Computations with Coleman integrals

Transcription

1 Computations with Coleman integrals Jennifer Balakrishnan Harvard University, Department of Mathematics AWM 40 Years and Counting (Number Theory Session) Saturday, September 17, 2011

2 Outline 1 Introduction p-adic (Coleman) integrals Hyperelliptic curves 2 Results Integrals that compute heights (joint with Besser) Application: a p-adic BSD conjecture (joint with Müller, Stein) 3 Conclusion Future work

3 Introduction p-adic (Coleman) integrals What can Coleman integrals tell us? Different integrals, different things: Experiments with Chabauty s method: find such that 0 ω = 0 Torsion points on curves (Coleman s original application, for curves of g > 1) Kim s nonabelian Chabauty method: use z b ω 0ω 1 to recover integral points on elliptic curves p-adic heights on curves: h p (D 1, D 2 ) = D 2 ω D1 Syntomic regulators on curves: for {f, g} K 2 (C), reg p ({f, g})(ω) = (f ) log(g)ω p-adic polylogarithms and multiple zeta values, following Besser-de Jeu Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 1 / 22

4 Introduction Hyperelliptic curves Why hyperelliptic curves? A hyperelliptic curve is a smooth projective curve C/K (char K 2) of genus g 1 with an affine model with deg f (x) 2g + 2. Why hyperelliptic curves? y 2 = f (x), Key step in the calculation of Coleman integrals: matrix of Frobenius on de Rham cohomology Kedlaya s algorithm does this (and has been implemented) for hyperelliptic curves, so hyperelliptic curves are a natural starting point Analogues of Kedlaya s algorithm exist for other classes of curves and varieties, so what we re describing today could be extended to these objects (more later) Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 2 / 22

5 Notation and setup X: genus g hyperelliptic curve (of the form y 2 = f (x), f monic of degree d) over K = Q p p: prime of good reduction X: special fibre of X X an C p : generic fibre of X (as a rigid analytic space) Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 3 / 22

6 Notation and setup, in pictures There is a natural reduction map from XC an p to X; the inverse image of any point of X is a subspace of XC an p isomorphic to an open unit disk. We call such a disk a residue disk of X. A wide open subspace of XC an p is the complement in XC an p of the union of a finite collection of disjoint closed disks of radius λ i < 1: -1 red () -1 red (S) red -1 red (R) R X an Cp S X λ1 λ2 1 1 Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 4 / 22

7 Computing tiny integrals We refer to any Coleman integral of the form Q ω in which, Q lie in the same residue disk as a tiny integral. To compute such an integral: Construct a linear interpolation from to Q. For instance, in a non-weierstrass residue disk, we may take x(t) = (1 t)x() + tx(q) y(t) = f (x(t)), where y(t) is expanded as a formal power series in t. Formally integrate the power series in t: Q ω = 1 0 ω(x(t), y(t)) dt. Q Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 5 / 22

8 roperties of the Coleman integral Coleman formulated an integration theory on wide open subspaces of curves over O, exhibiting no phenomena of path dependence. This allows us to define Q ω whenever ω is a meromorphic 1-form on X, and, Q X(Q p ) are points where ω is holomorphic. roperties of the Coleman integral include: Theorem (Coleman) Linearity: Q (αω 1 + βω 2 ) = α Q ω 1 + β Q ω 2. Additivity: R ω = Q ω + R Q ω. Change of variables: if X is another such curve, and f : U U is a rigid analytic map between wide opens, then Q f ω = f (Q) f () ω. Fundamental theorem of calculus: Q df = f (Q) f (). Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 6 / 22

9 roperties of the Coleman integral Coleman formulated an integration theory on wide open subspaces of curves over O, exhibiting no phenomena of path dependence. This allows us to define Q ω whenever ω is a meromorphic 1-form on X, and, Q X(Q p ) are points where ω is holomorphic. roperties of the Coleman integral include: Theorem (Coleman) Linearity: Q (αω 1 + βω 2 ) = α Q ω 1 + β Q ω 2. Additivity: R ω = Q ω + R Q ω. Change of variables: if X is another such curve, and f : U U is a rigid analytic map between wide opens, then Q f ω = f (Q) f () ω. Fundamental theorem of calculus: Q df = f (Q) f (). Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 6 / 22

