Jennifer S. Balakrishnan

Explicit Coleman Integration for Hyperelliptic Curves

Coleman’s theory of p-adic integration figures prominently in several number-theoretic applications, such as finding torsion and rational points on curves, and computing p-adic regulators in K-theory (including p-adic heights on elliptic curves). We describe an algorithm for computing Coleman integrals on hyperelliptic curves, and its implementation in Sage.

Appendix and erratum to “Massey products for elliptic curves of rank 1”

The paper [6] contains a few errors in the basic assumptions as well as in the formula of corollary 0.2. First of all, it should have been made clear at the outset that the regular model E for the elliptic curve E must be the minimal regular model, and X the complement of the origin in the regular minimal model. Similarly, the tangential base-point b must be integral, in that it is a Z−basis of the relative tangent space eTE/Z. It could also be an integral two-torsion point for the arguments of the paper to hold verbatim. The most significant error is in the contribution of the local terms at l 6= p, that is, Lemma 1.2. The problem is that a point that is integral on X may not be integral on a smooth model over a field of good reduction. As it stands, the lemma will only apply to points that are integral in this stronger sense. However, to get immediate examples, one can replace the lemma by

Quadratic Chabauty and rational points, I:

We give the first explicit examples beyond the Chabauty-Coleman method where Kims nonabelian Chabauty program determines the set of rational points of a curve defined over

p

\mathbb{Q}

-adic heights

or a quadratic number field. We accomplish this by studying the role of

A p-adic analogue of the conjecture of Birch and Swinnerton-Dyer for modular abelian varieties

p

Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks

-adic heights in explicit nonabelian Chabauty.

Computing integral points on hyperelliptic curves using quadratic Chabauty

Mazur, Tate, and Teitelbaum gave a p-adic analogue of the Birch and Swinnerton-Dyer conjecture for elliptic curves. We provide a generalization of their conjecture in the good ordinary case to higher dimensional modular abelian varieties over the rationals by constructing the p- adic L-function of a modular abelian variety and showing it satises the appropriate interpolation property. We describe the techniques used to formulate the conjecture and give evidence supporting the conjecture in the case when the modular abelian variety is of dimension 2.

Constructing genus 3 hyperelliptic Jacobians with CM

Most systematic tables of data associated to ranks of elliptic curves order the curves by conductor. Recent developments, led by work of Bhargava-Shankar studying the average sizes of

-adic heights of Heegner points and Λ-adic regulators

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