O-minimality and Diophantine Geometry

Transcription

1 O-minimality and Diophantine Geometry Jonathan Pila University of Oxford ICM 2014, Seoul

2 Themes Rational points on definable sets in o-minimal structures and applications to some Diophantine problems comprehended within the Zilber-Pink conjecture. Model theory of (complex and real) exponentiation. Schanuel s conjecture in transcendental number theory.

3 Plan Diophantine geometry: Mordell conjecture to Zilber-Pink Transcendental number theory: Schanuel and Ax-Schanuel Model theory of exponentiation, o-minimal structures Counting rational points Diophantine applications: ingredients and problems

4 Diophantine geometry: Mordell s conjecture Diophantine problems: solving systems of algebraic equations F i (x 1,..., x n ) = 0, i = 1,..., m, F i Z[x 1,..., x n ] for x i in integers, rational numbers,... When does a system have only finitely many solutions? Conjecture (Mordell,1922; proved by Faltings, 1983) A plane algebraic curve F (x, y) = 0 of genus 2 defined over Q has only finitely many rational points. E.g. any (projectively) non-singular curve of degree 4 or greater. Likewise over a number field K, [K, Q] <.

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

6 Step 1: From Mordell to Mordell-Lang 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.

7 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).

8 Step 2:...to André-Oort... We consider V X where X is a Shimura variety... 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

9 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

10 Fundamental domains for the modular function

11 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.

12 Some further examples X = C n Special subvarieties: irreducible components of algebraic sets defined by some equations Φ N (x k, x l ) = 0 with various N and x h = σ with various 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.

13 Step 3:...to Zilber-Pink 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, Bombieri-Masser-Zannier, Pink. Open even for X = (C ) n. Implies MM, ML, AO, and much more: A Mordell conjecture for the 21st century.

14 ZP 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, BMZ, 2009) If C (C ) n, defined over C, is not atypical then there are only finitely many atypical P C. Theorem (Habegger-P, 201?) Let C A be a curve in an abelian variety, both over Q. If C is not atypical then there are only finitely many atypical P C. Habegger-P, 2012: a partial result for modular setting C C n.

15 Implies: Baker s theorem Four exponentials conjecture Algebraic independence of logarithms of multiplicatively independent algebraic numbers And much more, but open for n 2 Transcendence theory: Schanuel s conjecture Theorem (Lindemann (-Weierstrass) Theorem, 1873/5) If z i Q are l.i. over Q then e z 1,..., e zn are algebraically independent over Q. Conjecture (Schanuel, circa 1966) If z 1,..., z n C are l.i. over Q then tr.deg. Q Q(z 1,..., z n, e z 1,..., e zn ) n.

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

17 Ax-Schanuel K a field of locally meromorphic functions with derivations D i, C = ker D i, containing functions x 1,..., x n and e x 1,..., e xn. Definition The x 1,..., x n are called linearly independent over Q mod C if there are no non-trivial equations q i x i = c, q i Q, c C. Theorem ( Ax-Schanuel, Ax, 1971) If x 1,..., x n are linearly independent over Q mod C, then tr.deg. C C(x 1,..., x n, e x 1,..., e xn ) n + rank K (D k x l ).

18 Ax-Lindemann Consider algebraic W C n with z 1,..., z n, e z 1,..., e zn K the field of functions meromorphic on U W. Then z 1,..., z n fail to be linearly independent over Q mod C iff W is contained in a proper weakly special subvariety. Theorem ( Ax-Lindemann ) If W is not contained in a proper weakly special subvariety then exp(w ) is Zariski-dense in (C ) n. Else exp(w ) V for some proper V (C ) n. Another form: Theorem ( Ax-Lindemann ) For algebraic V (C ) n, a maximal algebraic W exp 1 (V ) is weakly special.

19 Model theory: 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: an infinitary formulation of the theory C with an exponential satisfying SC and exponential algebraic closedness is well-behaved. See work of: D Aquino, Macintyre, Terzo, Shkop, Kirby, Onshuus,.. Zilber formulated CIT = ZP for (C ) n (and semiabelian vars).

20 CIT and the Uniform Schanuel Conjecture 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. Theorem (Zilber, 2002) SC plus CIT implies USC. Also USC implies CIT (and SC).

21 Model theory: real exponentiation Question (Tarski) Is Th(R exp ) decidable? Led to development of o-minimality by van den Dries and Pillay-Steinhorn. Theorem (Wilkie, 1993) Th(R exp ) is (model-complete and) o-minimal. Theorem (Macintyre-Wilkie, 1996) Assume SC. Then Th(R exp ) is decidable.

22 O-minimality R with some relations and functions e.g. <,, +, f 1,..., R j,..., and consider definable sets in this language, i.e. sets Z R n, n = 1, 2,... with a definition Z = {z R n : φ(z) holds } with φ(z) a formula built using <, +,, f i, R j, constants,, and quantifiers, ranging over R. E.g. Z = {(x, y, z) R 3 : u v ( u = exp v) (xu 2 + yu + z = v) ) }. Definition Such a structure (R, <,, +, f i, R j,...) is o-minimal if every definable subset of R is a finite union of points and intervals. E.g. fails with sin x. Key points: This condition is very strong; such a structure is tame. Examples: R exp, R an exp are o-minimal.

