$p$-进伴随 $L$-函数与 Hilbert 模特征簇的分歧轨迹

Galois branched covers with fixed ramification locus

We examine conditions under which there exists a non-constant family of Galois branched covers of curves over an algebraically closed field $k$ of fixed degree and fixed ramification locus, under a notion of equivalence derived from considering linear series on a fixed smooth proper source curve $X$. We show such a family exists precisely when the following conditions are satisfied: $\operatorname{char}(k)=p>0$, $X$ is isomorphic to $\mathbb{P}^1_k$, there is a unique ramification point, and the Galois group is $(\mathbb{Z}/p\mathbb{Z})^m$ for some integer $m>0$.

Topology and Geometry of the Berkovich Ramification Locus for Rational Functions, II

This article is the second installment in a series on the Berkovich ramification locus for nonconstant rational functions f: P^1 -> P^1. Here we show the ramification locus of f is contained in a strong tubular neighborhood of finite radius around the connected hull of the critical points if and only if f is tamely ramified at all of its critical points. When the ground field has characteristic zero, this bound may be chosen to depend only on the residue characteristic. We give two applications to classical non-Archimedean analysis, including a new version of Rolle's theorem for rational funct…

On the number of connected components of the ramification locus of a morphism of Berkovich curves

Let $k$ be a complete, nontrivially valued non-archimedean field. Given a finite morphism of quasi-smooth $k$-analytic curves that admit finite triangulations, we provide upper bounds for the number of connected components of the ramification locus in terms of topological invariants of the source curve such as its topological genus, the number of points in the boundary and the number of open ends.

Topology and Geometry of the Berkovich Ramification Locus for Rational Functions

Given a nonconstant holomorphic map f: X -> Y between compact Riemann surfaces, one of the first objects we learn to construct is its ramification divisor R_f, which describes the locus at which f fails to be locally injective. The divisor R_f is a finite formal linear combination of points of X that is combinatorially constrained by the Hurwitz formula. Now let k be an algebraically closed field that is complete with respect to a nontrivial non-Archimedean absolute value. For example, k = C_p. Here the role of a Riemann surface is played by a projective Berkovich analytic curve. As these curv…

Radius of convergence of p-adic connections and the Berkovich ramification locus

We apply the theory of the radius of convergence of a p-adic connection to the special case of the direct image of the constant connection via a finite morphism of compact p-adic curves, smooth in the sense of rigid geometry. In the case of an etale covering of curves with good reduction, we get a lower bound for that radius and obtain a new geometric proof of a variant of the p-adic Rolle theorem of Robert and Berkovich. We take this opportunity to clarify the relation between our notion of radius of convergencand the more intrinsic one used by Kedlaya

关于 Bianchi 尖点形式的 $p$-进伴随 $L$-函数：$p$ 分裂情形

On $p$-adic adjoint $L$-functions for Bianchi cuspforms: the $p$-split case

本文通过 Bianchi 三维簇过收敛上同调上的 Hecke 等变配对，在 $p$ 分裂情形构造 Bianchi 尖点形式的 $p$-进伴随 $L$-函数，刻画特征簇分歧轨迹并证明其在特定点非零。

Ramification theory for varieties over a local field

We define generalizations of classical invariants of wild ramification for coverings on a variety of arbitrary dimension over a local field. For an l-adic sheaf, we define its Swan class as a 0-cycle class supported on the wild ramification locus. We prove a formula of Riemann-Roch type for the Swan conductor of cohomology together with its relative version, assuming that the local field is of mixed characteristic. We also prove the integrality of the Swan class for curves over a local field as a generalization of the Hasse-Arf theorem. We derive a proof of a conjecture of Serre on the Artin c…

p进群的弱阿瑟包的分歧

Ramification of weak Arthur packets for p-adic groups

本文研究p进群的弱阿瑟包，明确其分解为内射转移产生的阿瑟包的方式，引入弱球性概念并揭示其与Lusztig典范商空间的匹配关系，为相关朗兰兹理论研究提供支撑。

Hilbert reciprocity using K-theory localization

Usually the boundary map in K-theory localization only gives the tame symbol at $K_{2}$. It sees the tamely ramified part of the Hilbert symbol, but no wild ramification. Gillet has shown how to prove Weil reciprocity using such boundary maps. This implies Hilbert reciprocity for curves over finite fields. However, phrasing Hilbert reciprocity for number fields in a similar way fails because it crucially hinges on wild ramification effects. We resolve this issue, except at p=2. Our idea is to pinch singularities near the ramification locus. This fattens up K-theory and makes the wild symbol vi…

超椭圆曲线线包的算术折点

Arithmetic inflection of superelliptic curves

本文研究了超椭圆曲线线包的折点行为，探讨了折点集与牛顿多面体的关系。

Division Algebra Cyclicity in Prime Degree over a p-Adic Curve

We reprove two results of Saltman, Theorem 5.1 and Corollary 5.2 of [Sa07]: If F is the function field of a smooth p-adic curve and D is an F-division algebra of prime degree l\neq p, then D is Z/l-cyclic, and that if D is an F-division algebra of prime period l\neq p then D has index l if and only if its ramification locus on a suitable 2-dimensional model forF has no "hot points".