remarks. - A g and its special subvarieties share a common geometric nature: they are Shimura varieties g “ 1 : C H Ó ℘ Ó j A 1 – H { SL 2 p Z q E – C {p Z ω 1 ` Z ω 2 q - “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 A g ) 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... 13
Connections between AO and PC . Early circle of ideas which gave rise to 1. the AO conjecture. AO G ´ fct PC G -functions Ø periods Variational approach to PC (for abelian peri- ods) through G -functions? Example. E λ : y 2 “ x p x ´ 1 qp x ´ λ q , ω 1 p λ q „ πF p λ q , η 1 „ πF 1 p λ q , F “ F p 1 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)). 14
For singular modulus (ie special point), λ F p λ qp F 1 p λ q ` αF p λ qq “ 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 of dim. g ą 1 instead parametrized by a curve in A g (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 8 ly 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. 15
2. Curves in products of modular curves Case of C Ă A 1 ˆ A 1 Ă A 2 (A. (’93) - first, and only, unconditional case of AO , until Pila (2011)): AO A 1 ˆ A 1 : if C contains 8 ly many pairs of sin- gular moduli p j, j 1 q , C is either a vertical or hor- izontal line or some X 0 p N q . i) p j n , j 1 n q singular moduli on C, p D n , D 1 n q (dis- criminants of quadratic orders). Class field ? ? D 1 theory ❀ for n ąą 0 , Q p D n q “ Q p n q and D 1 n { D n takes finitely many values. ii) Linear forms in elliptic periods ❀ if 8 ly many special points on C , a branch of C n q Ñ p8 , j 1 “ if p j n “ j p τ n q , j 1 goes to p8 , 8q : then τ 1 “ ω 1 j p τ 1 qq , 1 { ω 1 2 is well-approximated by quadratic numbers τ n ; contradicts Masser’s lower bound for | ω 1 1 ´ τ n ω 1 2 | .] iii) analysis of Puiseux expansions. 16
3. Hypergeometric values a, b, c P Q , R p c q ą R p b q ą 0 , n “ den p a, b, c q , F p a, b, c ; λ q “ ř p a q m p b q m p c q m m ! λ m ş 1 0 x b ´ 1 p 1 ´ x q c ´ b ´ 1 p 1 ´ λx q ´ a dx “ B p b,c ´ b q satisfies HG diff. equation, monodromy = Schwarz triangle group ∆. numerator = period of J new n,a,b,c,λ y n “ x n p b ´ 1 q p 1 ´ x q n p c ´ b ´ 1 q p 1 ´ λx q ´ na denominator B p b, c ´ b q = period of simple CM quotient F b,c of Fermat jacobian. 17
Question (J. Wolfart): for which p a, b, c q are there 8 ly many λ P ¯ Q with F p a, b, c ; λ q P ¯ Q ? Answer (W¨ ustholz-Wolfart-Cohen-Edixhoven - Yafaev): iff ∆ finite or arithmetic. [“if” due to Wolfart. “Only if”: 3 steps: ustholz (special case of PC ): ¯ i) W¨ Q - linear rela- tions between periods of abelian periods come from endomorphisms Q q ñ J new ❀ p λ, F p a, b, c ; λ q P ¯ n,a,b,c,λ „ F b,c , ii) for P 1 zt 0 , 1 , 8u φ Ñ A g : λ ÞÑ J new n,a,b,c,λ , Im p φ q special iff ∆ finite or arithmetic. iii) AO ❀ J new n,a,b,c,λ has CM for 8 ly many λ ’s iff Im p φ q special.] 18
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