This article grew out of my talk in ‘The Legacy of Srinivasa Ramanujan’ conference where I spoke about some techniques to prove algebraicity results for the special values of symmetric cube L-functions attached to the Ramanujan ∆-function. If one wishes to compare these different techniques, then one needs to compare various automorphic periods attached to the symmetric cube transfer of ∆. Moti...

If F is a local field containing the group μn of n-th roots of unity, and if G is a split semisimple simply connected algebraic group, then Matsumoto [27] defined an n-fold covering group of G(F ), that is, a central extension of G(F ) by μn. Similarly if F is a global field with adele ring AF containing μn there is a cover G̃(AF ) of G(AF ) that splits over G(F ). The construction is built on i...

We investigate conditions under which geometrically-defined automorphic forms can be lifted from positive characteristic to characteristic zero.

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...