82 research outputs found
A database of genus 2 curves over the rational numbers
We describe the construction of a database of genus 2 curves of small
discriminant that includes geometric and arithmetic invariants of each curve,
its Jacobian, and the associated L-function. This data has been incorporated
into the L-Functions and Modular Forms Database (LMFDB).Comment: 15 pages, 7 tables; bibliography formatting and typos fixe
CM cycles on Kuga–Sato varieties over Shimura curves and Selmer groups
Given a modular form f of even weight larger than two and an imaginary quadratic field K
satisfying a relaxed Heegner hypothesis, we construct a collection of CM cycles on a Kuga–Sato
variety over a suitable Shimura curve which gives rise to a system of Galois cohomology classes
attached to f enjoying the compatibility properties of an Euler system. Then we use Kolyvagin’s
method [21], as adapted by Nekova´¿r [28] to higher weight modular forms, to bound the size of the relevant Selmer group associated to f and K and prove the finiteness of the (primary part) of the Shafarevich–Tate group, provided that a suitable cohomology class does not vanish.Peer ReviewedPostprint (author's final draft
On the Sato-Tate conjecture for non-generic abelian surfaces
We prove many non-generic cases of the Sato-Tate conjecture for abelian
surfaces as formulated by Fite, Kedlaya, Rotger and Sutherland, using the
potential automorphy theorems of Barnet-Lamb, Gee, Geraghty and Taylor.Comment: 21 pages. Minor changes and corrections. With an appendix by Francesc
Fit\'e. Essentially final version, to appear in Transactions of the AM
Monodromy groups of Jacobians with definite quaternionic multiplication
Let be an abelian variety over a number field. The connected monodromy
field of is the minimal field over which the images of all the -adic
torsion representations have connected Zariski closure. We show that for all
even , there exist infinitely many geometrically nonisogenous abelian
varieties over of dimension where the connected monodromy
field is strictly larger than the field of definition of the endomorphisms of
. Our construction arises from explicit families of hyperelliptic Jacobians
with definite quaternionic multiplication.Comment: 55 pages. v2: extended and improved the discussion of the moduli
space interpretation of our construction
- …