p-adic iterated integrals and rational points on curves

Transcription

1 Rational points on curves p-adic and computational aspects p-adic iterated integrals and rational points on curves Henri Darmon Oxford, September 28, 2012

2 (Joint with Alan Lauder and Victor Rotger )

3 Also based on work with Bertolini and Kartik Prasanna

4 Rational points on elliptic curves Let E be an elliptic curve over a number field K. BSD0: If L(E/K, 1) 0, then E(K) is finite. BSD1: If ord s=1 L(E/K, s) = 1, then rank(e(k)) = 1. BSDr: If ord s=1 L(E/K, s) = r, then rank(e(k)) = r. We will have nothing to say (hélas) about BSDr when r > 1. (For this see John Voight s lecture on Monday.)

5 Equivariant BSD Let E be an elliptic curve over Q; Let ρ : Gal(K/Q) GL n (C) be an Artin representation. Equivariant BSD conjecture (BSD ρ,r ). ord s=1 L(E, ρ, s) = r? dim C hom GQ (V ρ, E(K) C) = r. Question: if L(E, ρ, s) has a simple zero, produce the point in the ρ-isotypic part of Mordell-Weil predicted by the BSD conjecture.

6 Heegner points K = imaginary quadratic field, χ a ring class character of K, ρ = Ind Q K χ. Theorem. (Gross-Zagier, Kolyvagin, Bertolini-D, Zhang, Longo-Rotger-Vigni, Nekovar,...) BSD ρ,0 and BSD ρ,1 are true. Key ingredient in the proof: the collection of Heegner points on E defined over various ring class fields of K.

7 More general geometric constructions If V is a modular variety equipped with a (preferably infinite) supply { } of interesting algebraic cycles of codimension j, and Π : V E is a correspondence, inducing we may study the points Π( ). Π : CH j (V ) 0 E, Bertolini-Prasanna-D: Generalised Heegner cycles in V = W r A s ; (cf. Kartik Prasanna s lecture on Monday); Zhang, Rotger-Sols-D: Diagonal cycles, or more interesting exceptional cycles, on V = W r W s W t ; (cf. Victor Rotger s lecture on Thursday).

8 Stark-Heegner points K = real quadratic field, χ a ring class character of K, ρ = Ind Q K χ. Stark-Heegner points: Local points in E(C p ) defined (conjecturally) over the field H cut out by χ. They can be computed in practice, (cf. the lecture of Xavier Guitart on Monday reporting on his recent work with Marc Masdeu). Conjecture. L(E, ρ, s) has a simple zero at s = 1 if and only if hom(v ρ, E(H) C) is generated by Stark-Heegner points.

9 Stark-Heegner points The completely conjectural nature of Stark-Heegner points prevents a proof of BSD ρ,0 and BSD ρ,1 along the lines of the proof of Kolyvagin-Goss-Zagier when ρ is induced from a character of a real quadratic field. Goal: Describe a more indirect approach whose goal is to 1 Prove BSD ρ,0. 2 Construct the global cohomology classes κ E,ρ H 1 (Q, V p (E) V ρ ) which ought to arise from Stark-Heegner points via the connecting homomorphism of Kummer theory.

10 Λ-adic cohomology classes A Λ-adic Galois representation is a finite free module V over Λ equipped with a continuous action of G Q. Specialisations: ξ W := hom cts (Z p, C p ) = hom cts (Λ, C p ), ξ : V V ξ := V Λ,ξ Q p,ξ. Suppose there is a dense set of points Ω geom W and, for each ξ Ω geom, a class κ ξ H 1 fin (Q, V ξ). Definition The collection {κ ξ } ξ Ωgeom κ H 1 (Q, V ) such that interpolates p-adically if there exists ξ(κ) = κ ξ, for all ξ Ω geom.

11 p-adic limits of geometric constructions Suppose that V p (E) = H 1 et(e, Q p )(1) arises as the specialisation ξ E : V V p (E) for some ξ E not necessarily belonging to Ω geom. One may then consider the class κ E := ξ E (κ) H 1 (Q, V p (E)) and attempt to relate it to L(E, 1) and to the arithmetic of E. The class κ E is a p-adic limit of geometric classes, but need not itself admit a geometric construction.

12 p-adic limits of geometric constructions Coates-Wiles: V is induced from a family of Hecke characters of a quadratic imaginary field, Ω geom = {finite order Hecke characters }, the κ ξ arise from the images of elliptic units under the Kummer map, and ξ E corresponds to a Hecke character of infinity type (1, 0) attached to a CM elliptic curve E. Kato. V = V p (E)(1) Λ cyc, Ω geom = { finite order χ : Z p C p }, the κ χ H 1 (Q, V p (E)(1)(χ)) arise from the images of Beilinson elements in K 2 (X 1 (Np s )) (s =cond(χ)), and ξ E = 1.

