Darmon points for fields of mixed signature

Transcription

1 Darmon points for fields of mixed signature Number Theory Seminar, Cambridge University Xavier Guitart 1 Marc Masdeu 2 Mehmet Haluk Sengun 3 1 Institut für Experimentelle Mathematik 2,3 University of Warwick May 13, 2014 Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

2 The Hasse-Weil L-function Let F be a number field. Let E /F be an elliptic curve of conductor N = N E. Let K/F be a quadratic extension of F. Assume that N is square-free, coprime to disc(k/f ). a p (E) = 1 + p #E(F p ). Hasse-Weil L-function of the base change of E to K (R(s) >> 0) L(E/K, s) = ( 1 ap p s) 1 ( 1 a p p s + p 1 2s) 1. p N p N Assume that E is modular = Analytic continuation of L(E/K, s) to C. Functional equation relating s 2 s. Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

3 The BSD conjecture Bryan Birch Coarse version of BSD conjecture Sir Peter Swinnerton-Dyer ord s=1 L(E/K, s) = rk Z E(K). So L(E/K, 1) = 0 BSD = P K E(K) of infinite order. Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

4 The main tool for BSD: Heegner points Kurt Heegner Exist for F totally real and K/F totally complex (CM extension). I recall the definition of Heegner points in the simplest setting: F = Q (and K/Q imaginary quadratic), and Heegner hypothesis: l N = l split in K. These ensure that ord s=1 L(E/K, s) is odd (so 1). Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

5 Heegner Points (K/Q imaginary quadratic) Γ 0 (N) = { ( ) a b c d SL2 (Z): N c} Attach to E a holomorphic 1-form on H = {z C : I(z) > 0}. Φ E = 2πif E (z)dz = 2πi a n e 2πinz dz H 0 (Γ 0 (N), Ω 1 H). n 1 τ Given τ K H, set J τ = Φ E C. i { ( )} Well-defined up to the lattice Λ E = γ Φ E γ H 1 Γ 0 (N)\H, Z. There exists an isogeny (Weierstrass uniformization) η : C/Λ E E(C). Set P τ = η(j τ ) E(C). Fact: P τ E(H τ ), where H τ /K is a ring class field attached to τ. Theorem (Gross-Zagier) P K = Tr Hτ /K(P τ ) nontorsion L (E/K, 1) 0. Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

6 Heegner Points: revealing the trick Why did this work? 1 The Riemann surface Γ 0 (N)\H has an algebraic model X 0 (N) /Q. 2 There is a morphism φ defined over Q: φ: Jac(X 0 (N)) E. 3 The CM point (τ) ( ) Jac(X 0 (N))(H τ ) gets mapped to: φ((τ) ( )) = P τ E(H τ ). Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

7 Computing in practice: an example of Mark Watkins Let E be the elliptic curve of conductor N E = : E : y 2 + y = x x Watkins worked with 460 digits of precision and 600M terms of the L-series. Took less than a day (in 2006). The x-coordinate of the point has numerator: Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

8 Darmon s insight Henri Darmon Drop hypothesis of K/F being CM. Simplest case: F = Q, K real quadratic. However: There are no points on Jac(X 0 (N)) attached to such K. In general there is no morphism φ: Jac(X 0 (N)) E. When F is not totally real, even the curve X0 (N) is missing! Nevertheless, Darmon constructed local points in such cases and hoped that they were global. Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

9 Goals of this talk 1 Review some history. 2 Sketch a general construction of Darmon points. 3 Give some details of the construction. 4 Explain the algorithmic challenges we face in their computation. The fun of the subject seems to me to be in the examples. B. Gross, in a letter to B. Birch, Illustrate with fun examples. Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

10 Basic notation Consider an infinite place v F of F. If v is real, then: 1 It may extend to two real places of K (splits), or 2 It may extend to one complex place of K (ramifies). If v is complex, then it extends to two complex places of K (splits). n = #{v F : v splits in K}. K/F is CM n = 0. If n = 1 we call K/F quasi-cm. S(E, K) = { v N F : v not split in K }, s = #S(E, K). Sign of functional equation for L(E/K, s) should be ( 1) #S(E,K). From now on, we assume that s is odd. Fix a place ν S(E, K). 1 If ν = p is finite = non-archimedean case. 2 If ν is infinite = archimedean case. Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

11 Goals of this talk 1 Review some history. 2 Sketch a general construction of Darmon points. 3 Give some details of the construction. 4 Explain the algorithmic challenges we face in their computation. The fun of the subject seems to me to be in the examples. B. Gross, in a letter to B. Birch, Illustrate with fun examples. Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

