Klein-Four covers of the projective line in characteristic two
Darren Glass
本文研究了定义在特征为2的域上的曲线，它们是射影直线的$(\ZZ/2\ZZ)^2$ -覆盖。特别是，我们证明了这2秩发生在这样的曲线的一个给定的属，并在可能的情况下，我们给出明确的方程，这样的曲线。
In this paper we examine curves defined over a field of characteristic 2 which are $(\ZZ/2\ZZ)^2$-covers of the projective line. In particular, we prove which 2-ranks occur for such curves of a given genus and where possible we give explicit equations for such curves.
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In this paper we compute the first vertex of a generic Newton polygon in some special cases, and the corresponding Hasse polynomial. This allows us to show the nonexistence of $p$-cyclic coverings of the projective line in characteristic $p$ with supersingular jacobian for some (infinite families of) genera.
We use tools from combinatorial group theory in order to study actions of three types on groups acting on a curve, namely the automorphism group of a compact Riemann surface, the mapping class group acting on a surface (which now is allowed to have some points removed) and the absolute Galois group $\mathrm{Gal}(\bar{\Q}/\Q)$ in the case of cyclic covers of the projective line.
Counting rational points on Hirzebruch-Kleinschmidt varieties over number fields
本文研究了数域上 Picard 秩为 2 的光滑射影分裂环面簇在 Arakelov 高度下的有理点渐近增长，证明其可分解为有限个不相交子簇并给出显式渐近公式，同时提供了包含 Hirzebruch 曲面的实例。
Generic ordinarity for abelian coverings of the projective line
研究射影直线阿贝尔覆盖在特征$p$下的平凡性，通过计算与射影直线上乘法特征和相关的$L$函数的一般牛顿多边形，证明赫维茨空间的一般牛顿多边形与包含其像的最小志村簇的一致。
Let $k$ be an algebraically closed field of characteristic $p$ and let $X$ the projective line over $k$ with three points removed. We investigate which finite groups $G$ can arise as the monodromy group of finite étale covers of $X$ that are tamely ramified over the three removed points. This provides new information about the tame fundamental group of the projective line. In particular, we show that for each prime $p\ge 5$, there are families of tamely ramified covers with monodromy the symmetric group $S_n$ or alternating group $A_n$ for infinitely many $n$. These covers come from the moduli…
Mod p Monodromy of Cyclic Covers of the Projective Line
本文证明了对于模p系数的项目线循环覆盖的monodromy的大monodromy定理，扩展了Achter和Pries关于二阶和三阶循环覆盖的结果，采用积分上同调的证明方法。
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.
Zeta functions of totally ramified p-covers of the projective line
本文证明了在具有任意k个指定阶极点的一元有理函数空间中，存在定义在有理数Q上的一个Zebraki稠密开子集U，使得对于U（Qbar）$中的每一个几何点f，f在素数p处的指数和的L-函数有Newton多边形，当p趋于无穷大时，该Newton多边形逼近Hodge多边形.作为代数几何中的一个应用，我们证明了在任意k个指定阶点完全分歧的射影直线的p-覆盖的zeta函数的p进牛顿多边形有一个渐近通有下界.
Tschirnhausen bundles of covers of the projective line
研究光滑曲线覆盖射影直线时，结构层前推得到的向量丛的可能类型，对度数为4的情形给出完整答案，并提出一个多面体猜想以刻画本原覆盖的丛，证明所有本原覆盖的丛落在此多面体内且正比例可实现，同时展示本原性假设的必要性及Hurwitz空间像在推广下不保持。
Let $\mathcal{P}: \cdots \rightarrow C_2\rightarrow C_1\rightarrow {\mathbb P}^1$ be a $\mathbb{Z}_p$-cover of the projective line over a finite field of cardinality $q$ and characteristic $p$ which ramifies at exactly one rational point, and is unramified at other points. In this paper, we study the $q$-adic valuations of the reciprocal roots in $\mathbb{C}_p$ of $L$-functions associated to characters of the Galois group of $\mathcal{P}$. We show that for all covers $\mathcal{P}$ such that the genus of $C_n$ is a quadratic polynomial in $p^n$ for $n$ large, the valuations of these reciprocal …
Klein Quartic Curve and its Modularity
本文通过将克莱因四次曲线实现为 7 次费马曲线的商来计算其局部 Zeta 函数，并借助伽罗瓦表示给出相应的模形式。
We provide a Boseck-type basis of the space of holomorphic differentials for a large class of solvable covers of the projective line with perfect field of constants of characteristic $p > 0$. Within this class, we also describe the Galois module structure of holomorphic differentials for abelian covers.
A smooth geometrically connected curve over the finite field $\mathbb{F}_q$ with gonality $γ$ has at most ${γ(q+1)}$ rational points. The first author and Grantham conjectured that there exist curves of every sufficiently large genus with gonality $γ$ that achieve this bound. In this paper, we show that this bound can be achieved for an infinite sequence of genera using abelian covers of the projective line. We also argue that abelian covers will not suffice to prove the full conjecture.
Numerical calculation of three-point branched covers of the projective line
本文提出一种数值方法来计算复射影直线的三次分支覆盖，通过开发处理Fuchsian三角群及其有限指数子群的算法，计算这些群上的模形式的幂级数展开。
Let X be a smooth projective curve of positive genus defined over a number field K. Assume given a Galois covering map x from X to the projective line over K and a place v of K. We introduce a local canonical height on the set of K_v-valued points of X associated to x as an integral with logarithmic integrand, generalizing Tate's local Neron function on an elliptic curve. The resulting global height can be viewed as a 'Mahler measure' associated to x. We prove that the local canonical height can be obtained by averaging, and taking a limit, over divisors of higher order Weierstrass points on X…
We review the Shimura-Taniyama method for computing the Newton polygon of an abelian variety with complex multiplication. We apply this method to cyclic covers of the projective line branched at three points. As an application, we produce multiple new examples of Newton polygons that occur for Jacobians of smooth curves in characteristic $p$. Under certain congruence conditions on $p$, these include: the supersingular Newton polygon for each genus $g$ with $4 \leq g \leq 11$; nine non-supersingular Newton polygons with $p$-rank $0$ with $4 \leq g \leq 11$; and, for all $g \geq 5$, the Newton p…
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