The Schanuel paradigm

Transcription

1 The Schanuel paradigm Jonathan Pila University of Oxford 2014 Clay Research Conference, Oxford

2 Themes Schanuel s conjecture, analogues, in particular Functional analogues and their connections with certain Diophantine problems

3 Transcendental number theory: exponentials Hermite (1873): e is a transcendental number Hermite-Lindemann (1882): If α Q then e α is transcendental Definition Complex numbers z 1,..., z n are algebraically dependent over Q if there is a non-zero polynomial P Q[Z 1,..., Z n ] with P(z 1,..., z n ) = 0. Otherwise, algebraically independent. α is an algebraic number if α is algebraically dependent over Q. These form a field denoted Q C. Theorem (Lindemann (-Weierstrass) 1882(-85)) Let α 1,..., α n Q be linearly independent over Q. Then e α 1,..., e αn are algebraically independent over Q.

4 Transcendental number theory: logarithms Theorem (Gelfond-Schneider, 1934) If a 0, 1 is algebraic and b is algebraic and irrational then a b is transcendental. Affirms Hilbert s problem 7. E.g. 2 2, e π = (e iπ ) i = ( 1) i Conjecture (Gelfond) Let α 1,..., α n Q be non-zero and multiplicatively independent. Then (any determination of) log α 1,..., log α n are algebraically independent over Q. Theorem (Baker, 1964) Let α 1,..., α n Q be non-zero and multiplicatively independent. Then (any determination of) log α 1,..., log α n are linearly independent over Q.

5 Transcendental number theory: algebraic powers Conjecture (Gelfond) Suppose β Q has degree d over Q, and α Q with α 0, 1. Then α β, α β2,..., α βd 1 are algebraically independent over Q. Theorem (Gelfond, 1948) This is true for d = 3 (and d = 2 by Gelfond-Schneider). Theorem (Philippon; Nesterenko, Diaz 1989) With notation as above and d 3, at least [ d+1 2 ] of them are algebraically independent over Q.

6 Schanuel s conjecture Conjecture (Schanuel, circa 1966) Suppose z 1,..., z n C are linearly independent over Q. Then tr.deg. Q Q(z 1,..., z n, e z 1,..., e zn ) n. Exercise: This implies all the previous theorems and conjectures (and much more: e.g. algebraic independence of π, e π (theorem of Nesterenko), and of π, e). True for n = 1 (Hermite-Lindemann); open for n = 2

7 Schanuel s Conjecture in Lang, Intro. Trans. Nbrs. (1966); J. Ax, Annals of Mathematics 93 (1971),

8 Ax-Schanuel Consider: and exp : C n (C ) n, exp(z 1,..., z n ) = (e z 1,..., e zn ), Γ C n (C ) n graph of exp V C n (C ) n an algebraic variety (defined by pols). How do V and Γ interact? Theorem (Ax 1971; Ax-Schanuel ) Let A be a component of V Γ. Then dim A dim V + dim Γ dim C n (C ) n = dim V n unless A is contained in a translate of a rational linear sbspc of C n. Condition on A: q i z i constant on A for some q i Q, not all 0. Reformulation: For z i, e z i as functions on A: tr.deg. C C(z 1,..., z n, e z 1,..., e zn ) n + dim A, unless...

9 Ax-Lindemann Definition A weakly special subvariety of C n is a subvariety L defined by finitely many equations q ij z i = c j, j = 1,..., k, q ij Q, c j C. Then T = exp L is a coset of a subtorus of (C ) n. If W exp 1 (Y ), AS for W Y : dw d(w Y ) n unless.. Theorem ( Ax-Lindemann, special case of Ax-Schanuel) For algebraic subvariety W C n, exp W is Zariski-dense in (C ) n unless W is contained in a proper weakly special subvariety. Zariski-dense: no proper subvariety of (C ) n contains V. If L is smallest weakly special W L, exp W is Z-dense in exp L. Theorem ( Ax-Lindemann, reformulation) For V (C ) n, maximal algebraic W exp 1 (V ) is wkly special.

