Related papers: Differential operators and hyperelliptic curves ov…

We show that for all odd primes $p$, there exist ordinary elliptic curves over $\bar{\mathbb{F}}_p(x)$ with arbitrarily high rank and constant $j$-invariant. This shows in particular that there are elliptic curves with arbitrarily high rank…

Cais, Ellenberg and Zureick-Brown recently observed that over finite fields of characteristic two, all sufficiently general smooth plane projective curves of a given odd degree admit a non-trivial rational 2-torsion point on their Jacobian.…

This paper determines the normal forms of hyperelliptic supersingular curves of genus g over an algebraically closed field F of characteristic 2 for 0 < g< 9. We also show that every hyperelliptic supersingular curve of genus 9 over F has…

We show that for any elliptic curve (with j invariant not 0 or 1728) over any field of characteristic different from 2 and 3, there exists an hyperelliptic curve H of genus 5 with two independent maps to the given elliptic curve. We also…

In this paper we prove that there are no hyperelliptic supersingular curves over F_2bar of genus 2^n-1 for any integer n>1. Let g be a natural number, and h=floor(log_2(g+1)+1). Let X be a hyperelliptic curve over F_2bar of genus g>2 and…

A superelliptic curve $\X$ of genus $g\geq 2$ is not necessarily defined over its field of moduli but it can be defined over a quadratic extension of it. While a lot of work has been done by many authors to determine which hyperelliptic…

This paper presents algorithmic approaches to study superspecial hyperelliptic curves. The algorithms proposed in this paper are: an algorithm to enumerate superspecial hyperelliptic curves of genus $g$ over finite fields $\mathbb{F}_q$,…

We prove that for any pair of integers 0\leq r\leq g such that g\geq 3 or r>0, there exists a (hyper)elliptic curve C over F_2 of genus g and 2-rank r whose automorphism group consists of only identity and the (hyper)elliptic involution. As…

We study genus $g$ hyperelliptic curves with reduced automorphism group $A_5$ and give equations $y^2=f(x)$ for such curves in both cases where $f(x)$ is a decomposable polynomial in $x^2$ or $x^5$. For any fixed genus the locus of such…

In the classical case of irreducible smooth algebraic curves every genus $2$ curve is hyperelliptic, or in other words there is a complete linear series $g_2^1$ on them. On the other hand if $g > 2$, then a generic smooth curve of genus $2$…

In this paper, we show that there exist families of curves (defined over an algebraically closed field $k$ of characteristic $p >2$) whose Jacobians have interesting $p$-torsion. For example, for every $0 \leq f \leq g$, we find the…

We prove results about the intersection of the p-rank strata and the boundary of the moduli space of hyperelliptic curves in characteristic p > 2. Using this, we prove that the Z/\ell-monodromy of every irreducible component of the stratum…

In this work, we investigate hyperelliptic curves of type $C: y^2 = x^{2g+1} + ax^{g+1} + bx$ over the finite field $\mathbb{F}_q, q = p^n, p > 2$. For the case of $g = 3$ and $4$ we propose algorithms to compute the number of points on the…

We prove that for any number field $K$ and any fixed genus $g \geq 2$, there are infinitely many non-isomorphic hyperelliptic curves of genus $g$ over $K$ whose Jacobians have rank over $K$ equal to each of 0, 1, or 2. As an example of our…

Let $C: y^2=f(x)$ be a hyperelliptic curve of genus $g\geq 1$, defined over a complete discretely valued field $K$, with ring of integers $O_K$. Under certain conditions on $C$, mild when residue characteristic is not $2$, we explicitly…

For a hyperelliptic curve of genus $g$, a divisor in general position of degree $g+1$ is given by polynomial equations. There is an action from an algebraic group on the representations of divisors by polynomials which fixes divisor…

Given a polynomial $f$ with coefficients in a field of prime characteristic $p$, it is known that there exists a differential operator that raises $1/f$ to its $p$th power. We first discuss a relation between the `level' of this…

Let $K$ be an algebraically closed field of characteristic different from 2, $g$ a positive integer, $f(x)$ a degree $(2g+1)$ polynomial with coefficients in $K$ and without multiple roots, $C: y^2=f(x)$ the corresponding genus $g$…

We determine all genus 2 curves, defined over $\mathbb C$, which have simultaneously degree 2 and 3 elliptic subcovers. The locus of such curves has three irreducible 1-dimensional genus zero components in $\mathcal M_2$. For each component…

Let $K$ be a field of characteristic different from $2$, $\bar{K}$ its algebraic closure. Let $n \ge 3$ be an odd integer. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$ and without repeated roots. Let us consider…