Period polynomials have long been fruitful tools for the study of values of L-functions in the context of major outstanding conjectures. In this paper, we survey some facets of this study from the perspective of Eichler cohomology. We discuss ways to incorporate non-cuspidal modular forms and values of derivatives of L-functions into the same framework. We further review investigations of the l...

(0.1) Let F be a totally real number field of degree d. It is well known that one can associate to any cuspidal Hilbert eigenform f over F of parallel weight 2 a compatible system of two-dimensional l-adic Galois representations Vl(f) of ΓF = Gal(Q/F ) over Ql (having fixed embeddings Q ↪→ C and Q ↪→ Ql). (0.2) On the other hand, the Shimura variety X associated to RF/QGL(2)F has reflex field Q...

We prove a geometric version of a classical result on the characterization of an irreducible cuspidal automorphic representation of GLn(AE ) being the base change of a stable cuspidal representation of the quasi-split unitary group associated to the quadratic extension E/F , via the nonvanishing of certain period integrals, called being distinguished. We show that certain cohomology of an autom...

For certain algebraic Hecke characters χ of an imaginary quadratic field F we define an Eisenstein ideal in a p-adic Hecke algebra acting on cuspidal automorphic forms of GL2/F . By finding congruences between Eisenstein cohomology classes (in the sense of G. Harder) and cuspidal classes we prove a lower bound for the index of the Eisenstein ideal in the Hecke algebra in terms of the special L-...

For certain algebraic Hecke characters χ of an imaginary quadratic field F we define an Eisenstein ideal in a p-adic Hecke algebra acting on cuspidal automorphic forms of GL2/F . By finding congruences between Eisenstein cohomology classes (in the sense of G. Harder) and cuspidal classes we prove a lower bound for the index of the Eisenstein ideal in the Hecke algebra in terms of the special L-...

Following G. Laumon [12], to a nonramified `-adic local system E of rank n on a curve X one associates a complex of `-adic sheaves nKE on the moduli stack of rank n vector bundles on X with a section, which is cuspidal and satisfies the Hecke property for E. This is a geometric counterpart of the well-known construction due to J. Shalika [19] and I. Piatetski-Shapiro [18]. We express the cohomo...

Let F be a non-Archimedean local field, and πi (i = 1, 2, 3) irreducible admissible representations of G = GL2(F ), such that the product of their central characters is trivial. In [8], Prasad shows that there exists, up to a scalar factor, at most one G-invariant linear form on π1 ⊗ π2 ⊗ π3, and determines exactly when such a form exists. These results have been used by Harris and Kudla [6] in...

Building on the work of Kudla and Millson we obtain a lifting of cuspidal cohomology classes for the symmetric space associated to GO(V ) for an indefinite rational quadratic space V of even dimension to holomorphic Siegel modular forms on GSpn(A). For n = 2 we prove Thom’s Lemma for hyperbolic 3space, which together with results of Kudla and Millson imply an interpretation of the Fourier coeff...