We derive the equations necessary to perform a two-descent on the Jacobians of curves of genus two with rational Weierstrass points. We compute the Mordell-Weil rank of the Jacobian of some genus two curves defined over the rationals, and discuss the practicality of using this method. Introduction Let C he a curve of genus two defined over a number field K, and J its Jacobian variety. The Morde...

This paper studies adaptive regularized methods for nonlinear least-squares problems where the model of the objective function used at each iteration is either the Euclidean residual regularized by a quadratic term or the Gauss-Newton model regularized by a cubic term. For suitable choices of the regularization parameter the role of the regularization term is to provide global convergence. In t...

In this paper, we study bounds for the number of rational points on twists C ′ of a fixed curve C over a number field K, under the condition that the group of K-rational points on the Jacobian J ′ of C ′ has rank smaller than the genus of C ′. The main result is that with some explicitly given finitely many possible exceptions, we have a bound of the form 2r + c, where r is the rank of J ′(K) a...

This paper is the second of a series devoted to the study of the rank of J0(q) (the Jacobian of the modular curve X0(q)), from the analytic point of view stemming from the Birch and Swinnerton-Dyer conjecture, which is tantamount to the study, on average, of the order of vanishing at the central critical point of the L-functions of primitive weight two forms f of level q (q prime). We prove tha...

This paper addresses the problem of evaluating a subset of the range of a vector-valued function. It is based on a work by Goldsztejn and Jaulin which provides methods based on interval analysis to address this problem when the dimension of the domain and co-domain of the function are equal. This paper extends this result to vector-valued functions with domain and co-domain of different dimensi...

We derive a number of local identities of arbitrary rank involving Jacobi elliptic functions and use them to obtain several new results. First, we present an alternative, simpler derivation of the cyclic identities discovered by us recently, along with an extension to several new cyclic identities of arbitrary rank. Second, we obtain a generalization to cyclic identities in which successive ter...

For every genus g, we construct a smooth, complete, rational polarized algebraic variety DMg together with a normal crossing divisor D = ⋃ Di, such that for every moduli space MΣ(2, 0) of semistable topologically trivial vector bundles of rank 2 on an algebraic curve Σ of genus g there exists a holomorphic isomorphism f : MΣ(2, 0) \ K2 → DMg \ D, where K2 is the Kummer variety of the Jacobian o...