12 Non-archimedean History These constructions are also known as Stark-Heegner points. H. Darmon (1999): F = Q, quasi-cm, s = 1. Darmon-Green (2001): special cases, used Riemann products. Darmon-Pollack (2002): same cases, overconvergent methods. Guitart-M. (2012): all cases, overconvergent methods. M. Trifkovic (2006): F imag. quadratic ( = quasi-cm)), s = 1. Trifkovic (2006): F euclidean, E of prime conductor. Guitart-M. (2013): F arbitrary, E arbitrary. M. Greenberg (2008): F totally real, arbitrary ramification, s 1. Guitart-M. (2013): F = Q, quasi-cm case, s 1. Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

13 Archimedean History Initially called Almost Totally Real (ATR) points. But this name only makes sense in the original setting of Darmon. H. Darmon (2000): F totally real, s = 1. Darmon-Logan (2003): F quadratic norm-euclidean, NE trivial. Guitart-M. (2011): F quadratic and arbitrary, N E trivial. Guitart-M. (2012): F quadratic and arbitrary, N E arbitrary. J. Gartner (2010): F totally real, s 1.? Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

14 Goals of this talk 1 Review some history. 2 Sketch a general construction of Darmon points. 3 Give some details of the construction. 4 Explain the algorithmic challenges we face in their computation. The fun of the subject seems to me to be in the examples. B. Gross, in a letter to B. Birch, Illustrate with fun examples. Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

15 Our construction Xavier Guitart M. Haluk Sengun Available for arbitrary base number fields F (mixed signature). Comes in both archimedean and non-archimedean flavors. All of the previous constructions become particular cases. We can provide genuinely new numerical evidence. Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

16 Overview of the construction We define a quaternion algebra B /F and a group Γ SL 2 (F ν ). The group Γ acts (non-discretely in general) on H ν. We attach to E a cohomology class Φ E H n ( Γ, Ω 1 H ν ). We attach to each embedding ψ : K B a homology class Θ ψ H n ( Γ, Div 0 H ν ). Well defined up to the image of Hn+1 (Γ, Z) δ H n (Γ, Div 0 H ν ). Cap-product and integration on the coefficients yield an element: J ψ = Θ ψ, Φ E K ν. J ψ is well-defined up to a multiplicative lattice { } L = δ(θ), Φ E : θ H n+1 (Γ, Z). Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

17 Conjectures J ψ = Θ ψ, Φ E K ν /L. Conjecture 1 (Oda, Yoshida, Greenberg, Guitart-M-Sengun) There is an isogeny β : K ν /L E(K ν ). Dasgupta Greenberg, Rotger Longo Vigni: some non-arch. cases. Completely open in the archimedean case. The Darmon point attached to E and ψ : K B is: P ψ = β(j ψ ) E(K ν ). Conjecture 2 (Darmon, Greenberg, Trifkovic, Gartner, G-M-S) 1 The local point P ψ is global, and belongs to E(K ab ). 2 P ψ is nontorsion if and only if L (E/K, 1) 0. We predict also the exact number field over which P ψ is defined. Include a Shimura reciprocity law like that of Heegner points. Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

18 Aside: an interesting by-product Let Φ H n ( Γ, Ω 1 H ν ) be an eigenclass with integer eigenvalues. In favorable situations Φ comes from an elliptic curve E over F. No systematic construction of such curves for non totally real F. We can compute the lattice unram. quadratic ext. of F ν. { } L = δ(θ), Φ : θ H n+1 (Γ, Z) F. ν 2 Suppose that Conjecture 1 is true. From L one can find a Weierstrass equation E ν (F ν 2) = F ν 2 /L. Hopefully the equation can be descended to F. A similar technique (in the archimedean case) used by L. Dembélé to compute equations for elliptic curves with everywhere good reduction. Stay tuned! Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

19 Goals of this talk 1 Review some history. 2 Sketch a general construction of Darmon points. 3 Give some details of the construction. 4 Explain the algorithmic challenges we face in their computation. The fun of the subject seems to me to be in the examples. B. Gross, in a letter to B. Birch, Illustrate with fun examples. Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

