arXiv subjects
Publications and source records attributed to Chandrashekhar Khare.
We give a direct approach to recover some of the results of Wiles and Tayor on modularity of certain 2-dimensional p-adic representations of the absolute Galois group of Q.
In this paper we pose and answer the following question in a few different contexts: Given a homomorphism f:L_1 --> L_2 of a ``lattices'' that ``reduces mod p'' for almost all primes p, is f ``algebraic''? For instance the lattices may be the Mordell-Weil lattices of rational points of abelian varieties over Q, or arithmetic groups etc. Implicit in an affirmative answer to the question for Mordell-Weil lattices is a novel criterion for abelian varities to be isogenous.
This is a revised version of the preprint which has been available electronically for a while. The paper will now appear in J. Ramanujan Math. Soc.
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.
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.
We prove that the reduction mod \ell of the local Langlands correspondence between supercuspidal representations of GL_n(F), where F is a finite extension of Q_p, and representations of the Galois group of F is well-defined. The results and methods of this paper have been used by Vigneras to give a proof of the ``local Langlands conjecture mod \ell'', that was proved earlier by her by different methods (local harmonic analysis mod \ell) for \ell>n. Unlike the local methods used earlier by Vigneras, our more global method also generalises well to the characteristic p case.