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Journals and Conferences
In this note, we define a cryptosystem based on non-commutative properties of groups. The cryptosystem is based on the hardness of the problem of factoring over these groups. This problem, interestingly, boils down to discrete logarithm problem on some Abelian groups. Further, we illustrate this method in three different non-Abelian groups GLn(Fq), UTn(Fq)… (More)
We explicitly construct the canonical rational models of Shimura curves, both analytically in terms of modular forms and algebraically in terms of coefficients of genus 2 curves, in the cases of quaternion algebras of discriminant 6 and 10. This emulates the classical construction in the elliptic curve case. We also give families of genus 2 QM curves, whose… (More)
In this note, we show that if the characteristic polynomial of some Hecke operator Tn acting on the space of weight k cusp forms for the group SL2(Z ) is irreducible, then the same holds for Tp, where p runs through a density one set of primes. This proves that if Maeda’s conjecture is true for some Tn, then it is true for Tp for almost all primes p.
We construct certain elements in the integral motivic cohomology group H M (E × E,Q(2))Z, where E and E are elliptic curves over Q. When E is not isogenous to E these elements are analogous to ‘cyclotomic units’ in real quadratic fields as they come from modular parametrisations of the elliptic curves. We then find an analogue of the class number formula… (More)
We estimate the number of Fourier coefficients that determine a Hilbert modular cusp form of arbitrary weight and level. The method is spectral (Rayleigh quotient) and avoids the use of the maximum principle.
We show that any abelian surface with multiplication by the quaternion Q-algebra of discriminant 6, with field of moduli Q and which is a Jacobian in characteristic 2 and 3, has infinitely many primes of superspecial reduction. This is done by examining CM points in characteristic 0 and p and the values of a certain j-function on the associated moduli space… (More)