ITERATED COLEMAN INTEGRATION FOR HYPERELLIPTIC CURVES

Transcription

1 ITERATED COLEMAN INTEGRATION FOR HYERELLITIC CURVE JENNIFER BALAKRIHNAN Astract The Coleman integral is a p-adic line integral Doule Coleman integrals on elliptic curves appear in Kim s nonaelian Chaauty method, the first numerical examples of which were given y the author, Kedlaya, and Kim [3] This paper descries the algorithms used to produce those examples, as well as techniques to compute higher iterated integrals on hyperelliptic curves, uilding on previous joint work with Bradshaw and Kedlaya [2] 1 Introduction In a series of papers in the 198s, Coleman gave a p-adic theory of integration on the projective line [6], then on curves and aelian varieties [7, 8] This integration theory relies on locally defined antiderivatives that are extended analytically y the principle of Froenius equivariance In joint work with Bradshaw and Kedlaya [2], we made this construction explicit and gave algorithms to compute single Coleman integrals for hyperelliptic curves Having algorithms to compute Coleman integrals allows one to compute p-adic regulators in K-theory [6, 8], carry out the method of Chaauty-Coleman for finding rational points on higher genus curves [11], and utilize Kim s nonaelian analogue of the Chaauty method [1] Kim s method, in the case of rank 1 elliptic curves, allows one to find integral points via the computation of doule Coleman integrals Indeed, Coleman s theory of integration is not limited to single integrals; it gives rise to an entire class of locally analytic functions, the Coleman functions, on which antidifferentiation is well-defined In other words, one can define iterated p-adic integrals [4, 6] ξ n ξ 1 which ehave formally like iterated path integrals 1 t1 tn 1 f n (t n ) f 1 (t 1 ) dt n dt 1 Let us fix some notation Let C e a genus g hyperelliptic curve over an unramified extension K of p having good reduction Let k F q denote its residue field, where q p m We will assume that C is given y a model of the form y 2 f(x), where f is a monic separale polynomial with deg f 2g + 1 Our methods for computing iterated integrals are similar in spirit to those detailed in [2] We egin with algorithms for tiny iterated integrals, use Froenius equivariance to write down a linear system yielding the values of integrals etween points in different residue disks, and, if needed, use asic properties of integration to correct endpoints We egin with some asic properties of iterated path integrals 1

2 2 JENNIFER BALAKRIHNAN 2 Iterated path integrals We follow the convention of Kim [1] and define our integrals as follows: ξ 1 ξ 2 ξ n 1 ξ n : ξ 1 (R 1 ) 1 ξ 2 (R 2 ) n 2 ξ n 1 (R n 1 ) n 1 for a collection of dummy parameters R 1,, R n 1 and 1-forms ξ 1,, ξ n We egin y recalling some key formal properties satisfied y iterated path integrals [5] roposition 21 Let ξ 1,, ξ n e 1-forms, holomorphic at points, on C Then the following are true: (1) ξ 1ξ 2 ξ n, (2) all permutations σ (3) ω i 1 ω in ( 1) n ω i n ω i1 ω σ(i 1)ω σ(i2) ω σ(in) n j1 ω i j, As an easy corollary of roposition 21(2), we have Corollary 22 For a 1-form ω i and points, as efore, ( ) n ω i ω i ω i 1 ω i n! When possile, we will use this to write an iterated integral in terms of a single integral 3 p-adic cohomology We riefly recall some p-adic cohomology from [9], necessary for formulating the integration algorithms Let C e the affine curve otained y deleting the Weierstrass points from C, and let A K[x, y, z]/(y 2 f(x), yz 1) e the coordinate ring of C Let A denote the Monsky-Washnitzer weak completion of A; it is the ring consisting of infinite sums of the form { i } B i (x) y i, B i (x) K[x], deg B i 2g, further suject 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 the space over which we integrate, so we consider odd 1-forms written as ω g(x, y) dx 2y, g(x, y) A Any such differential can e written as (31) ω df + c ω + + c ω, with f A, c i K, and ω i x i dx 2y (i,, 2g 1) Namely, the set of differentials {ω i } i forms a asis of the odd part of the de Rham cohomology of A, which we denote as H 1 dr (C ) ξ n,

