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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Cyril Guyot ¹, Kiumars Kaveh ², Vijay M. Patankar ³

Affiliations
1 Kasten Chase Applied Research, Orbitor Place, 5100 Orbitor Drive, Missisauga, Ontario L4W 4Z4, CA
2 1984 Mathematics Road, Department of Mathematics, University of British Columbia, Vancouver, Bristish Columbia V6T 1Z2, CA
3 100 St. George Street, Department of Mathematics, University of Toronto, Toronto, Ontario M6G 2M6, CA

Abstract

We investigate efficient formulae to double and add divisors on the Jacobian of a hyperelliptic curve of genus 3. The main contributions of this paper are as follows: (1) Overall improvements in the complexity of the addition and doubling algorithms for both even and odd characteristics, (2) Algorithms applicable to almost all hyperelliptic curves of genus 3, and (3) Efficient computation of the resultant of two polynomials and of the inverse of one polynomial modulo another. This paper is specifically written in an implementation-ready format.

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