18,445 research outputs found
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.Comment: 14 pages, 5 tables, Cython implementation of algorithm included.
Revised version. To appear in the LMS Journal of Computation and Mathematic
Computations of vector-valued Siegel modular forms
We carry out some computations of vector valued Siegel modular forms of
degree two, weight (k,2) and level one. Our approach is based on Satoh's
description of the module of vector-valued Siegel modular forms of weight (k,
2) and an explicit description of the Hecke action on Fourier expansions. We
highlight three experimental results: (1) we identify a rational eigenform in a
three dimensional space of cusp forms, (2) we observe that non-cuspidal
eigenforms of level one are not always rational and (3) we verify a number of
cases of conjectures about congruences between classical modular forms and
Siegel modular forms.Comment: 18 pages, 2 table
Explicit computations of Hida families via overconvergent modular symbols
In [Pollack-Stevens 2011], efficient algorithms are given to compute with
overconvergent modular symbols. These algorithms then allow for the fast
computation of -adic -functions and have further been applied to compute
rational points on elliptic curves (e.g. [Darmon-Pollack 2006, Trifkovi\'c
2006]). In this paper, we generalize these algorithms to the case of families
of overconvergent modular symbols. As a consequence, we can compute -adic
families of Hecke-eigenvalues, two-variable -adic -functions,
-invariants, as well as the shape and structure of ordinary Hida-Hecke
algebras.Comment: 51 pages. To appear in Research in Number Theory. This version has
added some comments and clarifications, a new example, and further
explanations of the previous example
Modularity of the Consani-Scholten quintic
We prove that the Consani-Scholten quintic, a Calabi-Yau threefold over QQ,
is Hilbert modular. For this, we refine several techniques known from the
context of modular forms. Most notably, we extend the Faltings-Serre-Livne
method to induced four-dimensional Galois representations over QQ. We also need
a Sturm bound for Hilbert modular forms; this is developed in an appendix by
Jose Burgos Gil and the second author.Comment: 35 pages, one figure; with an appendix by Jose Burgos Gil and Ariel
Pacetti; v3: corrections and improvements thanks to the refere
Computing the number of certain Galois representations mod
Using the link between mod Galois representations of \qu and mod
modular forms established by Serre's Conjecture, we compute, for every prime
, a lower bound for the number of isomorphism classes of continuous
Galois representation of \qu on a two--dimensional vector space over \fbar
which are irreducible, odd, and unramified outside .Comment: 28 pages, 3 table
On Shimura's decomposition
Let be an odd integer and a positive integer such that . Let be an even Dirichlet character modulo . Shimura
decomposes the space of half-integral weight cusp forms as a
direct sum of (the subspace spanned by 1-variable theta- series)
and where runs through a certain family of
integral weight newforms. The explicit computation of this decomposition is
important for practical applications of a theorem of Waldspurger relating
critical values of -functions of quadratic twists of newforms of even weight
to coefficients of modular forms of half-integral weight.Comment: 12 pages, to appear in the International Journal of Number Theor
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