C. S. Rajan

Distinguished representations, base change, and reducibility for unitary groups

We show the equality of the local Asai L-functions defined via the Rankin-Selberg method and the Langlands-Shahidi method for a square-integrable representation of GL n (E). As a consequence, we characterize reducibility of certain induced representations of U(n,n), and the image of the base change map from U(n) to GL n (E) in terms of GL n (F)-distinguishedness. We also determine precisely when the Steinberg representation of GL n (E) is distinguished with respect to a character of F ∗ .

Frobenius splitting and ordinarity

We examine the relationship between the notion of Frobe- nius splitting and ordinarity for varieties. We show the following: a) The de Rham-Witt cohomology groups H i (X, W(OX)) of a smooth projec- tive Frobenius split variety are finitely generated over W(k). b) we provide counterexamples to a question of V. B. Mehta that Frobenius split varieties are ordinary or even Hodge-Witt. c) a Kummer K3 sur- face associated to an Abelian surface is F-split (ordinary) if and only if the associated Abelian surface is F-split (ordinary). d) for a K3-surface defined over a number field, there is a set of primes of density one in some finite extension of the base field, over which the surface acquires ordinary reduction.

On the vanishing of the measurable Schur cohomology groups of Weil groups

We generalize a theorem of Tate and show that the second cohomology of the Weil group of a global or local field with coefficients in C ∗ (or, more generally, with coefficients in the complex points of an algebraic torus over C) vanish, where the cohomology groups are defined using measurable cochains in the sense of Moore. We recover a theorem of Labesse stating that the admissible homomorphisms of a Weil group to the Langlands dual group of a reductive group can be lifted to an extension of the Langlands dual group by a torus.

Refinement of strong multiplicity one for automorphic representations of GL(n)

We state a qualitative form of strong multiplicity one for GL 1 . We derive refinements of strong multiplicity one for automorphic representations arising from Eisenstein series associated to a Borel subgroup on GL(n), and for the cuspidal representations on GL(n) induced from idele class characters of cyclic extensions of prime degree. These results are in accordance with a conjecture of D. Ramakrishnan. We also show that Ramakrishnans conjecture follows from a weak form of Ramanujans conjecture. We state a conjecture concerning the structural aspects of refinements of strong multiplicity one for a pair of general automorphic representations.

Recovering modular forms and representations from tensor and symmetric powers

We consider the problem of determining the relationship between two representations knowing that some tensor or symmetric power of the original representations coincide. Combined with refinements of strong multiplicity one, we show that if the characters of some tensor or symmetric powers of two absolutely irreducible l-adic representation with the algebraic envelope of the image being connected, agree at the Frobenius elements corresponding to a set of places of positive upper density, then the representations are twists of each other by a finite order character.

On the size of the Shafarevich--Tate group of elliptic curves over function fields

Let E be a nonconstant elliptic curve, over a global field K of positive, odd characterisitc. Assuming the finiteness of the Shafarevich-Tate group of E, we show that the order of theShafarevich-Tate group of E, is given by O(N1/2+6 log(2)/ log(q)), where N is the conductor of E,q isthe cardinality of the finite field of constants of K, and where the constant in the bound depends only on K. The method of proof is to workwith the geometric analog of the Birch-Swinnerton Dyer conjecture for thecorresponding elliptic surface over the finite field, as formulatedby Artin-Tate, and to examine the geometry of this elliptic surface.

On an Archimedean analogue of Tate's conjecture

We consider an Archimedean analogue of Tates conjecture, and verify the conjecture in the examples of isospectral Riemann surfaces constructed by Vigneras and Sunada. We prove a simple lemma in group theory which lies at the heart of T. Sunadas theorem about isospectral manifolds.

On isospectral arithmetical spaces

We study the relationship between the arithmetic and the spectrum of the Laplacian for manifolds arising from congruence arithmetic subgroups of SL(1,D), where D is an indefinite quaternion division algebra defined over a number field F. We give new examples of isospectral but nonisometric compact, arithmetically defined varieties, generalizing the class of examples constructed by Vigneras. These examples are based on an interplay between the simply connected and adjoint group and depend explicitly on the failure of strong approximation for the adjoint group. The examples can be considered as a geometric analogue and also as an application of the concept and results on L-indistinguishability for SL(1,D) due to Labesse and Langlands. We verify that the Hasse-Weil zeta functions are equal for the examples of isospectral pair of arithmetic varieties we construct giving further evidence for an archimedean analogue of Tate’s conjecture, which expects that the spectrum of the Laplacian determines the arithmetic of such spaces.

On strong multiplicity one for automorphic representations

We extend the strong multiplicity one theorem of Jacquet, Piatetski-Shapiro and Shalika. Let π be a unitary, cuspidal, automorphic representation of GLn(AK). Let S be a set of finite places of K, such that the sum ∑v∈SNv−2/(n2+1) is convergent. Then π is uniquely determined by the collection of the local components {πv|v∉S,v finite} of π. Combining this theorem with base change, it is possible to consider sets S of positive density, having appropriate splitting behavior with respect to a solvable extension L of K, and where π is determined up to twisting by a character of the Galois group of L over K.

Group cohomology and the symplectic structure on the moduli space of representations

The space of equivalence classes of irreducibleb representations of the fundamental group of a compact oriented surface of genus at least 2 in a Lie group has a natural symplectic form. In lAB], Atiyah and Bott described this symplectic structure using methods from gauge theory. Goldman [G] constructed the skew-symmetric pairing algebraically using methods from group cohomology. Using Poincar6 duality, Goldman showed that the pairing is nondegenerate and identified it with the symplectic structure given by gauge theory.