Let j be the modular invariant. For the primes p ≤ 23 the q-expansion coefficients of U(j− 744) are multiplicative as it was a Hecke eigenformmodulo a power of p which increases withm. This was conjectured by Atkin on the basis of extensive numerical experiments, and is proved in this paper. The cases p = 5, 7 and 11 are under special consideration in this paper.

In this paper, we present a design strategy of elliptic curves whose extension degrees needed for reduction attacks have a controllable lower boundary, based on the complex multiplication elds method of Atkin and Morain over prime elds.

A classical way to compute the number of points of elliptic curves defined over finite fields from partial data obtained in SEA (Schoof Elkies Atkin) algorithm is a so-called “Match and Sort” method due to Atkin. This method is a “baby step/giant step” way to find the number of points among C candidates with O( √ C) elliptic curve additions. Observing that the partial information modulo Atkin’s...

This is a joint project with Liqun Fang Ben Linowitz Andrew Rupinski Helena Verrill We give new examples of modular forms on noncongruence subgroups whose l-adic representations are modular and whose expansion coefficients satisfy Atkin-Swinnerton-Dyer congruences.

V Johnson, E Volikos, S E Halford, E T Eftekhar Sadat, S Popat, I Talbot, K Truninger, J Martin, J Jass, R Houlston, W Atkin, I P M Tomlinson, A R J Silver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....