Related papers: Generalized twisted Edwards curves over finite fie…

In this paper we present a new approach to counting the proportion of hyperelliptic curves of genus $g$ defined over a finite field $\mathbb{F}_q$ with a given $a$-number. In characteristic three this method gives exact probabilities for…

In this paper we explicitly compute equations for the twists of all the smooth plane quartic curves defined over a number field k. Since the plane quartic curves are non-hyperelliptic curves of genus 3 we can apply the method developed by…

Given an integer $\gamma\geq 2$ and an odd prime power $q$ we show that for every large genus $g$ there exists a non-singular curve $C$ defined over $\mathbb{F}_q$ of genus $g$ and gonality $\gamma$ and with exactly $\gamma(q+1)$…

We study the character sums \[\phi_{(m,n)}(a,b)=\sum_{x\in\mathbb{F}_q}\phi\left(x(x^{m}+a)(x^{n}+b)\right),\textrm{ and, } \psi_{(m,n)}(a,b)=\sum_{x\in\mathbb{F}_q}\phi\left((x^{m}+a)(x^{n}+b)\right)\] where $\phi$ is the quadratic…

We show that if f: X --> Y is a finite, separable morphism of smooth curves defined over a finite field F_q, where q is larger than an explicit constant depending only on the degree of f and the genus of X, then f maps X(F_q) surjectively…

We consider a family of genus 2 hyperelliptic curves of even order and obtain explicitly the system of 5 linear ODEs for periods of the corresponding Abelian integrals of first, second, and third kind, as functions of the parameters of the…

In this article, we show that in each of four standard families of hyperelliptic curves, there is a density-$1$ subset of members with the property that their Jacobians have adelic Galois representation with image as large as possible. This…

Faltings' theorem [Fal83],[Fal91] (formerly the Mordell conjecture [Mo22]) states that a curve of genus greater than one over any number field has only finitely many points. Again a natural question is how many points can such a curve have.…

A projective, smooth, absolutely irreducible algebraic curve X of genus g defined over a finite field F_q is called optimal if for every other such genus g curve Y over F_q one has $\#Y(F_q)\le \#X(F_q)$. In this paper we show that for…

This paper is devoted to abelian varieties arising from generalized Legendre curves. In particular, we consider their corresponding Galois representations, periods, and endomorphism algebras. For certain one parameter families of…

Let K be a field of characteristic zero, f(x) be a polynomial with coefficients in K and without multiple roots. We consider the superelliptic curve C_{f,q} defined by y^q=f(x), where q=p^r is a power of a prime p. We determine the Hodge…

On a general hypersurface of degree $d\leq n$ in $\mathbb P^n$ or $\mathbb P^n$ itself, we prove the existence of curves of any genus and high enough degree depending on the genus passing through the expected number $t$ of general points or…

We extend our method to compute division polynomials of Jacobians of curves over Q to curves over Q(t), in view of computing mod ell Galois representations occurring in the \'etale cohomology of surfaces over Q. Although the division…

We use twisted stable maps to answer the following question. Let E\subset P^2 be a smooth cubic. How many rational degree d curves pass through a general points of E, have b specified tangencies with E and c unspecified tangencies, and pass…

A generalized complex manifold which satisfies the $\partial \overline{\partial}$-lemma admits a Hodge decomposition in twisted cohomology. Using a Courant algebroid theoretic approach we study the behavior of the Hodge decomposition in…

Let $\mathscr{G}_{\rm CM}(d)$ denote the collection of groups (up to isomorphism) that appear as the torsion subgroup of a CM elliptic curve over a degree $d$ number field. We completely determine $\mathscr{G}_{\rm CM}(d)$ for odd integers…

The problem of constructing curves with many points over finite fields has received considerable attention in the recent years. Using the class field theory approach, we construct new examples of curves ameliorating some of the known…

We describe algebraic curves $ X : F(x, y) = 0 $ defined over $\overline{\mathbb{Q}}$ that satisfy the following property: there exist a number field $k$ and an infinite set $S \subset k$ such that, for every $y \in S$, the roots of the…

An investigation of morphisms that coincide topologically is used to generalize to all characteristics and partly reprove Tamagawa's theorem on the Grothendieck conjecture in anabelian geometry for affine hyperbolic curves. The theorem now…

Let D be a simply laced Dynkin diagram of rank r whose affinization has the shape of a star (i.e., D4,E6,E7,E8). To such a diagram one can attach a group G whose generators correspond to the legs of the affinization, have orders equal to…