3 ITERATED COLEMAN INTEGRATION FOR HYERELLITIC CURVE 3 To compute the p-power Froenius action φ on H 1 dr (C ), one does the following: Let φ K denote the unique automorphism lifting Froenius from F q to K Extend φ K to A y setting and use the relations φ(x) x p ( φ(y) y p 1 + φ(f)(xp ) f(x) p ) 1/2 f(x) p ( ) 1/2 (φ(f)(x y p p ) f(x) p ) i i y 2pi, i y 2 f(x) d(x i y j ) (2ix i 1 y j+1 + jx i f (x)y j 1 ) dx 2y to reduce large powers of x and large (in asolute value) powers of y to write φ (ω) in the form (31) This reduction process is known as Kedlaya s algorithm [9], and we will repeatedly use this algorithm to reduce iterated integrals involving ω A dx 2y to iterated integrals in terms of asis elements ω i 4 Integrals: lemmas Recall that we use Kedlaya s algorithm to compute single Coleman integrals as follows: Algorithm 41 (Coleman integration in non-weierstrass disks [2]) Input: The asis differentials (ω i ) i, points, C(C p) in non-weierstrass residue disks, and a positive integer m such that the residue fields of, are contained in F p m ( ) Output: The integrals ω i (41) i (1) Calculate the action of the m-th power of Froenius on each asis element (see Remark 42): (φ m ) ω i dh i + M ij ω j (2) By change of variales, we otain (M I) ij ω j h i ( ) h i () φ m ( ) ω i ω i φ m () (the fundamental linear system) ince the eigenvalues of the matrix M are algeraic integers of C-norm p m/2 1 (see [9, 2]), the matrix M I is invertile, and we may solve (41) to otain the integrals ω i

4 4 JENNIFER BALAKRIHNAN Remark 42 To compute the action of φ m, first carry out Kedlaya s algorithm to write φ ω i dg i + B ij ω j If we view h, g as column vectors and M, B as matrices, induction on m shows that h φ m 1 (g) + Bφ m 2 (g) + + Bφ K (B) φ m 2 K (B)g M Bφ K (B) φ m 1 K (B) Note, however, that when points, C(C p ) are in the same residue disk, the tiny Coleman integral etween them can e computed using a local parametrization, just as in the case of a real-valued line integral This is also true when the integrals are iterated (see ection 5) However, to compute general iterated integrals, we will need to employ the analogue of additivity in endpoints to link integrals etween different residue disks First, let us consider the case where we are reaking up the path y one point Lemma 43 Let,, e points on C such that a path is to e taken from to via Let ξ 1,, ξ n e a collection of 1-forms holomorphic at the points,, Then the following statement holds: ξ 1 ξ n n i ξ 1 ξ i ξ i+1 ξ n roof We proceed y induction The case n 1 is clear Let us suppose the statement holds for n k Then we have that (42) (43) (44) (45) ( ) ξ 1 ξ k+1 ξ 1 ξ k (R) ( k ξ 1 ξ i i ( ) ξ 1 ξ k (R) + ( ξ 1 ) ( ξ k+1 ξ i+1 ξ k ) (R) ξ k+1 ξ 2 ξ k ) (R) ( + ξ 1 ξ k 1 ( ) + ξ 1 ξ k (R) ξ k ) (R) ξ k+1 ξ k+1 ξ k+1 ξ k+1 Oserve that this last iterated integral (45) can e rewritten as ( ) ( ) R ξ 1 ξ k (R) ξ k+1 + ξ k+1,

5 ITERATED COLEMAN INTEGRATION FOR HYERELLITIC CURVE 5 and that further, the terms from (42) through (44) give us Thus we have as desired k 1 i k 1 ξ 1 ξ k+1 i ξ 1 ξ i ξ 1 ξ i ( ) ( + ξ 1 ξ k k+1 i ξ 1 ξ i ξ i+1 ξ k+1, ξ i+1 ξ k+1 ) ξ k+1 + ξ 1 ξ k+1 ξ i+1 ξ k+1, Applying Lemma 43 twice, we otain the following, which will e used to link integrals etween different residue disks: Lemma 44 (Link lemma) Let points,,, e on C such that a path is to e taken from to to to Let ξ 1,, ξ n e a collection of 1-forms holomorphic at the points,,, Then we have n n ξ 1 ξ n ξ 1 ξ i ξ i+1 ξ j ξ j+1 ξ n i ji Below we record aspecific case of the link lemma, which we shall use throughout this paper Example 45 (Link lemma for doule integrals) uppose we have two differentials ξ, ξ 1 Then we have ξ ξ 1 ξ ξ 1 + ξ ξ 1 + ξ ξ 1 + ξ 1 ξ + ξ 1 ξ 5 Tiny iterated integrals We egin with an algorithm to compute tiny iterated integrals Algorithm 51 (Tiny iterated integrals) Input: oints, C(C p ) in the same residue disk (neither equal to the point at infinity) and differentials ξ 1,, ξ n without poles in the disk of Output: The integral ξ 1ξ 2 ξ n (1) Compute a parametrization (x(t), y(t)) at in terms of a local coordinate t (2) For each k, write ξ k (x, y) in terms of t: ξ k (t) : ξ k (x(t), y(t)) (3) Let I n+1 (t) : 1