10 roperties of the Coleman integral Coleman formulated an integration theory on wide open subspaces of curves over O, exhibiting no phenomena of path dependence. This allows us to define Q ω whenever ω is a meromorphic 1-form on X, and, Q X(Q p ) are points where ω is holomorphic. roperties of the Coleman integral include: Theorem (Coleman) Linearity: Q (αω 1 + βω 2 ) = α Q ω 1 + β Q ω 2. Additivity: R ω = Q ω + R Q ω. Change of variables: if X is another such curve, and f : U U is a rigid analytic map between wide opens, then Q f ω = f (Q) f () ω. Fundamental theorem of calculus: Q df = f (Q) f (). Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 6 / 22

11 roperties of the Coleman integral Coleman formulated an integration theory on wide open subspaces of curves over O, exhibiting no phenomena of path dependence. This allows us to define Q ω whenever ω is a meromorphic 1-form on X, and, Q X(Q p ) are points where ω is holomorphic. roperties of the Coleman integral include: Theorem (Coleman) Linearity: Q (αω 1 + βω 2 ) = α Q ω 1 + β Q ω 2. Additivity: R ω = Q ω + R Q ω. Change of variables: if X is another such curve, and f : U U is a rigid analytic map between wide opens, then Q f ω = f (Q) f () ω. Fundamental theorem of calculus: Q df = f (Q) f (). Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 6 / 22

12 Coleman s construction How do we integrate if, Q aren t in the same residue disk? Coleman s key idea: use Frobenius to move between different residue disks (Dwork s analytic continuation along Frobenius ) Frobenius Q Q Tiny integral So we need to calculate the action of Frobenius on differentials. Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 7 / 22

13 Frobenius, MW-cohomology X : affine curve (X { Weierstrass points of X }) A: coordinate ring of X To discuss the differentials we will be integrating, we recall: The Monsky-Washnitzer (MW) weak completion of A is the ring A consisting of infinite sums of the form { i= B i (x) y i, B i (x) K[x], deg B i 2g further subject to the condition that v p (B i (x)) grows faster than a linear function of i as i ±. We make a ring out of these using the relation y 2 = f (x). These functions are holomorphic on wide opens, so we will integrate 1-forms ω = g(x, y) dx 2y, g(x, y) A. Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 8 / 22 },

14 Odd vs even degree models of the curve Any odd differential ω = g(x, y) dx 2y, g(x, y) A can be written as where f ω A, c i K and ω = df ω + c 0 ω c d 2 ω d 2, ω i = xi dx 2y (i = 0,..., d 2), with d = deg f (x). When d = 2g + 1, the set {ω i } d 2 i=0 forms a basis of the odd part of the de Rham cohomology of A. Let us suppose that d is odd. By linearity and the fundamental theorem of calculus, we reduce the integration of ω to the integration of the ω i. Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 9 / 22

15 Integrals between points in non-weierstrass disks Let φ denote Frobenius. Recall that a Teichmüller point of X is a point such that φ() =. One way to compute Coleman integrals Q ω i: Find the Teichmüller points, Q in the residue disks of, Q. Use Frobenius to compute Q ω i. Use additivity in endpoints to recover the integral: Q ω i = ω i + Q ω i + Q Q ω i. Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 10 / 22

16 Integrals between points in non-weierstrass disks Let φ denote Frobenius. Recall that a Teichmüller point of X is a point such that φ() =. One way to compute Coleman integrals Q ω i: Find the Teichmüller points, Q in the residue disks of, Q. Use Frobenius to compute Q ω i. Use additivity in endpoints to recover the integral: Q ω i = ω i + Q ω i + Q Q ω i. Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 10 / 22

