Related papers: A Thomae-like Formula: Algebraic Computations of T…
We construct the so-called theta vectors on noncommutative T^4, which correspond to the theta functions on commutative tori with complex structures. Following the method of Dieng and Schwarz, we first construct holomorphic connections and…
The scalar curvature for the noncommutative four torus $\mathbb{T}_\Theta^4$, where its flat geometry is conformally perturbed by a Weyl factor, is computed by making the use of a noncommutative residue that involves integration over the…
Let K be a field of characteristic different from 2 and C an elliptic curve over K given by a Weierstrass equation. To divide an element of the group C by 2, one must solve a certain quartic equation. We characterise the quartics arising…
This article concerns the computational complexity of a fundamental problem in number theory: counting points on curves and surfaces over finite fields. There is no subexponential-time algorithm known and it is unclear if it can be…
A log Calabi--Yau surface $(X,D)$ is given by a smooth projective surface $X$, together with an anti-canonical cycle of rational curves $D \subset X$. The homogeneous coordinate ring of the mirror to such a surface, or to the complement…
The discriminant of a smooth plane cubic curve over the complex numbers can be written as a product of theta functions. This provides an important connection between algebraic and analytic objects. In this paper, we perform a new approach…
To count bundles on curves, we study zetas of elliptic curves and their zeros. There are two types, i.e., the pure non-abelian zetas defined using moduli spaces of semi-stable bundles, and the group zetas defined for special linear groups.…
Let $X$ be an Abelian cover $\mathbb{CP}^{1}$ ramified at $mr$ points, $\lambda_1...\lambda_{mr}.$ we define a class of non positive divisors on $X$ of degree $g-1$ supported in the pre images of the branch points on $X$, such that the…
We give an algorithm to determine factorization types of primes in the number fields generated by a single point of odd order on an elliptic curve. We apply this to compute coefficients of the Dedekind zeta function of the field.
In this paper we prove a Thomae derivative formula for trigonal curves admitting a non-singular affine model. This formula relates the derivatives of theta functions with rational characteristics on the curve to explicit expressions in the…
We exhibit an algorithm to compute equations of an algebraic curve over a computable characteristic 0 field from the power series expansions of its regular 1-forms at a nonrational point of the curve, extending a 2005 algorithm of Baker,…
We continue our development of the invariant theory of genus one curves with the aim of computing certain twists of the universal family of elliptic curves parametrised by the modular curve X(n) for n = 2,3,4,5. Our construction makes use…
We introduce a new algorithm to compute the zeta function of a curve over a finite field. This method extends previous work of ours to all curves for which a good lift to characteristic zero is known. We develop all the necessary bounds,…
In this paper we classify the singular curves whose theta divisors in their generalized Jacobians are algebraic, meaning that they are cut out by polynomial analogs of theta functions. We also determine the degree of an algebraic theta…
We present a new method for computing the zeta function of an algebraic curve over a finite field. The algorithm relies on a trace formula of Harvey to count points on a plane model of the curve. The zeta function of the curve is then…
The present paper gives an explicit classification of the isomorphism classes of non-hyperelliptic genus 4 curves over an algebraically closed field of characteristic 0. A non-hyperelliptic genus 4 curve lies on a quadric in $\mathbb{P^3}$…
We study rational points on the elliptic surface given by the equation: $$y^2 = x^3 + AxQ(u,v)^2 + BQ(u,v)^3,$$ where $A,B\in \mathbb{Z}$ satisfy that $4A^3-27B^2\neq 0$ and $Q(u,v)$ is a positive-definite quadratic form. We prove…
It is a classical result (apparently due to Tate) that all elliptic curves with a torsion point of order n ($4 \leq n \leq 10$, or n = 12) lie in a one-parameter family. However, this fact does not appear to have been used ever for…
In this paper, we analyze the theta series associated to the quadratic form $Q(\mathbf{x}) := x_1^2 + x_2^2 + x_3^2 + x_4^2$ with congruence conditions on $x_i$ modulo $2, 3, 4$, and $6$. By employing special operators on modular,…
Let g be a positive integer congruent to 1 modulo 4 and K be an arbitrary number field. We construct infinitely many explicit one-parameter algebraic families of degree 4 del Pezzo surfaces and of genus g hyperelliptic curves such that each…