Explicit Coleman Integration in Larger Characteristic

Chabauty-Coleman computations on rank 1 Picard curves

We provably compute the full set of rational points on 1403 Picard curves defined over $\mathbb{Q}$ with Jacobians of Mordell-Weil rank $1$ using the Chabauty-Coleman method. To carry out this computation, we extend Magma code of Balakrishnan and Tuitman for Coleman integration. The new code computes $p$-adic (Coleman) integrals on curves to points defined over number fields where the prime $p$ splits completely and implements effective Chabauty for curves whose Jacobians have infinite order points that are not the image of a rational point under the Abel-Jacobi map. We discuss several interes…

p-adic integration on bad reduction hyperelliptic curves

In this paper, we introduce an algorithm for computing p-adic integrals on bad reduction hyperelliptic curves. For bad reduction curves, there are two notions of p-adic integration: Berkovich-Coleman integrals which can be performed locally; and abelian integrals with desirable number-theoretic properties. By covering a bad reduction hyperelliptic curve by annuli and basic wide open sets, we reduce the computation of Berkovich-Coleman integrals to the known algorithms on good reduction hyperelliptic curves. These are due to Balakrishnan, Bradshaw, and Kedlaya, and to Balakrishnan and Besser fo…

Vologodsky integration on curves with semi-stable reduction

We prove that the Vologodsky integral of a mermorphic one-form on a curve over a $p$-adic field with semi-stable reduction restrict to Coleman integrals on the rigid subdomains reducing to the components of the smooth part of the special fiber and that on the connecting annuli the differences of these Coleman integrals form a harmonic cochain on the edges of the dual graph of the special fiber. This determines the Vologodsky integral completely. We analyze the behavior of the integral on the connecting annuli and we explain the results in the case of a Tate elliptic curve.

半稳定曲线上的 p-adic 迭代积分

p-adic iterated integration on semistable curves

本文以幂零对数刚性基本群重构半稳定曲线上的 p-adic 迭代积分，借助 Frobenius 不变量与对偶图阐明 Berkovich–Coleman 积分，并将 Vologodsky 积分解释为沿组合典范路径的积分。

Explicit Vologodsky Integration for Hyperelliptic Curves

Vologodsky's theory of $p$-adic integration plays a central role in computing several interesting invariants in arithmetic geometry. In contrast with the theory developed by Coleman, it has the advantage of being insensitive to the reduction type at $p$. Building on recent work of Besser and Zerbes, we describe an algorithm for computing Vologodsky integrals on bad reduction hyperelliptic curves. This extends previous joint work with Katz to all meromorphic differential forms. We illustrate our algorithm with numerical examples computed in Sage.

p-adic Arakelov theory

We introduce the p-adic analogue of Arakelov intersection theory on arithmetic surfaces. The intersection pairing in an extension of the p-adic height pairing for divisors of degree 0 in the form described by Coleman and Gross. It also uses Coleman integration and is related to work of Colmez on p-adic Green functions. We introduce the p-adic version of a metrized line bundle and define the metric on the determinant of its cohomology in the style of Faltings. It is possible to prove in this theory analogues of the Adjunction formula and the Riemann-Roch formula.

The syntomic regulator for $K_4$ of curves

Let $C$ be a curve defined over a discrete valuation field of characteristic zero where the residue field has positive characteristic. Assuming that $C$ has good reduction over the residue field, we compute the syntomic regulator on a certain part of $K_4^[(3)}(C)$. The result can be expressed in terms of $p$-adic polylogarithms and Coleman integration. We also compute the syntomic regulator on a certain part of $K_4^[(3)}(F)$ for the function field $F$ of $C$. The result can be expressed in terms of $p$-adic polylogarithms and Coleman integration, or by using a trilinear map ("triple index") …

Diophantine and tropical geometry, and uniformity of rational points on curves

We describe recent work connecting combinatorics and tropical/non-Archimedean geometry to Diophantine geometry, particularly the uniformity conjectures for rational points on curves and for torsion packets of curves. The method of Chabauty--Coleman lies at the heart of this connection, and we emphasize the clarification that tropical geometry affords throughout the theory of $p$-adic integration, especially to the comparison of analytic continuations of $p$-adic integrals and to the analysis of zeros of integrals on domains admitting monodromy.

Motivic Mellin变换

Motivic Mellin transforms

本文将Mellin变换引入motivic积分领域，构建了在Mellin和Fourier变换下稳定的新函数类，提供Fubini定理与变量替换公式，并推广了前人理论与转移原理。