10 Model theory of complex exponentiation First-order theory of C is very well behaved: complete, decidable,... Th(C exp ) is complicated, as Z is definable: Z = {z C : w ( exp(w) = 1 exp(zw) = 1 ) }. Zilber showed: a suitable non-first order formulation of the theory of C with an exponential satisfying SC and exponential algebraic closedness is well-behaved, has a unique model of cardinality 2 ℵ 0. See work of: D Aquino, Macintyre, Terzo, Shkop, Kirby, Onshuus,.. including some talks at our workshop.. Zilber formulated CIT for (C ) n (and semiabelian varieties).

11 Uniform Schanuel Conjecture Conjecture (SC, reformulated) Let V C 2n be an algebraic set defined over Q with dim V < n. If (z 1,..., z n, e z 1,..., e zn ) V then there is a nontrivial linear dependence q i z i = 0, q i Q. Conjecture (Uniform Schanuel Conjecture, Zilber 2002) Let V C 2n be an algebraic set defined over Q with dim V < n. There exists a finite set µ(v ) of proper Q-linear subspaces of C n such that if (z 1,..., z n, e z 1,..., e zn ) V then there is M µ(v ) and k = (k 1,..., k n ) Z n such that z + 2πik M. Moreover if M has codimension one then k = 0.

12 An implication of SC Suppose V C n (C ) n is as above. Let W be the image of V under projection of C n (C ) n to (C ) n. Projection V W has generic dimension d outside some V V Suppose (z, e z ) = (z 1,..., z n, e z 1,..., e zn ) V V. On SC, z L C n rational subspace, dim L = l.d. Q (z i ). Let T = exp L, a mult. subgp, dim T = dim L = l.d. Q (z i ). Say exp z A T W some component A. Then dim T tr.deg.(z, e z ) d + dim A = dim V dim W + dim A. Since dim V < n, implies: dim T < n dim W dim A. That is, dim A > dim T + dim W dim(c ) n. I.e. exp z is in an atypical intersection of W with some such T.

13 Conjecture on Intersections with Tori ( CIT ) Sources: Zilber, Bombieri-Masser-Zannier. Conjecture ( CIT, Zilber, 2002) Let W (C ) n. There exists a finite set S W of proper subtori such that if A is an atypical component of W T for any subtorus T (C ) n then A S for some S S W. Theorem (Zilber, 2002) SC plus CIT implies USC. Also USC implies CIT (and SC): CIT is the difference between SC and USC. CIT is open in general for dim W 2.

14 CIT for curves An atypical component of a curve C (C ) n is either: A point P C satisfying two (or more) independent multiplicative conditions or The curve C itself if it satisfies a nontrivial multiplicative condition x k x n kn = 1. Theorem (Maurin 2008/Q, BMZ, 2008/C) If C (C ) n, defined over C, is not atypical then there are only finitely many atypical P C. Example: C = {(2, 3, t, t + 1, t ) : t C}; t = 1, 2, 3, 8, 9...

15 CIT implies Mordell-Lang More generally, let C (C ) 2 be a plane curve: F (x, y) = 0. For a 1,..., a k C let a 1,..., a k = { a n i i : n i Z}, the group generated by a 1,..., a k, and Γ = {z C : z n a 1,..., a k for some nonzero integer n} be its division group. The above implies as a very special case: Theorem (Liardet, 1974) There are only finitely many points (x, y) C with x and y in the division group Γ generated a 1,..., a n, unless C is of the form x n y m = a Γ. CIT implies Mordell-Lang, and is a far-reaching generalisation.

16 L.J. Mordell, Proc. Cam. Phil. Soc. 21 (1922), ; S. Lang, Ann. Mat. Pura. Appl. (4) 70 (1965),

17 Curve C (C ) 2 defined by F (x, y) = 0, F C[X, Y ]. Theorem (Ihara-Serre-Tate; Lang 1965) If infinitely many (ξ, η) C with ξ, η roots of unity then F is of form x n y m = ζ, with n, m Z, not both zero, ζ a root of unity....so C is a coset of a subgroup by a torsion point: torsion coset. Consider V (C ) n. If dim V 2 it may contain positive dimensional torsion-cosets: Theorem ( Multiplicative Manin-Mumford ; Mann/Laurent) V contains only finitely many maximal torsion cosets.