17 Integrals between points in non-weierstrass disks Let φ denote Frobenius. Recall that a Teichmüller point of X is a point such that φ() =. One way to compute Coleman integrals Q ω i: Find the Teichmüller points, Q in the residue disks of, Q. Use Frobenius to compute Q ω i. Use additivity in endpoints to recover the integral: Q ω i = ω i + Q ω i + Q Q ω i. Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 10 / 22

18 Using Frobenius Results More on Frobenius: Calculate the action of Frobenius φ on each basis differential, letting 2g 1 φ ω i = df i + M ij ω j. Compute Q ω j by solving a linear system As the eigenvalues of the matrix M are algebraic integers of C-norm p 1/2 1, the matrix M I is invertible, and we may solve the system to obtain the integrals Q ω i. j=0 Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 11 / 22

19 Using Frobenius Results More on Frobenius: Calculate the action of Frobenius φ on each basis differential, letting 2g 1 φ ω i = df i + M ij ω j. Compute Q ω j by solving a linear system As the eigenvalues of the matrix M are algebraic integers of C-norm p 1/2 1, the matrix M I is invertible, and we may solve the system to obtain the integrals Q ω i. j=0 Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 11 / 22

20 Using Frobenius Results More on Frobenius: Calculate the action of Frobenius φ on each basis differential, letting 2g 1 φ ω i = df i + M ij ω j. j=0 Compute Q ω j by solving a linear system Q ω i = φ(q ) ω i φ( ) As the eigenvalues of the matrix M are algebraic integers of C-norm p 1/2 1, the matrix M I is invertible, and we may solve the system to obtain the integrals Q ω i. Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 11 / 22

21 Using Frobenius Results More on Frobenius: Calculate the action of Frobenius φ on each basis differential, letting 2g 1 φ ω i = df i + M ij ω j. j=0 Compute Q ω j by solving a linear system Q ω i = Q φ ω i As the eigenvalues of the matrix M are algebraic integers of C-norm p 1/2 1, the matrix M I is invertible, and we may solve the system to obtain the integrals Q ω i. Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 11 / 22

22 Using Frobenius Results More on Frobenius: Calculate the action of Frobenius φ on each basis differential, letting 2g 1 φ ω i = df i + M ij ω j. j=0 Compute Q ω j by solving a linear system Q ω i = Q df i + 2g 1 j=0 M ij ω j As the eigenvalues of the matrix M are algebraic integers of C-norm p 1/2 1, the matrix M I is invertible, and we may solve the system to obtain the integrals Q ω i. Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 11 / 22

23 Using Frobenius Results More on Frobenius: Calculate the action of Frobenius φ on each basis differential, letting 2g 1 φ ω i = df i + M ij ω j. j=0 Compute Q ω j by solving a linear system Q ω i = Q df i + 2g 1 j=0 M ij Q As the eigenvalues of the matrix M are algebraic integers of C-norm p 1/2 1, the matrix M I is invertible, and we may solve the system to obtain the integrals Q ω i. Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 11 / 22 ω j

24 Using Frobenius Results More on Frobenius: Calculate the action of Frobenius φ on each basis differential, letting 2g 1 φ ω i = df i + M ij ω j. j=0 Compute Q ω j by solving a linear system Q ω i = f i (Q ) f i ( ) + 2g 1 j=0 M ij Q ω j. As the eigenvalues of the matrix M are algebraic integers of C-norm p 1/2 1, the matrix M I is invertible, and we may solve the system to obtain the integrals Q ω i. Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 11 / 22

25 Using Frobenius Results More on Frobenius: Calculate the action of Frobenius φ on each basis differential, letting 2g 1 φ ω i = df i + M ij ω j. j=0 Compute Q ω j by solving a linear system Q ω i = f i (Q ) f i ( ) + 2g 1 j=0 M ij Q ω j. As the eigenvalues of the matrix M are algebraic integers of C-norm p 1/2 1, the matrix M I is invertible, and we may solve the system to obtain the integrals Q ω i. Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 11 / 22

