Journal of the Ramanujan Mathematical Society

Vijay M. Patankar ¹, C. S. Rajan ²

Affiliations
1 Indian Statistical Institute c/o Society for Electronic Transactions and Security (SETS), MGR Knowledge City, C.I.T. Campus, Taramani, Chennai 600 113, IN
2 Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, IN

Abstract

We show that if two continuous semi-simple ℓ-adic Galois representations are locally potentially equivalent at a sufficiently large set of places then they are globaly potentially equivalent. We also prove an analogous result for arbitrarily varying powers of character values evaluated at the Frobenius conjugacy classes. In the context of modular forms, we prove: given two non-CM newforms f and g of weight at least two, such that a_p( f )^{n _p} = a_p(g)^{n _p} on a set of primes of positive upper density and for some set of natural numbers n_p, then f and g are twists of each other by a Dirichlet character.

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C. S. Rajan ¹

Affiliations
1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road Mumbai-400005, IN

Abstract

We show that the automorphism group of the curve X(11) is the Mathieu group over a field of characteristic 3, containing a 11^th ischolar_main of unity. As a consequence, we also exhibit a Galois covering of the affine line over characteristic 3, with Galois group M₁₁

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