6 6 JENNIFER BALAKRIHNAN (4) Compute, for k n,, 2, in descending order, I k (t) k 1 t(rk 1 ) ξ k I k+1 ξ k (u)i k+1 (u), with R k 1 in the disc of (5) Upon computing I 2 (t), we arrive at the desired integral: ξ 1 ξ 2 ξ n I 1 (t) t() ξ 1 (u)i 2 (u) We show how we carry out Algorithm 51 for doule integrals on an elliptic curve Example 52 (A tiny doule integral) Let C e the elliptic curve y 2 x(x 1)(x + 9), let p 7, and consider the points (9, 36), φ( ), and R (a + x( ), f(a + x( ))) so that R is in the same disk as and Furthermore, let ω dx 2y, ω 1 xdx 2y We compute the doule integral ω ω 1 First compute the local coordinates at : x(t) 9 + t + O(t 2 ) y(t) t t t t t5 + O(t 6 ) Then setting I 2 : x dx 2y, and making it a definite integral, we have I 2 R a x dx 2y x(t) dx(t) 2y(t) 1 8 a a a3 from which we arrive at I x() x( ) a a a6 + I 2 (a) dx(r(a)) 2y(R(a)) O(7 8 ) 6 Iterated integrals: linear system a7 + O(a 8 ), As in the case of computing single integrals, to compute general iterated Coleman integrals, we use Kedlaya s algorithm to calculate the action of Froenius on de Rham cohomology This gives us a linear system that allows us to solve for all (2g) n n-fold iterated integrals on asis differentials Theorem 61 Let, C(C p ) e non-weierstrass points such that the residue fields of, are contained in F p m Let M e the matrix of the action of the m-th power of Froenius on the asis differentials ω,, ω For constants c i,,i n 1

7 ITERATED COLEMAN INTEGRATION FOR HYERELLITIC CURVE 7 computale in terms of (n 1)-fold iterated integrals and n-fold tiny iterated integrals, the n-fold iterated Coleman integrals on asis differentials etween, can e computed via a linear system of the form ω i ω in 1 (I (2g) n (2g) n (M t ) n ) 1 c i i n 1 roof By the Link lemma (Lemma 44), we can reduce to the case where oth and are Teichmüller points (points fixed y some power of φ) Then we have (61) ω ii ω in φ m () φ m ( ) ω ii ω in (φ m ) (ω ii ω in ) (φ m ) (ω ii ) (φ m ) (ω in ) Recall that given ω,, ω a asis for H 1 dr (C ), we have (φ m ) ω il df il + M il jω j ustituting this expression in for each factor of (61) and expanding yields the linear system To illustrate our methods, in the next section, we present a more explicit version of this theorem, accompanied y algorithms, in the case of doule integrals We show how these are used in Kim s nonaelian Chaauty method in ection 8 7 Explicit doule integrals 71 The linear system for doule integrals etween Teichmüller points In this susection, we make explicit one aspect of Theorem 61: we give an algorithm to compute doule integrals etween Teichmüller points Algorithm 71 (Doule Coleman integration etween Teichmüller points) Input: The asis differentials (ω i ) i, Teichmüller points, C(C p) in non- Weierstrass residue disks, and a positive integer m such that the residue fields of, are contained in F p m Output: The doule integrals ( ) ω iω j i, (1) Calculate the action of the m-th power of Froenius on each asis element: (φ m ) ω i df i + M ijω j (2) Use Algorithm 41 to compute the single Coleman integrals ω j on all asis differentials (3) Use tep 2 and linearity to recover the other single Coleman integrals: df if k, M ijω j f k for each i, k