26 Integrals via Teichmüller, continued The linear system gives us the integral between different residue disks. utting it all together, we have Q ω i = ω i + Q ω i + Q Q ω i Q Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 12 / 22

27 Integrals via Teichmüller, continued The linear system gives us the integral between different residue disks. utting it all together, we have Q ω i = ω i + Q ω i + Q Q ω i Q Q Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 12 / 22

28 Integrals via Teichmüller, continued The linear system gives us the integral between different residue disks. utting it all together, we have Q ω i = ω i + Q ω i + Q Q ω i Q Q Tiny integral Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 12 / 22

29 Integrals via Teichmüller, continued The linear system gives us the integral between different residue disks. utting it all together, we have Q ω i = ω i + Q ω i + Q Q ω i Frobenius Q Q Tiny integral Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 12 / 22

30 Integrals via Teichmüller, continued The linear system gives us the integral between different residue disks. utting it all together, we have Q ω i = ω i + Q ω i + Q Q ω i Frobenius Q Q Tiny integral Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 12 / 22

31 Integrals that compute heights (joint with Besser) Why p-adic heights? Interpolating special values of the L-function in the usual BSD conjecture, we obtain a p-adic BSD conjecture: Conjecture (Mazur, Tate, and Teitelbaum) Let E be an elliptic curve over Q and let p be a prime of good ordinary reduction for E. Then the algebraic rank of E equals ord T (L p (E, T)) and L p(e, 0) = (1 α 1 ) 2 l c l X(E/Q) Reg p (E) E(Q) tors 2, with α the unit root of x 2 a p x + p and where the p-adic regulator Reg p (E) Q p is defined analogously to the classical regulator: Take 1,... r be a basis for E(Q) modulo torsion. Then Reg p (E) = det(m), where M ij = i, j p, with a p-adic height pairing, p : E(Q) E(Q) Q p. Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 13 / 22

32 Integrals that compute heights (joint with Besser) Notation C: genus g hyperelliptic curve of the form y 2 = f (x), with deg f (x) = 2g + 1 K: number field k: local field (char 0) with valuation ring O, uniformizer π F: residue field, O/πO, with F = q. J: Jacobian of C over k We ll assume C has good ordinary reduction at π. Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 14 / 22

33 Integrals that compute heights (joint with Besser) The Coleman-Gross height pairing is a symmetric bilinear pairing h : Div 0 (C) Div 0 (C) Q p, which can be written as a sum of local height pairings h = v h v over all finite places v of K. Techniques to compute h v depend on char(f): char(f) p: intersection theory, as in h.d. thesis of Müller ( 10) char(f) = p: logarithms, normalized differentials, Coleman integration Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 15 / 22

34 Local height: residue characteristic p Integrals that compute heights (joint with Besser) Suppose char(f) = p, and that k = K v, the completion at v p. Definition Let D 1, D 2 Div 0 (C) have disjoint support and ω D1 a normalized differential associated to D 1. The local height pairing at v above p is given by ) h v (D 1, D 2 ) = tr k/qp ω D1. ( D 2 Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 16 / 22

35 Integrals that compute heights (joint with Besser) Local height: residue characteristic p Constructing and using ω D1 : subtle, requires more cohomology In brief: a map Ψ that plays the role of a logarithm and aids in splitting H 1 dr (C) (which gives us ω D 1 ) new tricks to compute Coleman integrals of differentials with poles in non-weierstrass discs Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 17 / 22

36 A higher-dimensional analogue of MTT? Application: a p-adic BSD conjecture (joint with Müller, Stein) Conjecture Let A f be a modular abelian variety of dimension 2 attached to a newform f and p a prime of good ordinary reduction for A f. Let K f be the real quadratic field containing the Hecke eigenvalue a p. The Mordell-Weil rank of A f equals ord T (L p (A f, T)) and L p(a f, 0) N (Kf Q p )/Q p ((1 α 1 ) 2 ) X(A f ) Reg p (A f ) l N c l A f (Q) tors A f (Q) tors (log(1 + p)) r, where L p(a f, 0) is the leading coefficient of the p-adic L-series L p (A f, T), α is the unit root,and is an equality up to a sign c = ±1. Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 18 / 22

