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

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

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

The density of ramified primes in semisimple p-adic Galois representations

We prove that the density of ramified primes in semisimple p-adic representations of Galois groups of number fields is 0. Ravi Ramakrishna has produced examples of such representations that are infinitely ramified

On Heegner points of large conductors

Given a parametrisation of an elliptic curve over Q by a Shimura curve, we show that the images of almost all Heegner points are of infinite order. For parametrisations of elliptic curves by modular curves this was proven earlier by Nekovar and Schappacher by a different method