662 research outputs found
On the ramification of modular parametrizations at the cusps
We investigate the ramification of modular parametrizations of elliptic
curves over Q at the cusps. We prove that if the modular form associated to the
elliptic curve has minimal level among its twists by Dirichlet characters, then
the modular parametrization is unramified at the cusps. The proof uses
Bushnell's formula for the Godement-Jacquet local constant of a cuspidal
automorphic representation of GL(2). We also report on numerical computations
indicating that in general, the ramification index at a cusp seems to be a
divisor of 24
L-functions for holomorphic forms on GSp(4) x GL(2) and their special values
We provide an explicit integral representation for L-functions of pairs (F,g)
where F is a holomorphic genus 2 Siegel newform and g a holomorphic elliptic
newform, both of squarefree levels and of equal weights. When F,g have level
one, this was earlier known by the work of Furusawa. The extension is not
straightforward. Our methods involve precise double-coset and volume
computations as well as an explicit formula for the Bessel model for GSp(4) in
the Steinberg case; the latter is possibly of independent interest. We apply
our integral representation to prove an algebraicity result for a critical
special value of L(s, F \times g). This is in the spirit of known results on
critical values of triple product L-functions, also of degree 8, though there
are significant differences.Comment: 48 pages, typos corrected, some changes in Sections 6 and 7, other
minor change
On Ihara's lemma for Hilbert Modular Varieties
Let \rho be a modulo p representation of the absolute Galois group of a
totally real number field. Under the assumptions that \rho has large image and
admits a low weight crystalline modular deformation we show that any low weight
crystalline deformation of \rho unramified outside a finite set of primes will
be modular. We follow the approach of Wiles as generalized by Fujiwara. The
main new ingredient is an Ihara type lemma for the local component at \rho of
the middle degree cohomology of a Hilbert modular variety. As an application we
relate the algebraic p-part of the value at 1 of the adjoint L-function
associated to a Hilbert modular newform to the cardinality of the corresponding
Selmer group.Comment: This is a completely revised version (30 pages
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