8 8 JENNIFER BALAKRIHNAN c ik (4) Use the results of the aove two steps to write down, for each i, k, the constant + f i () df i (R)(f k (R)) f k ( )(f i () f i ( )) + M kj ω j f i (R)( M kj ω j (R)) M ij ω j (R)(f k (R) f k ( )) (5) Recover the doule integrals (see Remark 72 elow) via the linear system ω ω c ω ω 1 (I 4g 2 4g 2 (M t ) 2 ) 1 c 1 c, ω ω Remark 72 We otain the linear system in the following manner ince, are Teichmüller, we have (71) ω i ω k φ m () φ m ( ) ω i ω k (φ m ) (ω i ω k ) We egin y expanding the right side of (71) Recall that given ω,, ω a asis for H 1 dr (C ), we have Thus we have (φ m ) (ω i ω k ) (φ m ) ω i df i + M ij ω j (φ m ) (ω i )(φ m ) (ω k ) (df i + df i df k + ( M ij ω j )(df k + M kj ω j ) M ij ω j )df k + df i ( We expand the first three quantities separately First, we have df i df k df i (R) df k df i (R)(f k (R) f k ( )) df i (R)(f k (R)) f k ( ) M kj ω j ) + ( df i (R) df i (R)(f k (R)) f k ( )(f i () f i ( )) M ij ω j )( M kj ω j )

9 ITERATED COLEMAN INTEGRATION FOR HYERELLITIC CURVE 9 c ik Next, we have ( M ij ω j )df k The third term (via integration y parts) is df i ( M kj ω j ) df i (R) f i (R) f i () ( ( M ij ω j (R) df k M ij ω j (R)(f k (R) f k ( )) M kj ω j ) M kj ω j ) R R M kj ω j Denote the sum of these terms y c ik ; in other words, + f i () df i (R)(f k (R)) f k ( )(f i () f i ( )) + M kj ω j f i (R)( f i (R)( M kj ω j (R)) f i (R)( M kj ω j (R)) Then rearranging terms, our linear system reads ω ω ω ω 1 ω ω (I 4g 2 4g 2 (M t ) 2 ) 1 M kj ω j (R)) M ij ω j (R)(f k (R) f k ( )) c c 1 c, 72 Linking doule integrals Let and e in the disks of and, respectively Using the Link lemma for doule integrals (Example 45), we may link doule integrals etween different residue disks: (72) ω i ω k ω i ω k + ω i ω k + ω i ω k + ω k ω i + ω k ω i Algorithm 73 (Doule Coleman integration using intermediary Teichmüller points) Input: The asis differentials (ω i ) i, points, C(C p) in non-weierstrass residue disks ( ) Output: The doule integrals ω iω j i, (1) Compute Teichmüller points, in the disks of,, respectively

10 1 JENNIFER BALAKRIHNAN (2) Use Algorithm 41 to compute the single integrals ω i, ω i, ω i for all i (3) Use Algorithm 51 to compute the tiny doule integrals ω i ω k, ω i ω k (4) Use Algorithm 71 to compute the doule integrals { (5) Correct endpoints using ω i ω j } i, ω i ω k ω i ω k + ω i ω k + ω i ω k + ω k ω i + ω k ω i 73 Without Teichmüller points Alternatively, instead of finding Teichmüller points and correcting endpoints, we can directly compute doule integrals using a slightly different linear system Indeed, using the Link lemma for doule integrals, we take φ( ) and φ() to e the points in the disks of and, respectively, which gives (73) φ( ) φ() φ( ) φ() ω i ω k ω i ω k + ω i ω k + ω i ω k + ω k ω i + ω k ω i φ( ) φ() φ( ) φ( ) φ() To write down a linear system without Teichmüller points, we egin as efore, with (74) φ() φ( ) ω i ω k φ (ω i ω k ) c ik + A ij ω j A kj ω j utting together (73) and (74), we get (75) ω iω k (I 4g 2 4g 2 (M c t ) 2 ) 1 ik φ( ) ω iω k ( φ() ( ω i ) ( φ() φ( ) ω k ) ( ) ω i φ( ) ω k ) + φ() ω iω k This gives us the following alternative to Algorithm 71: Algorithm 74 (Doule Coleman integration) Input: The asis differentials (ω i ) i, points, C( p) in non-weierstrass residue disks or in Weierstrass disks in the region of convergence ( ) Output: The doule integrals ω iω j i, (1) Use Algorithm 41 to compute the single integrals ω i, φ() φ( ) ω i for all i (2) Use Algorithm 51 to compute φ( ) ω iω k, φ() ω iω k for all i, k (3) As in tep 4 of Algorithm 71, compute the constants c ik for all i, k