20 The group Γ Let B /F = quaternion algebra with Ram(B) = S(E, K) {ν}. B = M2 (F ) (split case) s = 1. Otherwise, we are in the quaternionic case. E and K determine a certain {ν}-arithmetic subgroup Γ SL 2 (F ν ): Let m = l N, split in K l. Let R D 0 (m) be an Eichler order of level m inside B. Fix an embedding ιν : R D 0 (m) M 2 (Z F,ν ). ) Γ = ι ν (R0 D (m)[1/ν] 1 SL 2 (F ν ). e.g. S(E, K) = {p} and ν = p give Γ SL2 ( O F [ 1 p ] ). e.g. S(E, K) = { } and ν = give Γ SL2 (O F ). Remark: We also write Γ D 0 (m) = RD 0 (m) 1. Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

21 Path integrals: archimedean setting H = (P 1 (C) P 1 (R)) + has a complex-analytic structure. SL 2 (R) acts on H through fractional linear transformations: ( a b ) az + b c d z = cz + d, z H. We consider holomorphic 1-forms ω Ω 1 H. Given two points τ 1 and τ 2 in H, define: τ2 τ 1 ω = usual path integral. Compatibility with the action of SL 2 (R) on H: γτ2 γτ 1 ω = τ2 τ 1 γ ω. Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

22 Path integrals: non-archimedean setting H p = P 1 (K p ) P 1 (F p ) has a rigid-analytic structure. SL 2 (F p ) acts on H p through fractional linear transformations: ( a b ) az + b c d z = cz + d, z H p. We consider rigid-analytic 1-forms ω Ω 1 H p. Given two points τ 1 and τ 2 in H p, define: τ2 τ 1 ω = Coleman integral. Compatibility with the action of SL 2 (F p ) on H p : γτ2 γτ 1 ω = Q P γ ω. Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

23 Coleman Integration Coleman integration on H p can be defined as: τ2 ( ) t τ2 ω = log p dµ ω (t) = lim τ 1 P 1 (F p) t τ 1 U U U log p ( tu τ 2 t U τ 1 ) res A(U) (ω). Bruhat-Tits tree of GL 2 (F p ), p = 2. H p having the Bruhat-Tits as retract. Annuli A(U) for a covering of size p 3. t U is any point in U P 1 (F p ). U P 1 (Fp) P 1 (F p) If res A(U) (ω) Z for all U, then have a multiplicative refinement: τ2 ( ) tu τ resa(u) (ω) 2 ω = lim K U p. τ 1 t U τ 1 U U Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

24 Cohomology Recall that S(E, K) and ν determine: ) Γ = ι ν (R0 D (m)[1/ν] 1 SL 2 (F ν ). Choose signs at infinity ε 1,..., ε n {±1}. Theorem (Darmon, Greenberg, Trifkovic, Gartner, G. M. S.) There exists a unique (up to sign) class Φ E H n ( Γ, Ω 1 ) H ν such that: 1 T q Φ E = a q Φ E for all q N. 2 U q Φ E = a q Φ E for all q N. 3 W σi Φ E = ε i Φ E for all embeddings σ i : F R which split in K. 4 Φ E is integrally valued. Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

25 Homology Let ψ : O R0 D (m) be an embedding of an order O of K. Which is optimal: ψ(o) = R D 0 (m) ψ(k). Consider the group O 1 = {u O : Nm K/F (u) = 1}. rank(o 1 ) = rank(o ) rank(o F ) = n. Choose a basis u 1,..., u n O 1 for the non-torsion units. ψ = ψ(u 1 ) ψ(u n ) H n (Γ, Z). ψ K acts on H ν through K B ιν GL 2 (F ν ). Let τψ be the (unique) fixed point of K on H ν. Have the exact sequence δ H n+1 (Γ, Z) H n (Γ, Div 0 H ν ) H n (Γ, Div H ν ) deg H n (Γ, Z) Θ ψ? [ ψ τ ψ ] [ ψ ] Fact: [ ψ ] is torsion. Can pull back a multiple of [ ψ τ ψ ] to Θ ψ H n (Γ, Div 0 H ν ). Well defined up to δ(hn+1 (Γ, Z)). Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

26 Goals of this talk 1 Review some history. 2 Sketch a general construction of Darmon points. 3 Give some details of the construction. 4 Explain the algorithmic challenges we face in their computation. The fun of the subject seems to me to be in the examples. B. Gross, in a letter to B. Birch, Illustrate with fun examples. Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