13 The Perrin-Riou philosophy Perrin-Riou. p-adic families of global cohomology classes are a powerful tool for studying p-adic L-functions. I will illustrate this philosophy in the following contexts: 1 Classes arising from Beilinson-Kato elements, and the Mazur-Swinnerton-Dyer p-adic L-function (as described in Massimo Bertolini s lecture); 2 Classes arising from diagonal cycles and the Harris-Tilouine triple product p-adic L-function (as discussed in Victor Rotger s lecture).

14 Modular units Manin-Drinfeld: the group O Y 1 (N) /C has maximal possible /C rank, namely #(X 1 (N) Y 1 (N)) 1. The logarithmic derivative gives a surjective map dlog : O Y 1 (N) Q Eis /Q(µN 2(Γ 1 (N), Q) ) to the space of weight two Eisenstein series. Let u χ O Y 1 (N) Q χ be the modular unit characterised by dlog u χ = G 2,χ, G 2,χ = 2 1 L(χ, 1) + σ χ (n)q n, σ χ (n) = χ(d)d. n=1 d n

15 Beilinson elements Given χ of conductor Np s, α χ := δ(u χ ) Het(X 1 1 (Np s ), Z p (1)), β χ := δ(w ζ u χ ) Het(X 1 1 (Np s ) Q(µNp s ), Z p (1)) κ χ := α χ β χ Het(X 2 1 (Np s ) Q(µNp s ), Z p (2)), κ χ := its image in H 1 (Q(µ Np s ), Het(X 1 1 (Np s ) Q, Z p(2))). The latter descends to a class κ χ H 1 (Q, Het(X 1 1 (Np s ) Q, Z p(2))(χ 1 )). Let X 1 (N) E be a modular elliptic curve, and κ E (G 2,χ, G 2,χ ) H 1 (Q, V p (E)(1)(χ 1 )) be the natural image.

16 Kato s Λ-adic class Key Remark: The Eisenstein series G 2,χ0 χ (with f χ = p s ) are the weight two specialisations of a Hida family G χ0. Theorem (Kato) There is a Λ-adic cohomology class satisfying κ E (G χ0, G χ0 ) H 1 (Q, V p (E)(χ 0 ) Λ cyc ( 1)), ξ 2,χ (κ E (G χ0, G χ0 )) = κ E (G 2,χ0 χ, G 2,χ0 χ) at all weight two specialisations ξ 2,χ.

17 The Kato Perrin-Riou class We can now specialise the Λ-adic cohomology class κ E (G χ0, G χ0 ) to Eisenstein series of weight one. κ E (G 1,χ0, G 1,χ0 ) := ν 1 (κ E (G χ0, G χ0 )). Theorem (Kato) The class κ E (G 1,χ0, G 1,χ0 ) is cristalline if and only if L(E, 1)L(E, χ 1 0, 1) = 0. Corollary BSD χ,0 is true for E.

18 Hida families To prove BSD ρ,0 for larger classes of ρ, we will 1 replace the Beilinson elements κ E (G 2,χ, G 2,χ ) H 1 (Q, V p (E)(1)(χ 1 )) by geometric elements κ E (g, h) H 1 (Q, V p (E) V g V h (k 1)) attached to a pair of cusp forms g and h of the same weight k 2. 2 Interpolate these classes in Hida families κ E (g, h). 3 Consider the weight one specialisations κ E (g 1, h 1 ) H 1 (Q, V p (E) V ρg1 V ρh1 ). Of special interest is the case where ρ g1 and ρ g2 are Artin representations.

19 Gross-Kudla-Schoen diagonal classes étale Abel-Jacobi map: AJ et : CH 2 (X 1 (N) 3 ) 0 Het(X 4 1 (N) 3, Q p (2)) 0 H 1 (Q, Het(X 3 1 (N) 3, Q p (2))) H 1 (Q, Het(X 1 1 (N), Q p ) 3 (2)) Gross-Kudla Schoen class: κ E (g, h) := AJ et ( ) f,g,h H 1 (Q, V p (E) V g V h (1)).

20 Hida Families Weight space: Ω := hom(λ, C p ) hom((1 + pz p ), C p ). The integers form a dense subset of Ω via k (x x k ). Classical weights: Ω cl := Z 2 Ω. If Λ is a finite flat extension of Λ, let X = hom( Λ, C p ) and let κ : X Ω be the natural projection to weight space. Classical points: X cl := {x X such that κ(x) Ω cl }.

