Bonnie Huggins
We determine conditions that guarantee that a hyperelliptic or plane curve over a field of characteristic not equal to 2 can be defined over its field of moduli. We also give new examples of curves not definable over their fields of moduli.
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In this paper we investigate the cohomological obstruction for the field of moduli of a $G$-cover to be a field of definition, in the case of local fields and covers with tame admissible reduction. This applies in particular to $p$-adic fields where $p$ does not divide the order of the group $G$. We give examples of $G$-covers with field of moduli $\QQ_p$ that cannot be defined over $\QQ_p$, for all primes $p>5$.
Let M_2 be the moduli space that classifies genus 2 curves. If a curve C is defined over a field k, the corresponding moduli point P=[C] is defined over k. Mestre solved the converse problem for curves with Aut(C) isomorphic to C_2. Given a moduli point defined over k, Mestre finds an obstruction to the existence of a corresponding curve defined over k, that is an element in Br_2(k) not always trivial. In this paper we prove that for all the other possibilities of Aut(C), every moduli point defined over k is represented by a curve defined over k. We also give an explicit construction of such a…
Given a smooth plane curve $\overline{C}$ of genus $g\geq 3$ over an algebraically closed field $\overline{k}$, a field $L\subseteq\overline{k}$ is said to be a \emph{plane model-field of definition for $\overline{C}$} if $L$ is a field of definition for $\overline{C}$, i.e. $\exists$ a smooth curve $C'$ defined over $L$ where $C'\times_L\overline{k}\cong \overline{C}$, and such that $C'$ is $L$-isomorphic to a non-singular plane model $F(X,Y,Z)=0$ in $\mathbb{P}^2_{L}$. {In this short note, we construct a smooth plane curve $\overline{C}$ over $\overline{\mathbb{Q}}$, such that the field of m…
This is a survey paper dealing with moduli aspects of curves over finite fields. It discusses counting points of moduli spaces, relations with modular forms and stratifications on moduli spaces.
We compute the effective cone for the moduli space of stable rational curves with at most six marked points.
We show how to calculate the Euler characteristic of a local system associated to an irreducible representation of the symplectic group of genus 3 on the moduli space of curves of genus 3 and the moduli space of principally polarized abelian varieties of dimension 3.
We describe the invariants of plane quartic curves -- nonhyperelliptic genus 3 curves in their canonical model -- as determined by Dixmier and Ohno, with application to the classification of curves with given structure. In particular, we determine modular equations for the strata in the moduli space $\Me_3$ of plane quartics which have at least seven hyperflexes, and obtain an computational characterization of curves in these strata.
The Gross-Zagier formula on singular moduli for Shimura curves
研究志村曲线上关于奇异模的格罗斯 - 扎吉尔公式，通过证明将其在模曲线积分模型上的相关结果推广到志村曲线，得到了该公式的一种推广情形。
In the present paper, we study a new kind of anabelian phenomenon concerning the smooth pointed stable curves in positive characteristic. It shows that the topological structures of moduli spaces of curves can be understood from the viewpoint of anabelian geometry. We formulate some new anabelian-geometric conjectures relating the tame fundamental groups of curves over algebraically closed fields of characteristic $p>0$ to the moduli spaces of curves. These conjectures are generalized versions of the weak Isom-version of the Grothendieck conjecture for curves over algebraically closed fields o…
Arithmetic quantum local systems over the moduli of curves
本研究构建了曲线模空间上量子局部系统的算术类似物并探究其基本结构，该结构可统一赋值数域上光滑真曲线第一几何平展上同调的伽罗瓦上同调类。
Let F be an algebraically closed field with char(F) not equal to 2, let F/K be a Galois extension, and let X be a hyperelliptic curve defined over F. Let ιbe the hyperelliptic involution of X. We show that X can be defined over its field of moduli relative to the extension F/K if Aut(X)/<ι> is not cyclic. We construct explicit examples of hyperelliptic curves not definable over their field of moduli when Aut(X)/<ι> is cyclic.
On the fields of definition of genus-one covers of $\mathbb{P}^1$
本文研究 Belyi 对的定义域相对于模场的扩张次数，证明该次数可任意大，并由此给出 Belyi 对局部-整体原理的反例。
Let $f:\mathbb{P}^N\to\mathbb{P}^N$ be an endomorphism of degree $d\ge2$ defined over $\overline{\mathbb{Q}}$ or $\overline{\mathbb{Q}}_p$, and let $K$ be the field of moduli of $f$. We prove that there is a field of definition $L$ for $f$ whose degree $[L:K]$ is bounded solely in terms of $N$ and $d$.
For any finite abelian group G, we study the moduli space of abelian $G$-covers of elliptic curves, in particular identifying the irreducible components of the moduli space. We prove that, in the totally ramified case, the moduli space has trivial rational Picard group, and it is birational to the moduli space M_{1,n}, where n is the number of branch points. In the particular case of moduli of bielliptic curves, we also prove that the boundary divisors are a basis of the rational Picard group of the admissible covers compactification of the moduli space. Our methods are entirely algebro-geomet…
We investigate the arithmetic of algebraic curves on coarse moduli spaces for special linear rank two local systems on surfaces with fixed boundary traces. We prove a structure theorem for morphisms from the affine line into the moduli space. We show that the set of integral points on any nondegenerate algebraic curve on the moduli space can be effectively determined.
These informal notes are an expanded version of lectures on the moduli space of elliptic curves given at Zhejiang University in July, 2008. Their goal is to introduce and motivate basic concepts and constructions (such as orbifolds and stacks) important in the study of moduli spaces of curves and abelian varieties through the example of elliptic curves. The reason for working with elliptic curves is that most constructions are elementary and explicit in this case. All four approaches to moduli spaces of curves -- complex analytic, topological, algebro-geometric, and number theoretic -- are con…
Let Hom^N_d be the set of morphisms of degree d from P^N to itself. For f an element of PGL_{N+1}, let phi^f represent the conjugation action f^{-1} phi f. Let M^N_d = Hom_d^N/PGL_{N+1} be the moduli space of degree d morphisms of P^N. A field of definition for class of morphisms is a field over which at least one morphism in the class is defined. The field of moduli for a class of morphisms is the fixed field of the set of Galois elements fixing that class. Every field of definition contains the field of moduli. In this article, we give a sufficient condition for the field of moduli to be a f…
We review the following subjects: 1. Basic theory on algebraic curves and their moduli space, 2. Schottky uniformization theory of Riemann surfaces, and its extension called arithmetic uniformization theory, 3. Application to these theories to the arithmetic of the moduli space of algebraic curves, especially to automorphic forms on this space.
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