27 Cycle Decomposition Goal H 2 (Γ, Z) δ H 1 (Γ, Div 0 H p ) H 1 (Γ, Div H p ) deg H 1 (Γ, Z) Θ ψ? [γ ψ τ ψ ] [γ ψ ] Theorem (word problem) Given a presentation F Γ giving Γ = g 1,..., g s r 1,..., r t, There is an algorithm to write γ Γ as a word in the g i s. Effective version for quaternionic groups: John Voight, Aurel Page. γ [Γ, Γ] = γ has word representation W, with W [F, F ]. We use gh D g D + h g 1 D (modulo 1-boundaries). Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

28 Cycle Decomposition: example G = R 1, R maximal order on B = B 6. F = X, Y G = x, y x 2 = y 3 = 1. Goal: write g τ as g i D i, with D i of degree 0. Take for instance g = yxyxy. Note that wt(x) = 2 and wt(y) = 3. First, trivialize in F ab : g = yxyxyx 2 y 3. To simplify γ τ 0 in H 1 (Γ, Div H p ), use: 1 gh D g D + h g 1 D. 2 g 1 D g gd. Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

29 Overconvergent Method (I) (F = Q, p = fixed prime) We have attached to E a cohomology class Φ H 1 (Γ, Ω 1 H p ). Goal: to compute integrals τ 2 τ 1 Φ γ, for γ Γ. Recall that τ2 ( ) t τ1 Φ γ = log p dµ γ (t). τ 1 P 1 (Q p) t τ 2 Expand the integrand into power series and change variables. We are reduced to calculating the moments: t i dµ γ (t) for all γ Γ. Z p Note: Γ Γ D 0 (m) ΓD 0 (pm). Technical lemma: All these integrals can be recovered from { } Z p t i dµ γ (t): γ Γ D 0 (pm). Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

30 Overconvergent Method (II) D = {locally analytic Z p -valued distributions on Z p }. ϕ D maps a locally-analytic function h on Z p to ϕ(h) Z p. D is naturally a Γ D 0 (pm)-module. The map ϕ ϕ(1 Zp ) induces a projection: ρ: H 1 (Γ D 0 (pm), D) H 1 (Γ D 0 (pm), Z p ). Shapiro s lemma allows to associate ϕ E H 1 (Γ D 0 (pm), Z p) to Φ E : ϕ E (γ) = µ γ (t). Z p Theorem (Pollack-Stevens, Pollack-Pollack) There exists a unique U p -eigenclass Φ lifting ϕ E. Moreover, Φ is explicitly computable by iterating the U p -operator. Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

31 Overconvergent Method (III) But we wanted to compute the moments of a system of measures... Proposition Consider the map Ψ: Γ D 0 (pm) D: [ ] γ h(t) h(t)dµ γ (t). Z p 1 Ψ belongs to H 1 (Γ D 0 (pm), D). 2 Ψ is a lift of µ. 3 Ψ is a U p -eigenclass. Corollary The explicitly computed Φ knows the above integrals. Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

32 Examples Where are the examples?? Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

33 Non-archimedean cubic Darmon point (I) F = Q(r), with r 3 r 2 r + 2 = 0. F has signature (1, 1) and discriminant 59. Consider the elliptic curve E /F given by the equation: E /F : y 2 + ( r 1) xy + ( r 1) y = x 3 rx 2 + ( r 1) x. E has conductor N E = ( r ) = p 17 q 2, where p 17 = ( r 2 + 2r + 1 ), q 2 = (r). Consider K = F (α), where α = 3r 2 + 9r 6. The quaternion algebra B/F has discriminant D = q 2 : B = F i, j, k, i 2 = 1, j 2 = r, ij = ji = k. Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

34 Non-archimedean cubic Darmon point (II) The maximal order of K is generated by w K, a root of the polynomial x 2 + (r + 1)x + 7r2 r One can embed O K in the Eichler order of level p 17 by: w K ( r 2 + r)i + ( r + 2)j + rk. We obtain γ ψ = 6r r+3 2 i + 2r2 +3r 2 j + 5r2 7 2 k, and τ ψ = (12g+8)+(7g+13)17+(12g+10)17 2 +(2g+9)17 3 +(4g+2) After integrating we obtain: J ψ = O(17 21 ), which corresponds to: P ψ = 3 ( 2 72 r 1, α + ) r2 + r E(K). 2 Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