18 Manin-Mumford and Mordell-Lang If (C ) n is replaced by an Abelian variety X = Λ\C n : Conjecture (Manin-Mumford, 1960 s; proved by Raynaud, 1983) A subvariety V of an abelian variety X contains only finitely many maximal torsion cosets. Now we replace torsion points by the division group Γ of a finitely generated group, torsion cosets by Γ-cosets of abelian subvarieties. Theorem (Mordell-Lang conjecture; Faltings, Vojta, McQuillan et al ) A subvariety V of an abelian variety X contains only finitely many maximal Γ-torsion cosets. Implies Mordell s conjecture, as a curve embeds in its Jacobian and group of rational points is finitely generated (Mordell, 1922-Weil).

19 The André-Oort Conjecture Analogue of Manin-Mumford for a Shimura variety X..... certain arithmetic quotients, e.g. moduli spaces; example below. Such X has a collection {T } of special subvarieties, including special points. Conjecture (André-Oort) Let X be a Shimura variety and V X a subvariety. Then V contains only finitely many maximal special subvarieties. André (1998): X = C 2, unconditional Edixhoven, Klingler, Ullmo, Yafaev ( ): on GRH O-minimality and point-counting (2011 ): unconditional cases

20 C as a Shimura variety Background: elliptic curves and their moduli: Lattice Λ = Z + Zτ, τ H = {τ C : Imτ > 0} Elliptic curve: E τ = Λ\C, has structure of an algebraic curve. j(e) determines E up to isomorphism over C. Key for us is the modular function a.k.a. j-invariant, j-function: j : H C, j(τ) = j(e τ ). Basic arithmetic properties: SL 2 (Z) invariance: ( ) ( ) a b a b j(gτ) = j(τ), g = SL c d 2 (Z), τ = aτ + b c d cτ + d. For g GL + 2 (Q), Φ N(j(τ), j(gτ)) = 0, modular polynomial Φ N. Special points: j(τ) where [Q(τ) : Q] = 2, CM points

21 Fundamental domain for the modular function ( ) 2 0 E.g. X = j(z) and Y = j(2z) = j( z) are related by = Φ 2 (X, Y ) = X 2 Y (X 2 Y + XY 2 ) + Y (X 2 +Y 2 ) XY (X +Y )

22 Generalisations of Schanuel Grothendieck-André period conjecture: but just say a bit about j. Theorem ( Modular HL ; Schneider, 1937) If α, j(α) Q then [Q(α) : Q] = 2. Conjecture ( Modular SC ) If z 1,..., z n H are not quadratic and no modular relations (gz k z l, all k, l, g GL + 2 (Q)) then tr.deg. Q Q(α 1,..., α n, j(α 1 ),..., j(α n )) n. Open for n 2 (even the Lindemann case that all z i Q).

23 X = C 2 as a Shimura variety Special subvarieties of X = C 2 : Dimension 2: X. Dimension 1: {(x, y) : x = j 0 } and {(x, y) : y = j 0 } where j 0 is special, and modular curves Φ N (x, y) = 0. Dimension 0: special points (j 0, j 1 ) with j 0, j 1 special. Theorem (André, 1998) A curve C C 2 containing infinitely many special points is special. Edixhoven 1998: proof on GRH. Kühne, Bilu-Masser-Zannier, 2012/13: effective.

24 Some further examples X = C n Moduli space of n-tuples of elliptic curves. Special subvarieties: hybrids of what happens in C 2 : irreducible components of algebraic sets defined by some equations of the forms: Φ N (x k, x l ) = 0 with various k, l, N and x h = σ with various h special σ. Special points: n-tuples of special points in C. Weakly special subvarieties: allow x h = c, any c C. X = A g Moduli space of pp abelian varieties of dimension g. Special points: The moduli points of CM abelian varieties.

25 The Zilber-Pink Conjecture Let X be a mixed Shimura variety, V X a subvariety Definition A subvariety A V is atypical if there is a special subvariety T X such that A V T and dim A > dim V + dim T dim X. Conjecture (Zilber-Pink) V X contains only finitely many maximal atypical subvarieties. Sources: Zilber, BMZ, Pink. Can view in terms of USC Open even for X = (C ) n. Implies MM, ML, AO, and much more.

