13,468 research outputs found
The Saito-Kurokawa lifting and Darmon points
Let E_{/_\Q} be an elliptic curve of conductor with and let
be its associated newform of weight 2. Denote by the -adic
Hida family passing though , and by its -adic
Saito-Kurokawa lift. The -adic family of Siegel modular forms
admits a formal Fourier expansion, from which we can define a family of
normalized Fourier coefficients indexed by positive
definite symmetric half-integral matrices of size . We relate
explicitly certain global points on (coming from the theory of
Stark-Heegner points) with the values of these Fourier coefficients and of
their -adic derivatives, evaluated at weight .Comment: 14 pages. Title change
Permutations of Massive Vacua
We discuss the permutation group G of massive vacua of four-dimensional gauge
theories with N=1 supersymmetry that arises upon tracing loops in the space of
couplings. We concentrate on superconformal N=4 and N=2 theories with N=1
supersymmetry preserving mass deformations. The permutation group G of massive
vacua is the Galois group of characteristic polynomials for the vacuum
expectation values of chiral observables. We provide various techniques to
effectively compute characteristic polynomials in given theories, and we deduce
the existence of varying symmetry breaking patterns of the duality group
depending on the gauge algebra and matter content of the theory. Our examples
give rise to interesting field extensions of spaces of modular forms.Comment: 44 pages, 1 figur
Computing Hilbert Class Polynomials
We present and analyze two algorithms for computing the Hilbert class
polynomial . The first is a p-adic lifting algorithm for inert primes p
in the order of discriminant D < 0. The second is an improved Chinese remainder
algorithm which uses the class group action on CM-curves over finite fields.
Our run time analysis gives tighter bounds for the complexity of all known
algorithms for computing , and we show that all methods have comparable
run times
Hilbert modular surfaces for square discriminants and elliptic subfields of genus 2 function fields
We compute explicit rational models for some Hilbert modular surfaces
corresponding to square discriminants, by connecting them to moduli spaces of
elliptic K3 surfaces. Since they parametrize decomposable principally polarized
abelian surfaces, they are also moduli spaces for genus-2 curves covering
elliptic curves via a map of fixed degree. We thereby extend classical work of
Jacobi, Hermite, Bolza etc., and more recent work of Kuhn, Frey, Kani, Shaska,
V\"olklein, Magaard and others, producing explicit families of reducible
Jacobians. In particular, we produce a birational model for the moduli space of
pairs (C,E) of a genus 2 curve C and elliptic curve E with a map of degree n
from C to E, as well as a tautological family over the base, for 2 <= n <= 11.
We also analyze the resulting models from the point of view of arithmetic
geometry, and produce several interesting curves on them.Comment: 36 pages. Final versio
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