21 Hida families, cont d Definition A Hida family of tame level N is a triple (Λ, Ω, g), where 1 Λ g is a finite flat extension of Λ; 2 Ω g X g := hom(λ g, C p ) is a non-empty open subset (for the p-adic topology); 3 g = n a nq n Λ g [[q]] is a formal q-series, such that g(x) := n x(a n)q n is the q series of the ordinary p-stabilisation g x (p) of a normalised eigenform, denoted g x, of weight κ(x) on Γ 1 (N), for all x Ω g,cl := Ω g X g,cl.

22 Λ-adic Galois representations If g and h are Hida families, there are associated Λ-adic Galois representations V g and V h of rank two over Λ g and Λ h respectively (cf. Adrian Iovita s lecture on Thursday).

23 A p-adic family of global classes Theorem (Rotger-D) Let g and h be two Hida families. There is a Λ g Λ Λ h -adic cohomology class κ E (g, h) H 1 (Q, V p (E) (V g Λ V h ) Λ Λ cyc ( 1)), where V g, V h = Hida s Λ-adic representations attached to g and h, satisfying, for all weight two points (y, z) Ω g Ω h, ξ y,z (κ E (g, h)) = κ E (g y, g z ). This Λ-adic class generalises Kato s class, which one recovers when g and h are Hida families of Eisenstein series.

24 Generalised Kato Classes Lei, Loeffler and Zerbes are studying similar families of twisted Beilinson-Flach elements. There is a strong parallel between the three settings: 1 Beilinson-Kato elements, leading to the Kato class κ E (G 1,χ, G 1,χ ) H 1 (Q, V p (E)(χ 1 )); 2 Twisted diagonal cycles, leading to classes κ E (g, h) H 1 (Q, V p (E) V g V h ) where g and h are cusp forms of weight one with det(v g V h ) = 1; 3 The twisted Beilinson-Flach elements in David Loeffler s lecture, leading to classes κ E (g, G 1,χ ) H 1 (Q, V p (E) V g ), where V g is a not-necessarily-self-dual representation. All three will be called generalised Kato classes for E.

25 A reciprocity law for diagonal cycles As in Kato s reciprocity law, one can consider the specialisations of κ E (g, h) when g and h are evaluated at points of weight one. Theorem (Rotger-D; still in progress) Let (y, z) Ω g Ω h be points with wt(y) = wt(z) = 1. The class κ E (g y, h z ) is cristalline if and only if L(V p (E) g y h z, 1) = 0. Main ingredients: 1. The p-adic Gross-Zagier formula described in Rotger s lecture; 2. Perrin-Riou s theory of Bloch-Kato logarithms and dual exponential maps in p-adic families.

26 BSD ρ in analytic rank zero. Corollary Let E be an elliptic curve over Q and ρ 1, ρ 2 odd irreducible two-dimensional Galois representations. Then BSD ρ1 ρ 2,0 is true for E. Proof. Use the ramified class κ E (g, h) H 1 (Q, V p (E) ρ 1 ρ 2 ) to bound the image of the global points in the local points. Corollary Let χ be a dihedral character of a real quadratic field K, and let ρ = Ind Q K χ. Then BSD ρ,0 is true. Proof. Specialise to the case ρ 1 = Ind Q F χ 1 and ρ 2 = Ind Q F χ 2.

27 Analytic rank one, and Stark-Heegner points? Question. Assume that 1 g and h are attached to classical modular forms, and hence to Artin representations ρ g and ρ h ; 2 L(E, ρ g ρ h, 1) = 0, so that κ E (g, h) is cristalline. Project with Lauder and Rotger: Give an explicit, computable formula for log p (κ E (g, h)) (Ω 1 (E/Q p ) D(V ρg ) D(V ρh )). This would be useful both for theoretical and experimental purposes.

28 Perrin-Riou s formula for the log of the Kato class Recall there are two Mazur-Swinnerton-Dyer p-adic L-functions: L p,α (E/Q, s) and L p,β (E/Q, s) x 2 a p x + p = (x α)(x β), ord p (α) ord p (β). ord p (β) = 1: Kato-Perrin-Riou; Pollack-Stevens. (Cf. Bellaiche.) L p, (E, s) := ( 1 1 ) 2 ( L p,α (E, s) L p,β (E, s). β α)

29 Perrin-Riou s formula for the Kato class Theorem (Perrin-Riou) If L(E, χ, 1) = 0, there exists ω Ω 1 (E/Q) such that L α β p, (E, χ, 1) = [ϕω, ω] log p(κ E (G 1,χ, G 1,χ )). Conjecture (Perrin-Riou) If L p,α (E, 1) = 0, there exists a point P E(Q) and ω Ω 1 (E/Q) such that L α β p, (E, 1) = [ϕω, ω] log2 ω(p).

