Yuval Flicker -- Current Research

My research interests in Number Theory and Harmonic Analysis include p-adic automorphic forms, automorphic representations, algebraic groups, symmetric spaces, Galois cohomology, admissible representations of p-adic groups, trace formulae and cycle summation formulae, their stabilization, and the fundamental lemma.

My recent research interests concern an exploration of hoped for interrelations between the theory of liftings of (complex valued) automorphic representations and the theory of p-adic automorphic forms. While the former is a rich and established theory, governed by the hypothetical reciprocity law which predicts a parametrization of the automorphic representations of a reductive group G by means of homomorphisms of a (hypothetical) variant of the Weil group into the dual group of G, with which I have much experience, including a recent book, the latter is a relatively new theory -- in particular for me -- in which modular forms are varied p-adically according to their weights. This variation is parametrized by a rigid analytic variety, called the eigenvariety, introduced and studied by Coleman, Mazur, Buzzard. I believe that not only liftings of automorphic representations, named now ``classical'' by the p-adic community, could be extended by continuity to liftings of p-adic automorphic forms, but also that functorial maps on the underlying eigenvarieties can be used to establish ``classical'' lifting theorems, as the forms are classical when the slope is small, in some cases. This would show the power and relevance of the new p-adic developments in the better-known classical arena.

In my recent book ``Automorphic Forms and Shimura Varieties of PGSp(2)'' -- see book -- admissible and automorphic representations of the group in the title are classified by means of a definition of packets and quasi-packets, and liftings to PGL(4) and from the endoscopic group PGL(2)xPGL(2). Multiplicity one and rigidity theorem for packets are proven, as well as the (correct version of the) Ramanujan conjecture for geometric automorphic representations of this group. Moreover, Galois representations are attached to such automorphic representations.

In the recent 215-page thesis [Ch] ``Invariant Representations of GSp(2)'' of my student Ping-Shun Chan -- see Chan -- the admissible and automorphic representations of the group in the title which are invariant under twisting by a quadratic character are classified by means of liftings from two rank one twisted endoscopic groups. A host of questions is raised by this work.

The preprint ``Automorphic representations of low rank groups'' contains a rewrite of most of my work on the symmetric square lifting from SL(2) to PGL(3), which includes a proof of multiplicity one theorem for SL(2) and rigidity theorem for packets, a simple proof of the fundamental lemma, and of the unrestricted equality of the trace formulae, as well as the base change lifting from U(3) to GL(3,E), proof of multiplicity one, rigidity for packets, fundametal lemma, equality of trace formulae, and finally a proof of the Ramanujan conjecture for geometric automorphic representations of U(3) as well as a construction of Galois representations whose Hecke eigenvalues are related to the Hecke eigenvalues of these geometric automorphic representations. Many of the proofs are new. In particular, I point out that the attempted global proof of multiplicity one theorem for cuspidal representations of the unitary group U(3) is in fact incomplete, contrary to a belief dated 1983, and provide the details of a local proof, based on the meaning of the coefficients in the asymptotic expansion of the character.

My project: "Grothendieck theorem for non abelian H^2 and the Hasse principle" concerns a formally real generalization of the theorem of Grothendieck (found in Springer's Boulder article) on the vanishing of the non-abelian second cohomology set of an algebraic group over a field of cohomological dimension one. The generalization - to the context of such groups over fields of virtual cohomological dimension one - would take the form of a local-global principle with respect to the orderings of the field. This principle asserts that an element in the second non-abelian cohomology is neutral precisely when it is neutral in the real closure with respect to every ordering of the field. Applications: local-global principles on homogeneous spaces of algebraic groups over fields of virtual cohomological dimension one, thus existence of points at each real closure implies the existence of a point over the original field; the existence of a covering of the homogeneous space by a principal homogeneous space over the real closures implies the existence of a covering over the field itself. In fact, it suffice to make the local assumptions only at a dense subset of the real spectrum. This verifies a conjecture of Scheiderer. The techniques rely on those of (Grothendieck, Giraud and) Springer, and of Borovoi - who dealt with analogous questions in the case of number fields.

My project: "The fundamental lemma", concerns the study of the fundamental lemma for classical groups. This Lemma plays a key role in the comparison of trace formulae, which leads to liftings of automorphic forms, a main aim of the Langlands program. Progress has already been made - by means of a new technique - in the cases of the symmetric square lifting from SL(2) to PGL(3), the endoscopic lifting from U(2) to U(3), and the twisted endoscopic lifting from GSp(2) to GL(4). The technique is based on a description of a double coset decomposition H\G/K, for a subgroup H containing the torus T in question. Such a decomposition is available for classical groups by the work of Murase-Sugano. An attempt to generalize the technique to higher rank groups requires subtle analysis of twisted conjugacy classes. This already led to a new expression for the transfer factor. This direct and elementary technique is likely to succeed, independently of the highly sophisticated approaches

My project: "Langlands' bi-program", introduces a generalization of Langlands program to deal with periods of automorphic forms, and admissible representations with vectors fixed under a subgroup. It is based on a generalization of the trace formula to the context of a symmetric space G(E)/G(F) ([E:F]=2). Such a "bi-period summation formula" is introduced and stabilized. A transfer factor which is required in the stabilization and in the statement of the fundamental lemma, is introduced. The generalized transfer factor depends on a new definition of the transfer factor of Langlands-Shelstad in the case of standard conjugacy. This Langlands' bi-program opens a wide new territory for exploration, dealing with representations of G(E) with a period (or vector fixed by) G(F). Our next goal concerns the study of the relevant bi-orbital integrals.