71 research outputs found
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.Comment: 17 page
Shimura correspondence for level and the central values of -series II
Given a Hecke eigenform of weight and square-free level , by the
work of Kohnen, there is a unique weight modular form of level
mapping to 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 . Gross gave a construction of the half
integral weight form when 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 , for 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 ) which
gives two weight modular forms mapping to 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
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
A B\"ocherer-Type Conjecture for Paramodular Forms
In the 1980s B\"ocherer 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
Effective construction of Hilbert modular forms of half-integral weight
Given a Hilbert cuspidal newform we construct a family of modular forms
of half-integral weight whose Fourier coefficients give the central values of
the twisted -series of by fundamental discriminants.
The family is parametrized by quadratic conditions on the primes dividing the
level of , 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
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