Matthew Greenberg

STARK-HEEGNER POINTS AND THE COHOMOLOGY OF QUATERNIONIC SHIMURA VARIETIES

Let F be a totally real field of narrow class number one, and let E/F be a modular, semistable elliptic curve of conductor N � (1) .L etK/F be a non-CM quadratic extension with (Disc K, N) = 1 such that the sign in the functional equation of L(E/K, s) is −1. Suppose further that there is a prime p|N that is inert in K .W e describe a p-adic construction of points on E which we conjecture to be rational over ring class fields of K/F and satisfy a Shimura reciprocity law. These points are expected to behave like classical Heegner points and can be viewed as new instances of the Stark-Heegner point construction of [5]. The key idea in our construction is a reinterpretation of Darmon’s theory of modular symbols and mixed period integrals in terms of group cohomology.

COMPUTING SYSTEMS OF HECKE EIGENVALUES ASSOCIATED TO HILBERT MODULAR FORMS

We utilize effective algorithms for computing in the cohomology of a Shimura curve together with the Jacquet-Langlands correspondence to compute systems of Hecke eigenvalues associated to Hilbert modular forms over a totally real field F.

Lattice Methods for Algebraic Modular Forms on Classical Groups

We use Kneser’s neighbor method and isometry testing for lattices due to Plesken and Souveigner to compute systems of Hecke eigenvalues associated to definite forms of classical reductive algebraic groups.

Integral Eisenstein cocycles on GLn, II: Shintani's method

We define a cocycle on Gln using Shintanis method. It is closely related to cocycles defined earlier by Solomon and Hill, but differs in that the cocycle property is achieved through the introduction of an auxiliary perturbation vector Q. As a corollary of our result we obtain a new proof of a theorem of Diaz y Diaz and Friedman on signed fundamental domains, and give a cohomological reformulation of Shintanis proof of the Klingen-Siegel rationality theorem on partial zeta functions of totally real fields. Next we prove that the cohomology class represented by our Shintani cocycle is essentially equal to that represented by the Eisenstein cocycle introduced by Sczech. This generalizes a result of Sczech and Solomon in the case n=2. Finally we introduce an integral version of our Shintani cocycle by smoothing at an auxiliary prime ell. Applying the formalism of the first paper in this series, we prove that certain specializations of the smoothed class yield the p-adic L-functions of totally real fields. Combining our cohomological construction with a theorem of Spiess, we show that the order of vanishing of these p-adic L-functions is at least as large as the one predicted by a conjecture of Gross.

Nonsolvable number fields ramified only at 3 and 5

For p = 3 and p = 5, we exhibit a finite nonsolvable extension of Q which is ramified only at p, proving in the affirmative a conjecture of Gross. Our construction involves explicit computations with Hilbert modular forms.

Heegner point computations via numerical p -adic integration

Building on ideas of Pollack and Stevens, we present an efficient algorithm for integrating rigid analytic functions against measures obtained from automorphic forms on definite quaternion algebras. We then apply these methods, in conjunction with the Jacquet-Langlands correspondence and the Cerednik-Drinfeld theorem, to the computation of p-adic periods and Heegner points on elliptic curves defined over ℚ and

Rationality of secant zeta values

{\mathbb{Q}}(\sqrt{5})

Integral Eisenstein cocycles on GL n , II: Shintani's method

which are uniformized by Shimura curves.

Integral Eisenstein cocycles on

We use the theory of generalized