Wild Galois representations: a family of hyperelliptic curves with large inertia image

Towards the Generalized Purely Wild Inertia Conjecture for product of Alternating and Symmetric Groups

We obtain new evidence for the Purely Wild Inertia Conjecture posed by Abhyankar and for its generalization. We show that this generalized conjecture is true for any product of simple Alternating groups in odd characteristics, and for any product of certain Symmetric or Alternating groups in characteristic two. We also obtain important results towards the realization of the inertia groups which can be applied to more general set up. We further show that the Purely Wild Inertia Conjecture is true for any product of perfect quasi $p$-groups (groups generated by their Sylow $p$-subgroups) if the …

The 2-ranks of Hyperelliptic Curves with Extra Automorphisms

This paper examines the relationship between the automorphism group of a hyperelliptic curve defined over an algebraically closed field of characteristic two and the 2-rank of the curve. In particular, we exploit the wild ramification to use the Deuring-Shafarevich formula in order to analyze the ramification of hyperelliptic curves that admit extra automorphisms and use this data to impose restrictions on the genera and 2-ranks of such curves. We also show how some of the techniques and results carry over to the case where our base field is of characteristic $p>2$.

聚类、托里克秩和2-秩的超椭圆曲线在野情况下的研究

Clusters, toric ranks, and 2-ranks of hyperelliptic curves in the wild case

本文研究超椭圆曲线在特征为2的残差域下的聚类、托里克秩和2-秩性质，通过定义相对稳定模型并分析其特殊纤维的图结构，给出计算方法及阈值判定。

Root number of the Jacobian of $y^2=x^p+a$

Let $C/\mathbb{Q}$ be a hyperelliptic curve with an affine model of the form $y^2=x^p+a$. We explicitly determine the root number of the Jacobian of $C$, with particular focus on the local root number at $p$ where $C$ has wild ramification.

关于一族弗雷超椭圆曲线的体数

On the conductor of a family of Frey hyperelliptic curves

本文研究了一族弗雷超椭圆曲线的体数，通过双参数超椭圆曲线C(z,s)推广了已有结果，并提出了新的构造方法。

枚举$\u2119^1$上的奇次超椭圆曲线与阿贝尔曲面

Enumerating odd-degree hyperelliptic curves and abelian surfaces over $\mathbb{P}^1$

针对Venkatesh提出的数论渐近计数中低阶项拓扑本质的问题，研究者通过考察有限域上代数栈惯性栈的算术方面部分作答，并得到$\u2119_q(t)$上拟可容许奇次超椭圆曲线的新精确枚举结果。

wild情形下超椭圆曲线的簇与半稳定模型

Clusters and semistable models of hyperelliptic curves in the wild case

本文研究了超椭圆曲线在剩余特征p=2时的半稳定模型，提出了相对稳定模型的计算方法，并揭示了特殊纤维分支与根的簇之间的对应关系。

Wild Galois Representations: Elliptic curves over a $3$-adic field

Given an elliptic curve $E$ over a local field $K$ with residue characteristic $3$, we investigate the action of the absolute Galois group of $K$ in the case of potentially good reduction. In particular the only not completely known case is that of the $\ell$-adic Galois representation attached to an elliptic curve such that the image of inertia is non-cyclic, and isomorphic to $C_3 \rtimes C_4$. In this work we describe such a representation explicitly.

Constructing hyperelliptic curves with surjective Galois representations

In this paper we show how to explicitly write down equations of hyperelliptic curves over Q such that for all odd primes l the image of the mod l Galois representation is the general symplectic group. The proof relies on understanding the action of inertia groups on the l-torsion of the Jacobian, including at primes where the Jacobian has non-semistable reduction. We also give a framework for systematically dealing with primitivity of symplectic mod l Galois representations. The main result of the paper is the following. Suppose n=2g+2 is an even integer that can be written as a sum of two pri…

Wildly ramified covers with large genus

We study wildly ramified G-Galois covers $ϕ:Y \to X$ branched at B (defined over an algebraically closed field of characteristic p). We show that curves Y of arbitrarily high genus occur for such covers even when G, X, B and the inertia groups are fixed. The proof relies on a Galois action on covers of germs of curves and formal patching. As a corollary, we prove that for any nontrivial quasi-p group G and for any sufficiently large integer $σ$ with $p \nmid σ$, there exists a G-Galois étale cover of the affine line with conductor $σ$ above the point $\infty$.

关于具有单个剧烈分支跳跃的局部域的伽罗瓦扩张

On Galois Extensions of Local Fields with a Single Wild Ramification Jump

研究了具有单个剧烈分支跳跃的局部域伽罗瓦扩张的计数方法，结合卢宾-塔特扩张和规范子群的性质分析。

晶贝林形变环的不可约分支：极分歧情形

Irreducible components of crystabelline deformation rings: the wildly ramified case

本文针对p≥11的素数上可约非常一般伽罗瓦表示$\bar\rho$，利用ghost series研究其Kisin潜在晶形变环的不可约分支，确定分支总数及各分支特殊纤维的Hilbert-Samuel重数。

Tame multiplicity and conductor for local Galois representations

Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$. Let $σ$ be an irreducible smooth representation of the absolute Weil group $\Cal W_F$ of $F$ and $\sw(σ)$ the Swan exponent of $σ$. Assume $\sw(σ) \ge1$. Let $\Cal I_F$ be the inertia subgroup of $\Cal W_F$ and $\Cal P_F$ the wild inertia subgroup. There is an essentially unique, finite, cyclic group $\varSigma$, of order prime to $p$, so that $σ(\Cal I_F) = σ(\Cal P_F)\varSigma$. In response to a query of Mark Reeder, we show that the multiplicity in $σ$ of any character of $\varSigma$ is bounded by $\sw(σ)$.

Alternating group covers of the affine line

We prove Abhyankar's Inertia Conjecture for the alternating group A_{p+2} on p+2 letters when p = 2 mod 3, by showing that every possible inertia group occurs for a (wildly ramified) A_{p+2}-Galois cover of the projective k-line branched only at infinity where k is an algebraically closed field of characteristic p > 0. More generally, when 1 < s < p and gcd(p-1, s+1)=1, we prove that all but finitely many rational numbers which satisfy the obvious necessary conditions occur as the upper jump in the filtration of higher ramification groups of an A_{p+s}-Galois cover of the projective line branc…

$\overline{\mathbb{Z}}[\frac{1}{p}]$-块的 tame $p$-adic 群的参数化与降维到深度零

Parametrization and reduction to depth zero of $\overline{\mathbb{Z}}[\frac{1}{p}]$-blocks of tame $p$-adic groups

研究通过参数化和降维到深度零的方法，揭示了 tame $p$-adic 群的 $\overline{\mathbb{Z}}[\frac{1}{p}]$-块结构与局部朗兰兹对应的关系。