35 Archimedean cubic Darmon point (I) Let F = Q(r) with r 3 r = 0. F has discriminant 23, and is of signature (1, 1). Consider the elliptic curve E /F given by the equation: E /F : y 2 + (r 1) xy + ( r 2 r ) y = x 3 + ( r 2 1 ) x 2 + r 2 x. E has prime conductor N E = ( r ) of norm 89. K = F (w), with w 2 + (r + 1) w + 2r 2 3r + 3 = 0. K has class number 1, thus we expect the point to be defined over K. The computer tells us that rkz E(K) = 1 S(E, K) = {σ}, where σ : F R is the real embedding of F. Therefore the quaternion algebra B is just M 2 (F ). The arithmetic group to consider is Hyperbolic 3-space Γ = Γ 0 (N E ) SL 2 (O F ). Γ acts naturally on the symmetric space H H 3 : H H 3 = {(z, x, y): z H, x C, y R >0 }. Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

36 Archimedean cubic Darmon point (II) E ω E, an automorphic form with Fourier-Bessel expansion: ω E (z, x, y) = a (δα) (E)e 2πi(α 0 z+α 1 x+α 2 x) yh (α 1 y) H(t) = α δ 1 O F α 0 >0 ( i 2 eiθ K 1 (4πρ), K 0 (4πρ), i 2 e iθ K 1 (4πρ) ) ( dx d z dy d z d x d z t = ρe iθ. ) K0 and K 1 are hyperbolic Bessel functions of the second kind: K 0 (x) = e x cosh(t) dt, K 1 (x) = 0 0 e x cosh(t) cosh(t)dt. ω E is a 2-form on Γ\ (H H 3 ). The cocycle Φ E is defined as (γ Γ): Φ E (γ) = γ O O ω E (z, x, y) Ω 1 H with O = (0, 1) H 3. Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

37 Archimedean cubic Darmon point (III) Consider the embedding ψ : K M 2 (F ) given by: ( 2r w 2 ) + 3r r 3 r r 2 4r 1 Let γ ψ = ψ(u), where u is a fundamental norm-one unit of O K. γ ψ fixes τ ψ = i H. Construct Θ ψ = [γ ψ τ ψ ] H 1 (Γ, Div H). Θ ψ is equivalent to a cycle γ i (s i r i ) taking values in Div 0 H. J ψ = i si r i Φ E (γ i ) = i γi O si O r i ω E (z, x, y). We obtain, summing over all ideals (α) of norm up to 400, 000: J ψ = i P ψ E(C). Numerically (up to 32 decimal digits) we obtain: P ψ? = 10 ( r 1, w r 2 + 2r ) E(K). Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

38 Thank you! Bibliography, code and slides at: Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

39 Bibliography H. Darmon and A. Logan. Periods of Hilbert modular forms and rational points on elliptic curves. Int. Math. Res. Not. (2003), no. 40, H. Darmon and P. Green. Elliptic curves and class fields of real quadratic fields: Algorithms and evidence. Exp. Math., 11, No. 1, 37-55, H. Darmon and R. Pollack. Efficient calculation of Stark-Heegner points via overconvergent modular symbols. Israel J. Math., 153: , J. Gärtner. Darmon points and quaternionic Shimura varieties. Canad. J. Math. 64 (2012), no. 6. X. Guitart and M. Masdeu. Elementary matrix Decomposition and the computation of Darmon points with higher conductor. Math. Comp. (arxiv.org, ), X. Guitart and M. Masdeu. Computation of ATR Darmon points on non-geometrically modular elliptic curves. Exp. Math., X. Guitart and M. Masdeu. Computation of quaternionic p-adic Darmon points. (arxiv.org, ), X. Guitart, M. Masdeu and M.H. Sengun. Darmon points on elliptic curves over number fields of arbitrary signature. (arxiv.org, ), M. Greenberg. Stark-Heegner points and the cohomology of quaternionic Shimura varieties. Duke Math. J., 147(3): , D. Pollack and R. Pollack. A construction of rigid analytic cohomology classes for congruence subgroups of SL 3 (Z). Canad. J. Math., 61(3): , M. Trifkovic. Stark-Heegner points on elliptic curves defined over imaginary quadratic fields. Duke Math. J., 135, No. 3, , Marc Masdeu Darmon points for fields of mixed signature May 13, / 35

40 Available Code SAGE code for non-archimedean Darmon points when n = 1. Compute with quaternionic modular symbols. Need presentation for units of orders in B (J. Voight, A. Page). Implemented overconvergent method for arbitrary B. We obtain a method to find algebraic points. SAGE code for archimedean Darmon points (in restricted cases). Only for the split (B = M 2 (F )) cases, and: 1 F real quadratic, and K/F ATR (Hilbert modular forms) 2 F cubic (1, 1), and K/F totally complex (cubic automorphic forms). Marc Masdeu Darmon points for fields of mixed signature May 13, / 1