The p-adic height pairings of Coleman-Gross and of Nekovar

近高阶Coleman理论与$\mathrm{GSp}(4) \times \mathrm{GL}(2)$和$\mathrm{GSp}(4) \times \mathrm{GL}(2) \times \mathrm{GL}(2)$的$p$-adic $L$-函数

Nearly higher Coleman theory and p-adic L-functions for $\mathrm{GSp}(4) \times \mathrm{GL}(2)$ and $\mathrm{GSp}(4) \times \mathrm{GL}(2) \times \mathrm{GL}(2)$

本文构造了Siegel模形式与模形式扭曲的$p$-adic $L$-函数，利用近过收敛模形式空间和Gan-Gross-Prasad周期，在Coleman族中实现插值。

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.

$U_{n+1} \ imes U_n$ 的 Coleman 族的反循环 $p$-进 $L$-函数

Anticyclotomic $p$-adic $L$-functions for Coleman families of $U_{n+1} \times U_{n}$

本文利用定酉群球对分支律的 $p$-进插值，构造并在重数一假设下推广至 Coleman 族的反循环 $p$-进 $L$-函数，用以插值中心临界 $L$-值的平方根。

$p$-adic Gross-Zagier Formula for Heegner Points on Shimura Curves over Totally Real Fields

The main result of this text is a generalization of Perrin-Riou's p-adic Gross-Zagier formula to the case of Shimura curves over totally real fields. Let $F$ be a totally real field. Let $f$ be a Hilbert modular form over $F$ of parallel weight $2$, which is a new form and is ordinary at $p$. Let $E$ be a totally imaginary quadratic extension of $F$ of discriminant prime to $p$ and to the conductor of $f$. We may construct a $p$-adic $L$ function that interpolates special values of the complex $L$ functions associated to $f$, $E$ and finite order Hecke characters of $E$. The $p$-adic Gross-Zag…

On a common refinement of Stark units and Gross-Stark units

The purpose of this paper is to formulate and study a common refinement of a version of Stark's conjecture and its $p$-adic analogue, in terms of Fontaine's $p$-adic period ring and $p$-adic Hodge theory. We construct period-ring-valued functions under a generalization of Yoshida's conjecture on the transcendental parts of CM-periods. Then we conjecture a reciprocity law on their special values concerning the absolute Frobenius action. We show that our conjecture implies a part of Stark's conjecture when the base field is an arbitrary real field and the splitting place is its real place. It al…

任意约化类型超椭圆曲线上 $p$-进高度的算法

Algorithms for $p$-adic Heights on Hyperelliptic Curves of Arbitrary Reduction

本文提出计算任意约化类型超椭圆曲线上 Coleman--Gross $p$-进高度的算法，涵盖局部高度计算、Blakestad 分裂算法及数值实例，包括首个坏约化曲线的二次 Chabauty 示例。

p-Adic sigma functions and heights on Jacobians of genus 2 curves

Let $C$ be a genus $2$ hyperelliptic curve over a number field $K$, with a Weierstrass point $\infty$ at infinity, let $J$ be its Jacobian, let $Θ$ be the theta divisor with respect to $\infty$, and let $p$ be any prime number. We give an explicit construction of a $p$-adic height $h_p\colon J(\overline{\mathbb{Q}})\to \mathbb{Q}_p$ by means of $p$-adic analogues of Néron functions of divisor $2Θ$. We define such Néron functions using division polynomials and a generalisation of Blakestad's $p$-adic sigma function on the formal group of $J$. We prove that our $p$-adic Néron function $λ_v$ at a…

在双曲曲线上的p进高度计算

Computing p-adic heights on hyperelliptic curves

本文提出一种计算双曲曲线在p处的局部Coleman-Gross p进高度的算法，该算法比之前的算法更高效且适用于奇数和偶数次双曲曲线。

p-进解析度量与二次查巴乌 I

p-adic adelic metrics and Quadratic Chabauty I

本文提出了一种基于p-进阿雷克洛夫理论的新方法，用于构造abelian簇上的p-进高度，并简化了二次查巴乌方法用于计算有理点。