Nathan C. Ryan

Survey article: Characterizations of the Saito-Kurokawa lifting

There are a variety of characterizations of Saito-Kurokawa lifts from elliptic modular forms to Siegel modular forms of degree 2. In addition to giving a survey of known characterizations, we apply a recent result of Weissauer to provide a number of new and simpler characterizations of Saito-Kurokawa lifts.

LIFTING PUZZLES IN DEGREE FOUR

We identify the majority of Siegel modular eigenforms in degree four and weights up to 16 as being Duke–Imamoḡlu–Ikeda or Miyawaki–Ikeda lifts. We give two examples of eigenforms that are probably also lifts but of an undiscovered type. 2000 Mathematics subject classification: primary 11F46, 11F60; secondary 65D20, 65-04.

Numerical computation of a certain Dirichlet series attached to Siegel modular forms of degree two

The Rankin convolution type Dirichlet series DF,G(s) of Siegel modular forms F and G of degree two, which was introduced by Kohnen and the second author, is computed numerically for various F and G. In particular, we prove that the series DF,G(s), which share the same functional equation and analytic behavior with the spinor L-functions of eigenforms of the same weight are not linear combinations of those. In order to conduct these experiments a numerical method to compute the Petersson scalar products of Jacobi Forms is developed and discussed in detail.

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

We describe algorithms for computing central values of twists of

Computing Jacobi Forms

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Evaluating L-functions with few known coefficients.

-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.

Efficient verification of Tunnell’s criterion

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.

Computations of Vector-Valued Siegel Modular Forms

We address the problem of evaluating an

Inverting the Satake map for Spn and applications to Hecke operators

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A BÖCHERER-TYPE CONJECTURE FOR PARAMODULAR FORMS

-function when only a small number of its Dirichlet coefficients are known. We use the approximate functional equation in a new way and find that is possible to evaluate the