arXiv subjects

Gonzalo Tornaría

Publications and source records attributed to Gonzalo Tornaría.

15 recordsLinked to original sources

arXiv2024

Residual paramodularity of a certain Calabi-Yau threefold

We prove congruences of Hecke eigenvalues between cuspidal Hilbert newforms $f_{79}$ and $h_{79}$ over $F=\mathbb Q(\sqrt{5})$, of weights (2,2) and (2,4) respectively, level of norm 79. In the main example, the modulus is a divisor of 5 in some coefficient field, in the secondary example a divisor of 2. The former allows us to prove that the 4-dimensional mod-5 representation of $\mathrm{Gal}(\overline{\mathbb Q} / \mathbb Q)$ on the 3rd cohomology of a certain Calabi-Yau threefold comes from a Siegel modular form $F_{79}$ of genus 2, weight 3 and paramodular level 79. This is a weak form of a conjecture of Golyshev and van Straten. In aid of this, we prove also a congruence of Hecke eigenvalues between $F_{79}$ and the Johnson-Leung-Roberts lift $\mathrm{JR}(h_{79})$, which has weight 3 and paramodular level $79\times 5^2$.

math.NT↗

arXiv2023

On the $2$-Selmer group of Jacobians of hyperelliptic curves

Let $\mathcal{C}$ be a hyperelliptic curve $y^2 = p(x)$ defined over a number field $K$ with $p(x)$ integral of odd degree. The purpose of the present article is to prove lower and upper bounds for the $2$-Selmer group of the Jacobian of $\mathcal{C}$ in terms of the class group of the $K$-algebra $K[x]/(p(x))$. Our main result is a formula relating these two quantities under some mild hypothesis. We provide some examples that prove that our lower and upper bounds are as sharp as possible. As a first application, we study the rank distribution of the $2$-Selmer group in families of quadratic twists. Under some extra hypothesis we prove that among prime quadratic twists, a positive proportion has fixed $2$-Selmer group. As a second application, we study the family of octic twists of the genus $2$ curve $y^2 = x^5 + x$.

math.NT↗

arXiv2022

Effective construction of Hilbert modular forms of half-integral weight

Given a Hilbert cuspidal newform $g$ we construct a family of modular forms of half-integral weight whose Fourier coefficients give the central values of the twisted $L$-series of $g$ by fundamental discriminants. The family is parametrized by quadratic conditions on the primes dividing the level of $g$, where each form has coefficients supported on the discriminants satisfying the conditions. These modular forms are given as generalized theta series and thus their coefficients can be effectively computed. Our construction works over arbitrary totally real number fields, except that in the case of odd degree the square levels are excluded. It includes all discriminants except those divisible by primes whose square divides the level.

math.NT↗

arXiv2021

Quinary forms and paramodular forms

We work out the exact relationship between algebraic modular forms for a two-by-two general unitary group over a definite quaternion algebra, and those arising from genera of positive-definite quinary lattices, relating stabilisers of local lattices with specific open compact subgroups, paramodular at split places, and with Atkin-Lehner operators. Combining this with the recent work of Rösner and Weissauer, proving conjectures of Ibukiyama on Jacquet-Langlands type correspondences (mildly generalised here), provides an effective tool for computing Hecke eigenvalues for Siegel modular forms of degree two and paramodular level. It also enables us to prove examples of congruences of Hecke eigenvalues connecting Siegel modular forms of degrees two and one. These include some of a type conjectured by Harder at level one, supported by computations of Fretwell at higher levels, and a subtly different congruence discovered experimentally by Buzzard and Golyshev.

math.NT↗

arXiv2020

On $2$-Selmer groups and quadratic twists of elliptic curves

Let $K$ be a number field and $E/K$ be an elliptic curve with no $2$-torsion points. In the present article we give lower and upper bounds for the $2$-Selmer rank of $E$ in terms of the $2$-torsion of a narrow class group of a certain cubic extension of $K$ attached to $E$. As an application, we prove (under mild hypotheses) that a positive proportion of prime conductor quadratic twists of $E$ have the same $2$-Selmer group.

