Abstract:
We determine the harmonic volumes for all the hyperelliptic curves. This gives a geometric interpretation of a theorem established by A. Tanaka.

Abstract:
We assemble and reorganize the recent work in the area of hyperelliptic pairings: We survey the research on constructing hyperelliptic curves suitable for pairing-based cryptography. We also showcase the hyperelliptic pairings proposed to date, and develop a unifying framework. We discuss the techniques used to optimize the pairing computation on hyperelliptic curves, and present many directions for further research.

Abstract:
The main subject is the difference between the coarse moduli space and the stack of hyperelliptic curves. In particular, we compute their Picard groups, giving explicit description of the generators. We also study how many families of hyperelliptic curves may have the same modular map as well as the existence of the tautological family over the coarse moduli space of hyperelliptic curves.

Abstract:
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 \iota be 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)/<\iota> is not cyclic. We construct explicit examples of hyperelliptic curves not definable over their field of moduli when Aut(X)/<\iota> is cyclic.

Abstract:
We construct a new compactification of the moduli space H_g of smooth hyperelliptic curves of genus g. We compare our compactification with other well-known remarkable compactifications of H_g .

Abstract:
We find a closed formula for the number $\operatorname{hyp}(g)$ of hyperelliptic curves of genus $g$ over a finite field $k=\mathbb{F}_q$ of odd characteristic. These numbers $\operatorname{hyp}(g)$ are expressed as a polynomial in $q$ with integer coefficients that depend on the set of divisors of $q-1$ and $q+1$. As a by-product we obtain a closed formula for the number of self-dual curves of genus $g$. A hyperelliptic curve is self-dual if it is $k$-isomorphic to its own hyperelliptic twist.

Abstract:
Let $k$ be a number field, and let $S$ be a finite set of maximal ideals of the ring of integers of $k$. In his 1962 ICM address, Shafarevich asked if there are only finitely many $k$-isomorphism classes of algebraic curves of a fixed genus $g\ge 1$ with good reduction outside $S$. He verified this for $g=1$ by reducing the problem to Siegel's theorem. Parshin extended this argument to all hyperelliptic curves (cf. also the work of Oort). The general case was settled by Faltings' celebrated work. In this note we give a short proof of Shafarevich's conjecture for hyperelliptic curves, by reducing the problem to the case $g=1$ using the Theorem of de Franchis plus standard facts about discriminants of hyperelliptic equations.

Abstract:
The purpose of this paper is to study hyperelliptic curves with extra involutions. The locus $\L_g$ of such genus $g$ hyperelliptic curves is a $g$-dimensional subvariety of the moduli space of hyperelliptic curves $\H_g$. We discover a birational parametrization of $\L_g$ via dihedral invariants and show how these invariants can be used to determine the field of moduli of points $\p \in \L_g$. We conjecture that for $\p\in \H_g$ with $|\Aut(\p)| > 2$ the field of moduli is a field of definition and prove this conjecture for any point $\p\in \L_g$ such that the Klein 4-group is embedded in the reduced automorphism group of $\p$. Further, for $g=3$ we show that for every moduli point $\p \in \H_3$ such that $| \Aut (\p) | > 4$, the field of moduli is a field of definition and provide a rational model of the curve over its field of moduli.

Abstract:
Let $\mathcal{M}_{g,2}$ be the moduli space of curves of genus $g$ with a level-2 structure. We prove here that there is always a non hyperelliptic element in the intersection of four thetanull divisors in $\mathcal{M}_{6,2}$. We prove also that for all $g\geqslant3$, each component of the hyperelliptic locus in $\mathcal{M}_{g,2}$ is a connected component of the intersection of $g-2$ thetanull divisors.