11 ITERATED COLEMAN INTEGRATION FOR HYERELLITIC CURVE 11 (4) Recover the doule integrals using the linear system ω iω k (I 4g 2 4g 2 (M c t ) 2 ) 1 ik φ( ) ω iω k ( φ() ( ω i ) ( φ() φ( ) ω k ) ( ) ω i φ( ) ω k ) + φ() ω iω k Example 75 Let C e the genus 2 curve y 2 x 5 x 4 + x 3 + x 2 2x + 1 and let (1, 1), ( 1, 1) and p 7 We compute doule integrals on asis differentials: ω ω O(7 5 ) ω ω O(7 5 ) ω ω O(7 4 ) ω ω O(7 5 ) ω 1 ω O(7 5 ) ω 1 ω O(7 5 ) ω 1 ω O(7 5 ) ω 1 ω O(7 4 ) ω 2 ω O(7 4 ) ω 2 ω O(7 5 ) ω 2 ω O(7 3 ) ω 2 ω O(7 3 ) ω 3 ω O(7 5 ) ω 3 ω O(7 4 ) ω 3 ω O(7 3 ) ω 3 ω O(7 3 )

12 12 JENNIFER BALAKRIHNAN Example 76 Using the previous example, we verify the Fuini identity ( ) ( ) ω j ω i + ω i ω j ω i ω j We have ω O(7 6 ) ω O(7 6 ) ω O(7 6 ) ω O(7 6 ) We see, for example, ( ) ( ) ω ω 1 + ω 1 ω O(7 5 ) ω ω 1 ω 2 ω 3 + ( ) ( ) ω 3 ω O(7 3 ) ω 2 ω 3 74 Weierstrass points uppose one of or is a finite Weierstrass point Then directly using the linear system as aove fails, since the f i have essential singularities at finite Weierstrass points We remedy this as follows: roposition 77 Let e a non-weierstrass point, a finite Weierstrass point, and e a point in the residue disk of, near the oundary Then the integral from to can e computed as a sum of integrals: ω i ω k ω i ω k + ω i ω k + ω k ω i roof This follows from Lemma 43 in the case of n 2, where To compute tiny iterated integrals in a Weierstrass disk, we slightly modify Algorithm 51: Algorithm 78 (Tiny iterated integral in a Weierstrass disk) Input: a Weierstrass point, d the degree of totally ramified extension, ω i, ω j asis differentials Output: The integral ω i ω j ω i (R) ω j t1 t ω i (R) ut (1) Compute local coordinates (x(u), u) at (2) Let a p 1/d Rescale coordinates so that y : au, x : x(au) (3) Compute I 2 (u) x j dx 2y as a power series in u (4) Compute the appropriate definite integral using the step aove: R x j dx 2y t x(au) adu u I 2(t) u ω j

13 ITERATED COLEMAN INTEGRATION FOR HYERELLITIC CURVE 13 (where R (x(t), t)) Call this definite integral (now a power series in t) I 2 (5) Now since R (x(t), t), we have ω iω j 1 x(t)i dx(t) I 2 2t uppose is a finite Weierstrass point While one could compute the integral ω iω j directly using Algorithm 74 for all of the tiny doule integrals (and Algorithm 78 for the other doule integrals), in practice, that approach is expensive, as it requires the computation of several intermediate integrals with Froenius of points that are defined over ramified extensions This, in turn, makes the requisite degree d extension for convergence quite large Instead, the key idea is to compute a local parametrization at the finite Weierstrass point and to use this to compute the indefinite integral ω i Then to compute integrals involving oundary points, one can simply evaluate this indefinite integral at the appropriate points, instead of directly computing parametrizations, and thus integrals, over a totally ramified extension of p This idea is also used to evaluate doule integrals involving oundary points Algorithm 79 (Intermediary integrals for doule integrals with a Weierstrass endpoint) Input: finite Weierstrass point, non-weierstrass point, d the degree of totally ramified extension, n the precision of p, asis differentials ω i, ω j Output: Necessary things for the eventual computation of ω iω j (1) Compute (x(t), t) local coordinates at to precision nd (2) Let (x(a), a), where a p 1/d (3) Compute as a power series in t, I 2 (t) x(t) i dx(t) y(t) (4) Compute the definite integral ω i I 2 (a) (5) For all i < j, compute the definite integral ω iω j via Algorithm 51 Keep the intermediary indefinite integral (6) For all i j, use the fact that ( ω iω j i) ω to compute the doule integral in terms of the single integral (7) For all i > j, use the fact that ω iω j compute (8) Compute φ() ω jω i + ω i ω j to ω iω j (instead of directly computing it as a doule integral) ω i φ() ω i ω i y the indefinite integral in tep 3 Use this to deduce φ() ω i ω j for i j (9) Use the indefinite integral in tep 5 to get φ() ω i ω j for i < j (1) Repeat the trick in tep 7 to get φ() ω i ω j for i > j (11) Compute φ() ω i and use it to deduce φ() ω i ω j for i j (12) Compute φ() ω i ω j for i < j (13) Repeat the trick in tep 7 to get φ() (14) Use ω i ω i ω i to get ω i ω i ω j for i < j Algorithm 71 (Doule integrals from a Weierstrass endpoint) Input: finite Weierstrass point, non-weierstrass point, ω i, ω j asis differentials Output: The doule integrals ω iω j