30 Experimental evidence Numerical verifications have been carried out by Bernardi and Perrin-Riou, and pushed further by M. Kurihara and R. Pollack using the Pollack-Stevens theory of overconvergent modular symbols to compute log p (κ E (G 1,χ, G 1,χ )) p-adically when p is a supersingular prime. Example: The curve X 0 (17) is supersingular at p = 3. X 0 (17) 193 : y 2 + xy + y = x 3 x x (x, y) = ( , ) See M. Kurihara and R. Pollack, Two p-adic L-functions and rational points on elliptic curves with supersingular reduction, L-Functions and Galois Representations (Durham, 2007), , London Math Society LNS 320.

31 The logarithms of the generalised Kato classes Main idea: The p-adic logarithms of the (generalised) Kato classes should be expressed a limits of p-adic iterated integrals. Alan Lauder has devised highly efficient algorithms to compute these iterated integrals numerically. Caveat: The p-adic integrals that will be introduced in this talk are very different from the ones that arose in the lecture of Jennifer Balakrishnan on Monday. We are a bit baffled by this last fact.

32 The cohomological interpretation of modular forms A modular form g of weight k = 2 + r 2 can be interpreted as an element ω g H 0 (X, ω r Ω 1 X ), X = X 1(N). Gauss-Manin connection: ω r L r = sym r HdR 1 (E/X ). 0 L r Lr Ω 1 X Hodge filtration exact sequence 0. 0 H 0 (X, ω r Ω 1 X ) H1 dr (X, L r, ) H 1 (X, ω r ) 0.

33 p-adic modular forms p a prime not dividing N; A X the ordinary locus; W A a wide open neighborhod of this affinoid region. H 1 dr (X, L r, ) = H0 (W, L r Ω 1 X ) H 0 (W, L r ) = = H 0 (W, ω r Ω 1 X ) H 1 (W, L r ) H 0 (W, ω r Ω 1 X ) Mk oc(n) d 1+r M r oc (N).

34 The d operator Here d = q d dq d j ( n is the d operator on p-adic modular forms. a n q n n ) = nj a n q n if j 0; p n nj a n q n if j < 0. Fact: If g M oc k (N), then d 1 k g M oc 2 k (N).

35 p-adic iterated integrals: Type I Suppose γ H 1 dr (X /Q p), and g, h M k (N) (k = r + 2 2). Definition The p-adic iterated integral of g and h along γ is ω g ω h := γ, d 1 k g h X, γ where, X is the Poincaré duality on H 1 dr (X /Q p) = H 1 rig (W). Key fact: If γ HdR 1 (X )ur, the unit root subspace, then ω g ω h = γ, e ord (d 1 k g h) X, γ where e ord is Hida s ordinary projector.

36 p-adic iterated integrals: Type II Suppose γ H 1 dr (X, L r, ), and that f M 2 (N), g M k (N) (k = r + 2 2). Definition The p-adic iterated integral of f and g along γ is ω f ω g := η, d 1 f g r, γ where, r is the Poincaré duality on H 1 dr (X, L r, ).

37 Logarithms of Generalised Kato classes Consider these classes in the range of geometric interpolation : f is of weight two, attached to an elliptic curve E, and g and h are of weight k = r (and level prime to p). Then κ E (g, h) H 1 f (Q, V p(e) V g V h (r + 1)). Bloch-Kato logarithm: log p (κ E (g, h)) Fil 2r+3 (HdR 1 (X ) Hr+1 dr (W r ) H r+1 dr (W r )). Theorem (Rotger, D) 1. log p (κ E (g, h))(η ur f ω g ω h ) = η ur f ω g ω h. (This is an iterated integral of Type I.) 2. log p (κ E (g, h))(ω f ω g η ur h ) = η ur h ω f ω g. (This is an iterated integral of type II.)

38 The p-adic logarithms of generalised Kato classes Suppose now that g and h are of weight one, and that L(E, ρ g ρ h, 1) = 0, so that κ E (g, h) is cristalline. Goals of the current project with Lauder and Rotger. 1. Express the p-adic logarithms of the generalised Kato classes κ E (g, h) when g and h are of weight one in terms of (p-adic limits of) p-adic iterated integrals. 2. Compute these logarithms using Alan Lauder s fast algorithms for computing ordinary projections.

39 A final question When g and h are Eisenstein series of weight one, the relation between log p (κ E (G 1,χ, G 1,χ )) and p-adic iterated integrals suggests a strategy for proving Perrin-Riou s conjecture: log p (κ E (G 1,χ, G 1,χ ))? = log 2 p(p E,χ ), as described in Bertolini s lecture of yesterday. Question: When ρ g and ρ h are induced from characters of a real quadratic field K, show that log p (κ E (g, h)) = log 2 p(p g,h ), where P g,h are Stark-Heegner points attached to E and K. This would relate (a priori purely local, and poorly understood) Stark-Heegner points to the global classes κ E (g, h).

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