math.NT↗

arXiv2020

An explicit Waldspurger formula for Hilbert modular forms II

We describe a construction of preimages for the Shimura map on Hilbert modular forms using generalized theta series, and give an explicit Waldspurger type formula relating their Fourier coefficients to central values of twisted $L$-functions. This formula extends our previous work, allowing to compute these central values when the main central value vanishes.

math.NT↗

arXiv2018

An explicit Waldspurger formula for Hilbert modular forms

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.

math.NT↗

arXiv2016

Computing Jacobi Forms

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.

math.NT↗

arXiv2014

Shimura correspondence for level $p^2$ and the central values of $L$-series II

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 $p^2$, 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 $p^2$) 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.

math.NT↗

arXiv2012

Explicit computations of Siegel modular forms of degree two

Unlike classical modular forms, there is currently no general way to implement the computation of Siegel modular forms of arbitrary weight, level and character, even in degree two. There is however, a way to do it in a unified way. After providing a survey of known computations we describe the implementation of a class modeling Siegel modular forms of degree two in Sage. In particular, we describe algorithms to compute a variety of rings of Siegel modular forms, many of which are implemented in our class. A wide variety of Siegel modular forms (e.g., both vector- and scalar-valued) can be modeled via this class and we unify these via a construct we call a formal Siegel modular form. We define this notion and discuss it in detail.

math.NT↗

arXiv2012

Formulas for central critical values of twisted L-functions attached to paramodular forms

In the 1980s Böcherer formulated a conjecture relating the central values of the imaginary quadratic twists of the spin L-function attached to a Siegel modular form $F$ to the Fourier coefficients of $F$. This conjecture has been proved when $F$ is a lift. More recently, we formulated an analogous conjecture for paramodular forms $F$ of prime level, even weight and in the plus-space. In this paper, we examine generalizations of this conjecture. In particular, our new formulations relax the conditions on $F$ and allow for twists by real characters. Moreover, these formulation are more explicit than the earlier version. We prove the conjecture in the case of lifts and provide numerical evidence in the case of nonlifts.

math.NT↗

arXiv2010

A Böcherer-Type Conjecture for Paramodular Forms

In the 1980s Böcherer formulated a conjecture relating the central value of the quadratic twists of the spinor L-function attached to a Siegel modular form F to the coefficients of F . He proved the conjecture when F is a Saito-Kurokawa lift. Later Kohnen and Kuss gave numerical evidence for the conjecture in the case when F is a rational eigenform that is not a Saito-Kurokawa lift. In this paper we develop a conjecture relating the central value of the quadratic twists of the spinor L-function attached to a paramodular form and the coefficients of the form. We prove the conjecture in the case when the form is a Gritsenko lift and provide numerical evidence when it is not a lift.

math.NT↗

arXiv2006

Computing central values of twisted L-series: the case of composite levels

Let f be a newform of weight two and composite level N. We show how to compute weight 3/2 modular forms "associated" to f whose Fourier coefficients are related to the central values of quadratic twists (real and imaginary) of f. We will focus on examples for levels N=27, N=15, and N=75, which exhibit most aspects of our methods.

math.NT↗

arXiv2006

Computation of central value of quadratic twists of modular L-functions

Let f be a newform of weight two, prime level p. If D is a fundamental discriminant, define the twisted L-function L(f,D,s) to be the L-function associated to the twist of f by the quadratic character of conductor D. In this paper we consider the question of computing the family of twisted central values {L(f,D,1) : |D| <= x} for some x, by using an explicit version of Waldspurger's formula relating the central values L(f,D,1) to the |D|-th Fourier coefficient of weigth 3/2 modular forms in Shimura correspondence with f.

math.NT↗

arXiv2006

Shimura correspondence for level p^2 and the central values of L-series

Given a weight 2 and level p^2 modular form f, we construct two weight 3/2 modular forms (possibly zero) of level 4p^2 and non trivial character mapping to f via the Shimura correspondence. Then we relate the coefficients of the constructed forms to the central value of the L-series of certain imaginary quadratic twists of f. Furthermore, we give a general framework for our construction that applies to any order in definite quaternion algebras, with which one can, in principle, construct weight 3/2 modular forms of any level, provided one knows how to compute ideal classes representatives.

math.NT↗