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Properties of analytic vectors in representations of SL(2,R) are used to give new bounds for the triple products recently considered by P. Sarnak. A conjecture of Sarnak about such products is proved. The results of this paper generalize results of A. Good and M. Jutila about special cases, but the techniques are entirely different. One consequence of these results is a new estimate of the magnitude of the Fourier coefficients of cusp forms for non-arithmetic sub-groups of SL(2,R).

Properties of analytic vectors in representations of SL(2,R) are used to give new bounds for the triple products recently considered by P. Sarnak. A conjecture of Sarnak about such products is proved. The results of this paper generalize results of A. Good and M. Jutila about special cases, but the techniques are entirely different. One consequence of these results is a new estimate of the magnitude of the Fourier coefficients of cusp forms for non-arithmetic sub-groups of SL(2,R).

Given a number field we construct a canonical Galois gerbe over it and show that its cohomology provides a bridge between the refined local endoscopy introduced in arXiv:1304.3292 and classical global endoscopy. As particular applications, we express the canonical adelic transfer factor that governs the stabilization of the Arthur-Selberg trace formula as a product of normalized local transfer factors, we give an explicit construction of the pairing between an adelic L-packet and the corresponding S-group (based on the conjectural pairings in the local setting) that is the essential ingredient in the description of the discrete automorphic spectrum of a reductive group, and we give a proof of some expectations of Arthur.

Arthur has conjectured that the unitarity of a number of representations can be shown by finding appropriate automorphic realizations. This has been verified for classical groups by Moeglin and for the exceptional Chevalley group G_2 by Kim. In this paper we extend their results on spherical representations to the remaining exceptional groups E_6, E_7, E_8, and F_4. In particular we prove Arthur's conjecture that the spherical constituent of an unramified principal series of a Chevalley group over any local field of characteristic zero is unitarizable if its Langlands parameter coincides with half the marking of a coadjoint nilpotent orbit of the Langlands dual Lie algebra.

Let F be a number field with adele ring A_F, and \pi an isobaric, algebraic automorphic representation of GL_4(A_F) of a fixed archimedean weight, which is quasi-regular, meaning that at every archimedean place v of F, the 4-dimensional representation \sigma_v of the Weil group W_{F_v} attached to \pi_v is multiplicity free. Suppose there is an associated 4-dimensional, Hodge-Tate p-adic representation \rho of the absolute Galois group G_F, whose local L-factors agree with those of \pi (up to a shift) at almost all primes P of F. Then our first result is that the semisimplification of \rho does not contain any irreducible 2-dimensional Galois representation which is even. The second result is that if \pi is regular and \rho crystalline, then for sufficiently large p (see the article for a precise statement), the decomposition type of \rho is the same as the isobaric type of \pi. A consequence is that \rho is irreducible when \pi is cuspidal (and regular algebraic), which has also been proved by F. Calegari and T. Gee, in fact with no hypothesis on p. The third and final result, using Taylor's potential modularity theorem, is that given a pair (\sigma, \sigma') of odd, 2-dimensional p-adic representations of the same weight and distinct Hodge-Tate types, such that their direct sum is automorphic, the dimension of the G_F-invariants of the tensor product \eta of the dual of \sigma with \sigma' equals, for large p, the order of pole at s=1 of the L-function of \eta (with the bad factors removed). This is as predicted by the Tate conjecture when \sigma, \sigma' occur in p-adic etale cohmology of smooth projective varieties over F, and it also provides a useful link to a small piece of the work of C. Skinner and E. Urban.

In this paper we fully describe the cuspidal and the Eisenstein cohomology of the group $G=GL_2$ over a definite quaternion algebra $D/\Q$. Functoriality is used to show the existence of residual and cuspidal automorphic forms, having cohomology in degree 1. The latter ones turn out to be CAP-representations, though $G$ satisfies Strong Multiplicity One. A non-vanishing result on intertwining operators of induced representations will serve as a starting point for further investigations concerning rationality of critical $L$-values.

In this paper we fully describe the cuspidal and the Eisenstein cohomology of the group $G=GL_2$ over a definite quaternion algebra $D/\Q$. Functoriality is used to show the existence of residual and cuspidal automorphic forms, having cohomology in degree 1. The latter ones turn out to be CAP-representations, though $G$ satisfies Strong Multiplicity One. A non-vanishing result on intertwining operators of induced representations will serve as a starting point for further investigations concerning rationality of critical $L$-values.

In this paper we fully describe the cuspidal and the Eisenstein cohomology of the group $G=GL_2$ over a definite quaternion algebra $D/\Q$. Functoriality is used to show the existence of residual and cuspidal automorphic forms, having cohomology in degree 1. The latter ones turn out to be CAP-representations, though $G$ satisfies Strong Multiplicity One. A non-vanishing result on intertwining operators of induced representations will serve as a starting point for further investigations concerning rationality of critical $L$-values.

1Mermaid of Druid Lake and Other Stories

By Charles Weathers Bump

A collection of offbeat stories. Some are a bit out of the ordinary as suggested by the title story about a freshwater mermaid; some are not. - Summary by Amy Gramour