Let ζ5 be a primitive fifth root of unity, and let F = Q(ζ5). In this talk we describe recent computational work that investigates the modularity of elliptic curves over F . Here by modularity we mean that for a given elliptic curve E over F with conductor N there should exist an automorphic form f on GL2, also of conductor N , such that we have the equality of partial L-functions LS(s, f) = LS...

Let π be a cuspidal, automorphic representation of GSp4 attached to a Siegel modular form of degree 2. We refine the method of Furusawa [9] to obtain an integral representation for the degree-8 L-function L(s, π × τ), where τ runs through certain cuspidal, automorphic representation of GL2. Our calculations include the case of square-free level for the p-adic components of τ , and a wide class ...

This essay will explain the relationship between classical automorphic forms and representations of GL 2 (R). The classical theory of automorphic forms, in spite of initial appearances, is about the group GL 2 , not SL 2. The classical theory is concerned with functions on the upper half plane, which is acted on by fractional linear transformations in GL pos 2 (R), and it happens that the inter...

Conjecture 1 (Langlands). To each reductive group G over a number field K, each automorphic (complex) representation π of G, and each finite dimensional representation r of the (complex) group G, there is defined an automorphic L-function L(s, π, r), which enjoys an analytic continuation and functional equation generalizing the Riemann zeta function πΓ(s/2)ζ(s) (or Artin’s L-function L(s, σ), w...

is then a unitary representation of G(F) X G(F) on L2(G(F)). Kazhdan has suggested that there should be a local trace formula attached to R which is analogous to the global trace formula for automorphic forms. The purpose of this note is to discuss how one might go about proving such an identity, and to describe the ultimate form the identity is likely to take. To see the analogy with automorph...

we find a criterion for a morphism of coalgebras over a barr-exact category to be effective descent and determine (effective) descent morphisms for coalgebras over toposes in some cases. also, we study some exactness properties of endofunctors of arbitrary categories in connection with natural transformations between them as well as those of functors that these transformations induce between co...

We consider the automorphic forms which govern the gravitational threshold correction F1 in models of heterotic/IIA duality with N = 2 supersymmetry in four dimensions. In particular we derive the full nonperturbative formula for F1 for the dual pair originally considered by Ferrara, Harvey, Strominger and Vafa (FHSV). The answer involves an interesting automorphic product constructed by Borche...

Copyright q 2010 Hui-Sheng Ding et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A new and general existence and uniqueness theorem of almost automorphic solutions is obtained for the semilinear fractional differential equat...

We introduce a “geometric” method to bound periods of automorphic forms. The key features of this method are the use of equidistribution results in place of mean value theorems, and the systematic use of mixing and the spectral gap. Applications are given to equidistribution of sparse subsets of horocycles and to equidistribution of CM points; to subconvexity of the triple product period in the...