14 14 JENNIFER BALAKRIHNAN (1) Compute all of the integrals as in Algorithm 79 (2) Compute doule integrals ω iω j using the terms in tep 1 as appropriate in Algorithm 74 (ee Remark 711 for an additional improvement to this step) (3) Use additivity to recover the doule integrals ω iω j ω iω j + ω iω j + ω i ω j Remark 711 Note that in the case of g 1, the linear system only yields one doule integral not otainale through single integrals Indeed, for i, j 1, we have ( ) ω iω i ω i and ω iω j ω jω i + ω i ω j o it suffices to compute ω ω 1 Thus, rather than computing all of the constants c, c 1, c 1, c 11 and their correction factors (see (75)), if we pre-compute the two doule integrals that are expressile in terms of single integrals, as well as the product of single integrals that relates ω 1ω to ω ω 1, it suffices to compute c 1 (and its correction factor) to solve for the other three constants and ω ω 1 In other words, the linear system in Algorithm 74 tells us that (I 4 4 (M t ) 2 ) which we write as ω iω k A i v 1 s 1 v 1 i 11 c ik ( φ( ) ω iω k ( ) ( φ() φ() ω i φ( ) ω k x l 1 x 1 x 11, ) ( ) ω i φ( ) ω k ) + φ() ω iω k where the vector on the left consists of integrals (with i ω ω, i 11 ω 1ω 1, s 1 ω ω 1 all computed), and the vector on the right consists of constants (with l 1 computed) o we solve for x, x 1, x 11, v 1, since knowing v 1 ω ω 1 gives us the complete set of doule integrals on asis differentials: 1 x 1 (a 1 a 1 ) i x 1 x 11 (a 11 a 12 ) 1 (a 21 a 22 ) A c 1 l 1, v 1 1 (a 31 a 32 ) i 11 where A (a ij ) While this only gives a constant speed-up in terms of complexity, in practice, this helps when is defined over a highly ramified extension of p As numerical checks, one may use the following corollaries of roposition 77 Corollary 712 For, Weierstrass points and a third point, we have additivity in endpoints: ω iω j + ω iω j ω iω j Corollary 713 For, Weierstrass points, we have ω iω j + ω jω i,