26 Special point problems via point-counting Example: studying torsion points on V (C ) n e : C n (C ) n, e(z) = (e 2πiz 1,..., e 2πizn ) Invariance: Z n translations; Fundamental domain: F = F n Z, F Z = {z : 0 Re(z) 1}. Pre-image of torsion points are rational points in C n Strategy (Zannier): Look at rational points in Z = e 1 (V ) F. Play off a lower bound coming from Galois conjugates against an upper bound from Counting Theorem for rational points on definable sets in o-minimal structures (v.d.dries,pillay-steinhorn).

27 e : C n (C ) n, ctd Study: Z(Q, T ) = {P Z : P Q n, H(P) T }, Here: H(a/b) = max( a, b ), (a, b) = 1; H(q) = max H(q i ) Lower bounds: arithmetic [Q(exp(2πi/T )) : Q] = φ(t ) T Points in Z: a/t, a < 1, so H T. If ζ V has order T get T preimages in Z(Q, T ) Upper bound: point-counting (P-Wilkie) For Z definable : #Z (Q, T ) Z,ɛ T ɛ, all ɛ > 0, except semi-algebraic subsets. Algebraic subsets: functional transcendence What are the maximal algebraic subvarieties of π 1 (V )? Ax-Lindemann : They are weakly special An inductive argument gives new proof of Thm of Laurent/Mann: V (C ) n contains only finitely many maximal torsion cosets.

28 j : H n C n Same strategy can be applied to AO: j : H n C n, j(z) = (j(z 1 ),..., j(z n )) Invariance: SL 2 (Z) n action; Fundamental domain: F = FSL n 2 (Z), F SL 2 (Z) = {z : Re(z) 1/2, z 1} Pre-image of special points: quadratic points in H n Count quadratic points in Z = j 1 (V ) F Ingredient 1: Z is definable; gives upper bound Ingredients 2 (and 3): arithmetic: lower bound Ingredient 4: functional transcendence

29 L.Kronecker, Zur theorie der elliptischen functionen, 1883-; C.-L. Siegel, Acta Arithmetica 1 (1935), 83 86

30 Further applications to AO Setting: π : U X, V X. Always have: U C N : hermitian symmetric domain, semi-algebraic π is invariant under arithmetic group Γ pre-images of special pts are algebraic of bounded degree. Ingredients to prove AO (Ullmo, 2013): 1. Definability: Peterzil-Starchenko for A g (and X g ), Klingler-Ullmo-Yafaev and Gao in general (mixed Shimuras) 2. Galois lower bound: Tsimerman for A g, g 6 3. Height upper bound: Tsimerman for A g, all g 4. Ax-Lindemann: Ullmo-Yafaev for compact cases, P-Tsimerman for A g and Klingler-U-Y and Gao in general, by o-minimality and point-counting. Mok, non-arithmetic quotients, by complex geometry. AO known: for all A n g, g 6.

31 Ax-Lindemann For the modular function j : H n C n : Theorem (P, 2011) Let W H n be an algebraic subvariety. Then j(w ) is Zariski-dense in C n unless: some coordinate is constant on W, or some relation z k = gz l with g GL + 2 (Q) holds identically on W (and in that case it isn t Zariski dense). For a mixed Shimura variety X with π : U X : Theorem ( Ax-Lindemann ) Let W U be an algebraic subvariety. Then π(w ) is Zariski-dense in X unless W is contained in a proper weakly special subvariety of U (and then it isn t).

32 Ax-Schanuel for a Shimura variety Ax-Lindemann a special case of an Ax-Schanuel Conjecture. Conjecture (P-Tsimerman) Let Γ U X be the graph of π : U X. Let V U X be algebraic and A V Γ an irreducible component. Then dim A = dim V dim X unless A is contained in a proper weakly special subvariety. Approaches: differential algebra/model theory, monodromy, o-minimality and point-counting, complex geometry... Such results could be instrumental in proving special cases of ZP in presence of suitable arithmetic ingredients (hard to come by; some conditional results: P-Habegger).