638 research outputs found
Elliptic Curves over Real Quadratic Fields are Modular
We prove that all elliptic curves defined over real quadratic fields are
modular.Comment: 38 pages. Magma scripts available as ancillary files with this arXiv
versio
Cohomology and the Brauer groups of diagonal surfaces
We present a method for calculating the Brauer group of a surface given by a
diagonal equation in the projective space. For diagonal quartic surfaces with
coefficients in Q we determine the Brauer groups over Q and Q(i).Comment: 45 page
Constructing genus 3 hyperelliptic Jacobians with CM
Given a sextic CM field , we give an explicit method for finding all genus
3 hyperelliptic curves defined over whose Jacobians are simple and
have complex multiplication by the maximal order of this field, via an
approximation of their Rosenhain invariants. Building on the work of Weng, we
give an algorithm which works in complete generality, for any CM sextic field
, and computes minimal polynomials of the Rosenhain invariants for any
period matrix of the Jacobian. This algorithm can be used to generate genus 3
hyperelliptic curves over a finite field with a given zeta
function by finding roots of the Rosenhain minimal polynomials modulo .Comment: 20 pages; to appear in ANTS XI
- …