15 ITERATED COLEMAN INTEGRATION FOR HYERELLITIC CURVE 15 It is worth noting that in general, unlike in the case of a single Coleman integral, for and oth Weierstrass points, unless i k, the doule Coleman integral ω iω k is not necessarily However, in the case of i k, the integral can e computed as ω iω i 1 2 ( ω i) 2 Example 714 Consider the curve y 2 x(x 1)(x + 9), over 7, and the points 1 (1, ), 2 (, ), and ( 1, 4) We have 1 ω ω O(7 6 ) 1 ω ω 1 1 ω 1 ω O(7 6 ) O(7 6 ) 1 ω 1 ω O(7 6 ) 1 and 2 ω ω 2 ω ω 1 2 ω 1 ω 2 ω 1 ω O(7 6 ) O(7 6 ) O(7 6 ) O(7 6 ), from which we see that 2 1 ω ω 1 and likewise 2 1 ω 1 ω 8 Kim s nonaelian Chaauty method We now present the motivation for all of the algorithms thus far Let C/Z e the minimal regular model of an elliptic curve C/ of analytic rank 1 with Tamagawa numers all 1 Let X C { } and ω dx 2y, ω 1 xdx 2y Taking a tangential asepoint at (or letting e an integral 2-torsion point), we have the analytic functions log ω (z) With this setup, we have z ω, D 2 (z) z ω ω 1 Theorem 81 ([3, 1]) uppose is a point of infinite order in C(Z) X (Z) C(Z p ) is in the zero set of Then f(z) : (log ω ( )) 2 D 2 (z) (log ω (z)) 2 D 2 ( ) Corollary 82 ([3, 1]) The expression (81) D 2 ( ) (log ω ( )) 2 is independent of the point of infinite order in C(Z) Example 83 We revisit Example 1 in [3] Let E e the rank 1 elliptic curve y 2 x x , with minimal model E having Cremona lael 65a1 Consider the following points on E which are integral on E: (3, ), (39, 18), ( 33, 18), R (147, 1728) Using Algorithm 71, we compute the following

16 16 JENNIFER BALAKRIHNAN integrals: ω ω O(11 7 ) ω O(11 7 ) ω ω O(11 7 ) ω O(11 7 ) ω ω O(11 7 ) ω O(11 7 ), and we see that the ratio in Corollary 82 is constant on integral points: D 2 ( ) (log ω ( )) 2 D 2() (log ω ()) 2 D 2(R) (log ω (R)) O(11 5 ) However, for (13, 98), which is not integral on E, we see that ω ω O(11 7 ) ω O(11 7 ) D 2 () (log ω ()) O(11 5 ) Example 84 We give a variation on Example 4 in [3] Let E e the rank 1 elliptic curve y 2 x 3 16x + 16, with minimal model E having Cremona lael 37a1 Letting, e two fixed integral points on E, we can use the Link lemma to rewrite Theorem 81 so that the relevant doule integral is no longer from a tangential asepoint Indeed, integral points z occur in the zero set of ( z ) ( 2 ) 2 ω ω ω ω 1 + ω ( ω 1 ω ) 2 ( ω ) 2 ( z ω ω 1 + z ) ω ω 1 lightly modifying Algorithm 74 to take as endpoint a parameter z (see [1, 722] for more details), we can recover the integral points {(, ±4), (4, ±4), ( 4, ±4), (8, ±2), (24, ±116)} 9 acknowledgments The author would like to thank Kiran Kedlaya for several helpful conversations, William tein for access to the computer sagemathwashingtonedu, and the referees for useful suggestions This work was done as part of the author s hd thesis at MIT, during which she was supported y an NF Graduate Fellowship and an NDEG Fellowship This paper was prepared for sumission while the author was supported y NF grant DM-11383

17 ITERATED COLEMAN INTEGRATION FOR HYERELLITIC CURVE 17 References 1 J Balakrishnan, Coleman integration for hyperelliptic curves: algorithms and applications, MIT hd thesis (211) 2 J Balakrishnan, R W Bradshaw, and K Kedlaya, Explicit Coleman integration for hyperelliptic curves, Algorithmic Numer Theory (G Hanrot, F Morain, and E Thomé, eds), Lecture Notes in Computer cience, vol 6197, pringer, 21, pp J Balakrishnan, K Kedlaya, and M Kim, Appendix and erratum to Massey products for elliptic curves of rank 1, J Amer Math oc 24 (211), no 1, A Besser, Coleman integration using the Tannakian formalism, Math Ann 322 (22), K T Chen, Algeras of iterated path integrals and fundamental groups, Trans Amer Math oc 156 (1971), R F Coleman, Dilogarithms, regulators and p-adic L-functions, Invent Math 69 (1982), no 2, , Torsion points on curves and p-adic aelian integrals, Ann of Math (2) 121 (1985), no 1, R F Coleman and E de halit, p-adic regulators on curves and special values of p-adic L-functions, Invent Math 93 (1988), no 2, K Kedlaya, Counting points on hyperelliptic curves using Monsky-Washnitzer cohomology, J Ramanujan Math oc 16 (21), , erratum iid 18 (23), M Kim, Massey products for elliptic curves of rank 1, J Amer Math oc 23 (21), W McCallum and B oonen, The method of Chaauty and Coleman, preprint (21)