Prime-to-p étale covers of algebraic groups

$q$-étale covers of cyclic $p$-gonal covers

In this paper we study the Galois group of the Galois cover of the composition of a $q$-cyclic étale cover and a cyclic $p$-gonal cover for any odd prime $p$. Furthermore, we give properties of isogenous decompositions of certain Prym and Jacobian varieties associated to intermediate subcovers given by subgroups.

Etale double covers of cyclic p-gonal covers

This paper computes the Galois group of the Galois cover of the composition of an étale double cover of a cyclic $p$-gonal cover for any prime $p$. Moreover a relation between some of its Prym varieties and the Jacobian of a subcover is given. In a sense this generalizes the trigonal construction.

Double covers of EPW-sextics

EPW-sextics are special 4-dimensional sextic hypersurfaces (with 20 moduli) which come equipped with a double cover. We analyze the double cover of EPW-sextics parametrized by a certain prime divisor in the moduli space. We associate to the generic sextic parametrized by that divisor a K3 surface of genus 6 and we show that the double epw sextic is a contraction of the Hilbert square of the K3. This result has two consequences. First it gives a new proof of the following result of ours: smooth double EPW-sextics form a locally complete family of hyperkaehler projective deformations of the Hilb…

素数阶椭圆曲线的覆盖攻击

Cover attacks for elliptic curves with prime order

提出一种经 Weil 限制、$(\ell,\ell,\ell)$-同源迁移至三亏格 Jacobian 并用指标演算的新型覆盖攻击，可在 $\tilde{O}(q)$ 时间内求解部分素数阶椭圆曲线离散对数。

The Monodromy group of $pq$-covers

In this work we study the monodromy group of covers $φ\circ ψ$ of curves \linebreak $\mathcal{Y}\xrightarrow {\quad ψ} \mathcal{X} \xrightarrow {\quad φ} \mathbb{P}^{1}$, where $ψ$ is a $q$-fold cyclic étale cover and $φ$ is a totally ramified $p$-fold cover, with $p$ and $q$ different prime numbers with $p$ odd. We show that the Galois group $\mathcal{G}$ of the Galois closure $\mathcal{Z}$ of $φ\circ ψ$ is of the form $ \mathcal{G} = \mathbb{Z}_q^s \rtimes \mathcal{U}$, where $0 \leq s \leq p-1$ and $\mathcal{U}$ is a simple transitive permutation group of degree $p$. Since the simple transi…

Non-constant curves of genus 2 with infinite pro-Galois covers

For every odd prime number p, we give examples of non-constant smooth families of genus 2 curves over fields of characteristic p which have pro-Galois (pro-étale) covers of infinite degree with geometrically connected fibers. The Jacobians of the curves are isomorphic to products of elliptic curves.

The torsion in the cohomology of wild elliptic fibers

Given an elliptic fibration $f \colon X \to S$ over the spectrum of a complete discrete valuation ring with algebraically closed residue field, we use a Hochschild--Serre spectral sequence to express the torsion in $R^1f_\ast \mathscr{O}_X$ as the first group cohomology $H^1(G,H^0(S^\prime, \mathscr{O}_{S^\prime}))$. Here, $G$ is the Galois group of the maximal extension $K^\prime / K$ such that the normalization of $X \times_S S^\prime$ induces an étale covering of $X$, where $S^\prime$ is the normalization of $S$ in $K^\prime$. The case where $S$ is a Dedekind scheme is easily reduced to the…

论热带非分歧p-覆叠模空间之拓扑

On the topology of the moduli of tropical unramified p-covers

本文研究亏格g≥2热带曲线非分歧Z/p-覆叠模空间的拓扑，利用Chan–Galatius–Payne技巧证明其单连通性，并在亏格2时确定所有素数p下的同伦型，为素数阶循环étale覆叠的顶权上同调提供应用前景。

Finite groups scheme actions and incompressibility of Galois covers: beyond the ordinary case

Inspired by recent work of Farb, Kisin and Wolfson, we develop a method for using actions of finite group schemes over a mixed characteristic dvr R to get lower bounds for the essential dimension of a cover of a variety over K = Frac(R). We then apply this to prove p-incompressibility for congruence covers of a class of unitary Shimura varieties for primes p at which the reduction of the Shimura variety (at any prime of the reflex field over p) does not have any ordinary points. We also make some progress towards a conjecture of Brosnan on the p-incompressibility of the multiplication by p map…

The boundary of the $p$-rank $0$ stratum of the moduli space of cyclic covers of the projective line

We study the $p$-rank stratification of the moduli space of cyclic degree $\ell$ covers of the projective line in characteristic $p$ for distinct primes $p$ and $\ell$. The main result is about the intersection of the $p$-rank $0$ stratum with the boundary of the moduli space of curves. When $\ell=3$ and $p \equiv 2 \bmod 3$ is an odd prime, we prove that there exists a smooth trielliptic curve in characteristic $p$, with every genus $g$, signature type $(r,s)$ and $p$-rank $f$ satisfying the clear necessary conditions.

关于 $\mathbf{P}^1$ 在三个点处分支的阿贝尔覆盖的牛顿多边形与埃克萨尔-奥尔特类型

Ekedahl-Oort Types and Newton Polygons of Abelian Covers of $\mathbf{P}^1$ Branched at Three Points

本文研究了$\mathbf{P}^1$在三个点处分支的阿贝尔覆盖的牛顿多边形与埃克萨尔-奥尔特类型，揭示了超奇异和超特殊曲线的高频出现，并为Oort猜想提供了新的证据。

On étale fundamental groups of formal fibres of $p$-adic curves

We investigate a certain class of (geometric) finite (Galois) coverings of formal fibres of $p$-adic curves and the corresponding quotient of the (geometric) étale fundamental group. A key result in our investigation is that these (Galois) coverings can be compactified to finite (Galois) coverings of proper $p$-adic curves. We also prove that the maximal prime-to-$p$ quotient of the geometric étale fundamental group of a (geometrically connected) formal fibre of a $p$-adic curve is (pro-)prime-to-$p$ free of finite computable rank.

Cyclic covers of the projective line, their jacobians and endomorphisms

We study the endomorphism ring $End(J(C))$ of the complex jacobian $J(C)$ of a curve $y^p=f(x)$ where $p$ is an odd prime and $f(x)$ is a polynomial with complex coefficiens of degree $n>4$ and without multiple roots. Assume that all the coefficients of $f$ lie in a (sub)field $K$ and the Galois group of $f$ over $K$ is either the full symmetric group $S_n$ or the alternating group $A_n$. Then we prove that $End(J(C))$ is the ring of integers in the in the $p$th cyclotomic field, if $p$ is a Fermat prime (e.g., $p=3,5,17,257$). Similar results for $p=2$ (the case of hyperelliptic curves) were …

Lifting in Frattini Covers and A Characterization of Finite Solvable Groups

We prove a lifting theorem for odd Frattini covers of finite groups. Using this, we characterize solvable groups and more generally p-solvable groups in terms of containing a triple of elements of distinct prime power orders with product 1. This is also related to various questions about Fried's modular tower program and properties of Hurwitz spaces of covers of curves.