Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.20575

Arithmetic Siegel--Weil formula on the product of modular curves

Ryan C. Chen·Baiqing Zhu
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove the full generating series arithmetic Siegel--Weil formula on the product of two modular curves, with arbitrarily singular coefficients. This is the case $\operatorname{GSpin}(2,2)$ in Kudla's program.

Explore related subjects

math.NTmath.AGmath.RT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ryan C. Chen, Baiqing Zhu. 2026-10-14. Arithmetic Siegel--Weil formula on the product of modular curves. https://arxiv.org/abs/2610.20575

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Extensions of the Carlitz-McConnel and Blokhuis-Sziklai theorems for unions of cyclotomic classes

Let $p$ be a prime, let $q=p^n$, and let $D\subseteq \mathbb{F}_q^\ast$. A celebrated result of Carlitz and McConnel states that if $D$ is a proper subgroup of $\mathbb{F}_q^\ast$, and $f:\mathbb{F}_q\to\mathbb{F}_q$ is a function such that $(f(x)-f(y))/(x-y)\in D$ for all $x\neq y$, then $f$ must be of the form $f(x)=ax^{p^j}+b$. In this paper, we extend their result to the setting where $D$ is a union of cosets of a fixed subgroup of $\mathbb{F}_q^\ast$, under a mild assumption. In a similar spirit, we also investigate maximum cliques in related Cayley graphs over finite fields, strengthening several results of Blokhuis, Sziklai, and Asgarli and Yip.

math.NT↗
arXiv2026

Root numbers for twisted Fermat quotient curves II

This is a sequel to the previous work of the author Yanagihara (2025). Let $\ell$ be an odd prime, let $N \geq 1$ be an integer, and let $δ\geq 1$ be an $\ell^N$-th-power-free integer. Let $r,s,t>0$ be integers satisfying $r+s+t=\ell^N$. In Yanagihara (2025), the author computed the root number of the Fermat quotient curve $y^{\ell^N}=x^r(δ-x)^s$ under the assumptions that $\ell\nmid rst$ and that $\operatorname{ord}_{\ell}(δ)=0$ or $\ell\nmid \operatorname{ord}_{\ell}(δ)$. In this paper, we study the case where the technical assumption $\ell\nmid rst$ is dropped. As one such case, we compute the root number when $\ell^{N-1}\| r$ and $\ell\nmid stδ$.

math.NT↗
arXiv2026

Potential semistability of Finite height Galois representations: Relative case

Let $K$ be a $p$-adic field. We define the notion of finite height for an étale $\mathbb{Z}_p$-local system on a smooth affinoid adic space $\mathcal{X}$ over $K$ with semistable reduction. Using analytic prismatic $F$-crystals and purity results of Du-Liu-Moon-Shimizu (arXiv:2404.19603), we prove that if an étale $\mathbb{Z}_p$-local system over $\mathcal{X}$ is of finite height then its pullback along a finite étale cover of $\mathcal{X}$ is semistable. This answers a question of Tong Liu in the relative setting.

math.NT↗