An example of non-homeomorphic conjugate varieties

We give examples of smooth quasi-projective varieties over complex numbers, in the context of connected Shimura varieties, which are not homeomorphic to a conjugate of itself by an automorphism of the complex numbers.Comment: 4 page

Unique decomposition of tensor products of irreducible representations of simple algebraic groups

We show that a tensor product of irreducible, finite dimensional representations of a simple Lie algebra over a field of characteristic zero, determines the individual constituents uniquely. This is analogous to the uniqueness of prime factorisation of natural numbers.Comment: 23 pages, to appear in Annals of Mathematic

On strong multiplicity one for automorphic representations

We extend the strong multiplicity one theorem of Jacquet, Piatetski-Shapiro and Shalika. Let $\pi$ be a unitary, cuspidal, automorphic representation of GL_n(\A_K). Let $S$ be a set of finite places of $K$, such that the sum $\sum_{v\in S}Nv^{-2/(n^2+1)}$ is convergent. Then $\pi$ is uniquely determined by the collection of the local components $\{\pi_v\mid v\not\in S, ~v \~\text{finite}\}$ of $\pi$. Combining this theorem with base change, it is possible to consider sets $S$ of positive density, having appropriate splitting behavior with respect to solvable extensions of $K$, and where $\pi$ is determined upto twisting by a character of the Galois group of $L$ over $K$.Comment: 8 page

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 a tori over \C) vanish, where the cohomology groups are defined using measurable cochains in the sense of Moore. We recover a theorem of Labesse proving that admissible homomorphisms of a Weil group to the Langlands dual of a reductive group can be lifted to an extension of the Langlands dual group by a tori.Comment: To appear in Compositio Mathematica. Minor editing, and a conjecture in the earlier version on the vanishing of the higher cohomology groups of Weil groups is fals

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 represetations 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.Comment: 18 pages; this is a revised version of a paper submitted to the old Number Theory archive as ANT-035