Gonzalo Tornaría

Square Roots Modulo p

The algorithm of Tonelli and Shanks for computing square roots modulo a prime number is the most used, and probably the fastest among the known algorithms when averaged over all prime numbers. However, for some particular prime numbers, there are other algorithms which are considerably faster.In this paper we compare the algorithm of Tonelli and Shanks with an algorithm based in quadratic field extensions due to Cipolla, and give an explicit condition on a prime number to decide which algorithm is faster. Finally, we show that there exists an infinite sequence of prime numbers for which the algorithm of Tonelli and Shanks is asymptotically worse.

Ranks of Elliptic Curves and Random Matrix Theory: Examples of Shimura correspondence for level p 2 and real quadratic twists

We give examples of Shimura correspondence for rational modular forms f of weight 2 and level p^2, for primes p<=19, computed as an application of a method we introduced in \cite{Pacetti-Tornaria}. Furthermore, we verify in this examples a conjectural formula for the central values L(f,-pd,1) and, in case p = 3 (mod 4), a formula for the central values L(f,d,1) corresponding to the real quadratic twists of f.

Nonvanishing of twists of L-functions attached to Hilbert modular forms

We describe algorithms for computing central values of twists of

Computing Jacobi Forms

L

An explicit Waldspurger formula for Hilbert modular forms

-functions associated to Hilbert modular forms, carry out such computations for a number of examples, and compare the results of these computations to some heuristics and predictions from random matrix theory.

Shimura correspondence for level p2 and the central values of L-series, II

We describe an implementation for computing holomorphic and skew-holomorphic Jacobi forms of integral weight and scalar index on the full modular group. This implementation is based on formulas derived by one of the authors which express Jacobi forms in terms of modular symbols of elliptic modular forms. Since this method allows to generate a Jacobi eigenform directly from a given modular eigensymbol without reference to the whole ambient space of Jacobi forms it makes it possible to compute Jacobi Hecke eigenforms of large index. We illustrate our method with several examples.

Ranks of Elliptic Curves and Random Matrix Theory: Computation of central value of quadratic twists of modular L -functions

We describe a construction of preimages for the Shimura map on Hilbert modular forms, and give an explicit Waldspurger type formula relating their Fourier coefficients to central values of twisted L-functions. Our construction is inspired by that of Gross and applies to any nontrivial level and arbitrary base field, subject to certain conditions on the Atkin-Lehner eigenvalues and on the weight.

Shimura correspondence for level p2 and the central values of L-series

Given a Hecke eigenform f of weight 2 and square-free level N, by the work of Kohnen, there is a unique weight 3/2 modular form of level 4N mapping to f under the Shimura correspondence. Furthermore, by the work of Waldspurger the Fourier coefficients of such a form are related to the quadratic twists of the form f. Gross gave a construction of the half integral weight form when N is prime, and such construction was later generalized to square-free levels. However, in the non-square free case, the situation is more complicated since the natural construction is vacuous. The problem being that there are too many special points so that there is cancellation while trying to encode the information as a linear combination of theta series. In this paper, we concentrate in the case of level p2, for p > 2 a prime number, and show how the set of special points can be split into subsets (indexed by bilateral ideals for an order of reduced discriminant p2) which gives two weight 3/2 modular forms mapping to f under the Shimura correspondence. Moreover, the splitting has a geometric interpretation which allows to prove that the forms are indeed a linear combination of theta series associated to ternary quadratic forms. Once such interpretation is given, we extend the method of Gross–Zagier to the case where the level and the discriminant are not prime to each other to prove a Gross-type formula in this situation.