37 Application: a p-adic BSD conjecture (joint with Müller, Stein) Some candidate modular abelian varieties From Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves (Flynn et al. 01), a table of the associated curves and levels For each curve, the associated abelian variety has Mordell-Weil rank 2. N Equation 67 y 2 + (x 3 + x + 1)y = x 5 x 73 y 2 + (x 3 + x 2 + 1)y = x 5 2x 3 + x 85 y 2 + (x 3 + x 2 + x)y = x 4 + x 3 + 3x 2 2x y 2 + (x 3 + x 2 + 1)y = 2x 5 + x 4 + x y 2 + (x 3 + x 2 + 1)y = x 5 + x y 2 + (x 3 + x 2 + 1)y = x 4 x 2 x y 2 + (x 3 + x + 1)y = 2x 3 + x 2 + x 125,A y 2 + (x 3 + x + 1)y = x 5 + 2x 4 + 2x 3 + x 2 x 1 133,A y 2 + (x 3 + x 2 + 1)y = x 5 + x 4 2x 3 + 2x 2 2x 147 y 2 + (x 3 + x 2 + x)y = x 5 + 2x 4 + x 3 + x y 2 + (x 3 + x + 1)y = x 3 + 4x 2 + 4x y 2 + (x 3 + x 2 + x)y = x 5 + 2x 4 + 3x 3 + x 2 3x 167 y 2 + (x 3 + x + 1)y = x 5 x 3 x y 2 + (x 3 + x 2 + 1)y = x 5 + x 4 + x y 2 = x 5 x 4 + x 3 + x 2 2x y 2 + (x 3 + x + 1)y = x 3 + x 2 + x Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 19 / 22

38 Application: a p-adic BSD conjecture (joint with Müller, Stein) To numerically verify p-adic BSD, need to compute p-adic regulators and p-adic special values. For example, for N = 188, we have values of the p-adic regulator for various p: p p-adic regulator O(7 8 ) O(11 7 ) O(13 8 ) O(17 8 ) O(19 8 ) O(23 8 ) O(37 8 ) O(41 8 ) O(43 8 ) O(53 8 ) O(59 8 ) O(61 8 ) O(67 8 ) O(71 8 ) O(73 8 ) O(79 8 ) O(83 8 ) O(89 8 ) O(97 8 ) Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 20 / 22

39 Application: a p-adic BSD conjecture (joint with Müller, Stein) Here are the corresponding special values of the p-adic L-series: p p-adic special value O(7 4 ) O(11 4 ) O(13 4 ) O(17 4 ) O(19 4 ) O(23 4 ) O(37 3 ) O(41 3 ) O(43 3 ) O(53 3 ) O(59 3 ) O(61 2 ) O(67 2 ) O(71 2 ) O(73 2 ) O(79 2 ) O(83 2 ) O(89 2 ) O(97 2 ) Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 21 / 22

40 Future directions Conclusion Future work Heights after Harvey Our basic algorithm has linear runtime dependence on the prime p, arising from the corresponding dependence in Kedlaya s algorithm; could possibly follow Harvey s variant of Kedlaya s algorithm to reduce this to square-root dependence on p Beyond hyperelliptic curves Convert algorithms for computing Frobenius actions on de Rham cohomology (Gaudry-Gürel, Castryck-Denef-Vercauteren) into algorithms for computing Coleman integrals on such curves Use Abbott-Kedlaya-Roe (and Harvey s recent version) for smooth hypersurfaces in projective space (which can be generalized to nondegenerate hypersurfaces in toric varieties) to carry out Coleman integration on these surfaces Jennifer Balakrishnan (Harvard) Computations with Coleman integrals AWM 40 Years and Counting 22 / 22