23 Counting rational points on algebraic varieties Example: Expect there are no non-trivial integer solutions to x 5 + y 5 = w 5 + z 5. Trivial solutions: {x, y} = {w, z}. Known that trivial solutions out-number non-trivial ones: height = max( w, x, y, z ). Hooley (1964) proved #{nontrivial solutions up to height T } ɛ T 5/3+ɛ, while #{trivial solutions up to height T } 2T 2. Conjectures of Bombieri, Lang and the philosophy of Geometry governs arithmetic.

24 Counting rational points in definable sets Bombieri-P, 1989 : results for integer points on plane curves Heath-Brown 2002: p-adic method for rational points on vars What about higher-dimensional real analytic sets Z R n? Definition Let Z R n. The algebraic part Z alg of Z is the union of connected positive dimensional semi-algebraic sets contained in Z. The transcendental part of Z is Z trans = Z Z alg Semi-algebraic = finite boolean combinations of sets defined by algebraic equations and inequations. Height: H(a/b) = max( a, b ) for a, b Z, b 0, gcd(a, b) = 1. For q Q n, H(q) = max(h(q 1 ),..., H(q n )). Counting: For Z R n set Z(Q, T ) = {z Z Q n : H(z) T }.

25 The Counting Theorem Theorem (P-Wilkie, 2006) If Z R n is definable in some o-minimal structure and ɛ > 0 then there exists c(z, ɛ) such that #Z trans (Q, T ) c(z, ɛ)t ɛ. Method: elementary analysis; o-minimal Yomdin-Gromov paramaterizations of definable sets. Algebraic points of bounded degree: Z(k, T ) = {z Z : [Q(z i ) : Q] k, H(z i ) T, i = 1,..., n}. Under same conditions: #Z trans (k, H) c(z, k, ɛ)h ɛ.

26 Special point problems Setting: π : U X, V X, say V defined over Q. Examples: e : C n (C ) n, e(z) = (e 2πiz 1,..., e 2πizn ) Invariance: Z n translations; Fundamental domain: F = F n e, F e = {z : 0 Re(z) 1}. Ingredient 1: π F is definable in R an exp Pre-image of torsion points are rational points in C n Count rational points in Z = π 1 (V ) F, definable j : H n C n, j(z) = (j(z 1 ),..., j(z n )) Invariance: SL 2 (Z) n action; Fundamental domain: F = F n j, F j = {z : Re(z) 1/2, z 1} Ingredient 1: π F is definable in R an exp Pre-image of special points: quadratic points in H n Count quadratic points in Z = π 1 (V ) F, definable

27 e : C n (C ) n, ctd Ingredient 1: π F is definable Count rational points in definable Z = π 1 (V ) F Ingredients 2 and 3: Lower (Galois) and upper (height) bounds [Q(exp(2πi/N)) : Q] = φ(n) N If ζ V has order N get N preimages in Z(Q, N) Ingredient 4: Ax-Lindemann: maximal algebraic subvarieties of π 1 (V ) are weakly special: A special case of Ax-Schanuel Theorem Final inductive argument completes proof of: V (C ) n contains only finitely many maximal torsion cosets.

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

29 j : H n C n, ctd Ingredient 1: π F is definable Count quadratic points in definable Z = π 1 (V ) F Ingredients 2 and 3: Quadratic point τ H has discriminant (τ) = b 2 4ac where ax 2 + bx + c is the minimal polynomial for τ over Z. Classical CM theory: [Q(j(τ) : Q] = h(o ), class number of the corresponding quadratic order, is large (Landau-Siegel): h(o ) (τ) 1/2. Preimage in F has height (τ). So a special point of large discriminant in V gives many quadratic points in Z, most must lie in Z alg.

30 j : H n C n, ctd Last: characterise maximal (complex) algebraic subsets of j 1 (V ). Ingredient 4: Modular Ax-Lindemann Theorem (P, 2011) A maximal algebraic W j 1 (V ) is weakly special. Method: If W j 1 (V ) then so are many of its SL 2 (Z) n -translates. Apply the Counting Theorem to certain definable subsets of SL 2 (R) n with many points from SL 2 (Z) n. Inductive argument concludes (using more o-minimality): Theorem (P, 2011) AO for C n.

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

32 O-minimality and atypical intersections Turning to broader ZP problems (AO is very special): O-minimality and point-counting are applicable to other cases: Relative MM problems of Masser-Zannier± Bertrand-Pillay Results by (combinations of) Bays, Habegger, Orr, P Special subvarieties correspond to rational points on certain Grassmanian-type varieties (e.g. parameterising subavarieties of H n defined by real Mobius transformations). Given V X = C n, say, the set of subvarieties defined by Mobius transformations which intersect the definable set Z in a set of some given (atypical) dimension is definable, and we can apply the counting theorem.

33 Conditional Modular ZP LGO: Large Galois Orbit conjecture for suitable atypical intersection points WCA: Weak Complex Ax conjecture for the j-function Theorem (Habegger-P, 201?) Assume LGO and WCA. Then ZP holds for C n, WCA follows from a Modular Ax-Schanuel conjecture. Result announced by P-Tsimerman 2014.