Related papers: Hyperelliptic curves, L-polynomials, and random ma…

We consider generalizations of Szpiro's classical discriminant conjecture to hyperelliptic curves over a number field $K$, and to smooth, projective and geometrically connected curves $X$ over $K$ of genus at least one. The main results…

We combine the exact counting of all elliptic curves over $K = \mathbb{F}_q(t)$ with $\mathrm{char}(K) > 3$ by Bejleri, Satriano and the author, together with the torsion-free nature of most elliptic curves over global function fields…

We prove two related concentration inequalities concerning the number of rational points of hyperelliptic curves over subsets of a finite field. In particular, we investigate the probability of a large discrepancy between the numbers of…

The Sato-Tate distributions for genus 2 curves (conjecturally) describe the statistics of numbers of rational points on the curves. In this paper, we explicitly compute the auto-correlation functions of Sato-Tate distributions for genus 2…

In this paper we generalize results of P. Le Duff to genus n hyperelliptic curves. More precisely, let C/Q be a hyperelliptic genus n curve and let J(C) be the associated Jacobian variety. Assume that there exists a prime p such that J(C)…

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…

We determine the Sato-Tate groups and prove the generalized Sato-Tate conjecture for the Jacobians of curves of the form $$ y^2=x^p-1 \text{ and } y^2=x^{2p}-1,$$ where $p$ is an odd prime. Our results rely on the fact the Jacobians of…

Generalized unitarity cut of a Feynman diagram generates an algebraic system of polynomial equations. At high-loop levels, these equations may define a complex curve or a (hyper-)surface with complicated topology. We study the curve cases,…

A distribution whose normalization constant is an A-hypergeometric polynomial is called an A-hypergeometric distribution. Such a distribution is in turn a generalization of the generalized hypergeometric distribution on the contingency…

Using the ratios theorems, we calculate the leading order terms in $N$ for the following averages of the characteristic polynomial and its derivative: $\left< \left|\Lambda_A(1 )\right| ^{r} \frac{ \Lambda_A'(\mathrm{e}^{\mathrm{i} \phi})…

Let C be a supersingular genus-2 curve over an algebraically closed field of characteristic 3. We show that if C is not isomorphic to the curve y^2 = x^5 + 1 then up to isomorphism there are exactly 20 degree-3 maps phi from C to the…

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 $2$…

We review the main conjecture for an elliptic curve on $\Q$ having good supersingular reduction at $p$ and give some consequences of it. Then we define the notion of $\lambda$-invariant and of $\mu$- invariant in this situation,…

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 introduce a heuristic prediction for the distribution of the isomorphism class of the Galois group of the maximal pro-p extension of Q unramified outside a "random" set of primes. This is guided by reasoning similar to that governing the…

We consider certain families of automorphic representations over number fields arising from the principle of functoriality of Langlands. Let $G$ be a reductive group over a number field $F$ which admits discrete series representations at…

We propose a random matrix model for families of elliptic curve L-functions of finite conductor. A repulsion of the critical zeros of these L-functions away from the center of the critical strip was observed numerically by S. J. Miller in…

While the numbers of superspecial curves of genus at most 3 are well understood, and several computational approaches have been developed to count superspecial curves of genus 4 with large automorphism groups, much less is known in higher…

Boix, De Stefani and Vanzo have characterized ordinary/supersingular elliptic curves over $\mathbb{F}_p$ in terms of the level of the defining cubic homogenous polynomial. We extend their study to arbitrary genus, in particular we prove…

We characterize the distribution of 2-Selmer ranks of quadratic twists of elliptic curves over $\mathbb{Q}$ with full rational 2-torsion. We propose a new type of random alternating matrix model $M_{*,\mathbf…