Explicit Coleman integration for curves

Abstract

The Coleman integral is a p-adic line integral that plays a key role in computing several important invariants in arithmetic geometry. We give an algorithm for explicit Coleman integration on curves, using the algorithms of the second author [Tui16, Tui17] to compute the action of Frobenius on p-adic cohomology. We present a collection of examples computed with our implementation.

Cubic and quartic points on modular curves using generalised symmetric Chabauty

“…Such zero sets can be computed for d = 1 by evaluating Coleman integrals between Q prational points. In Sage on hyperelliptic curves, the computation of Coleman integrals over Q p has been implemented by Balakrishnan, Bradshaw, and Kedlaya [BBK10], [Bal15], while Balakrishnan and Tuitman [BT20] wrote a Magma implementation for all plane curves. For d > 1, however, one would need to evaluate Coleman integrals between points P, Q ∈ X(K) for extensions K/Q p of degree > 1, and this has not yet been done.…”

p-adic Integration on Hyperelliptic Curves of Bad Reduction: Algorithms and Applications

“…Some of these applications rely heavily on explicitly computing integrals. Coleman's construction is quite suitable for machine computation, and, when p is odd, there are practical algorithms due to Balakrishnan and others: see [BBK10, BB12, Bal15, Bes19] (single integrals on hyperelliptic curves), [Bal13,Bal15] (double integrals on hyperelliptic curves), [Bes21b] (single integrals on superelliptic curves), and [BT20] (single integrals on smooth curves). However, due to the highly abstract nature of Vologodsky's construction, Vologodsky integration has been, so far, difficult to compute.…”

“…In specific situations, we have implementations of various steps of this algorithm in Sage [The20] and/or Magma [BCP97], and using these, one can produce several interesting examples. In fact, we have mostly used Sage, because it includes implementations of the Coleman integration algorithms in [BBK10,BB12,Bal15], and such algorithms have only recently been implemented in Magma, see [BT20,BM]. Another reason for using Sage is that it is currently the only option to compute double Coleman integrals.…”

“…The method of Chabauty-Coleman [MP12] is a p-adic method that attempts to determine X(Q) under the condition that the Mordell-Weil rank r of the Jacobian of X is less than g. This method relies on the construction of an annihilating differential, which exists by the hypothesis r < g. Such a construction requires, among other things, computing Vologodsky integrals of regular 1-forms on X. In the special case that X has good reduction at p, this can be achieved using the algorithms developed by Balakrishnan-Tuitman [BT20]. In fact, explicit Chabauty-Coleman computations have been, so far, restricted to the good reduction case.…”

“…Algorithms to carry out integration on more general curves were developed in Balakrishnan-Tuitman [BT20] based on the work of Tuitman [Tui16,Tui17] that generalizes Kedlaya's algorithm to this setting. We note that these algorithms work only for meromorphic 1-forms that are holomorphic away from very bad points (see [BT20,Definition 2.8]).…”