Roberto Gómez,* Sascha Husa, Luis Lehner, and Jeffrey Winicour Pittsburgh Supercomputing Center, 4400 Fifth Avenue, Pittsburgh, Pennsylvania 15213 Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260 Albert Einstein Institute, Max Planck Gesellschaft, Haus 1, Am Mühlenberg, Golm, Germany Department of Physics and Astronomy, University of British Columbia...

We present a variant of an algorithm of Oliver Atkin for counting the number of points on an elliptic curve over a nite eld. We describe an implementation of this algorithm for prime elds. We report on the use of this implementation to count the number of points on a curve over IFp, where p is a 375-digit prime.

In 2003, Atkin and Garvan initiated the study of rank and crank moments for ordinary partitions. These moments satisfy a strict inequality. We prove that a strict inequality also holds for the first rank and crank moments of overpartitions and consider a new combinatorial interpretation in this setting.

We present a variant of an algorithm of Oliver Atkin for counting the number of points on an elliptic curve over a nite eld. We describe an implementation of this algorithm for prime elds. We report on the use of this implementation to count the the number points on a curve over F p , where p is a 210-digit prime.

Schoof’s classic algorithm allows point-counting for elliptic curves over finite fields in polynomial time. This algorithm was subsequently improved by Atkin, using factorizations of modular polynomials, and by Elkies, using a theory of explicit isogenies. Moving to Jacobians of genus-2 curves, the current state of the art for point counting is a generalization of Schoof’s algorithm. While we a...

Elliptic curves with a known number of points over a given prime field Fn are often needed for use in cryptography. In the context of primality proving, Atkin and Morain suggested the use of the theory of complex multiplication to construct such curves. One of the steps in this method is the calculation of a root modulo n of the Hilbert class polynomial HD(X) for a fundamental discriminant D. T...

In this paper we give an example of a noncongruence subgroup whose three-dimensional space of cusp forms of weight 3 has the following properties. For each of the four residue classes of odd primes modulo 8 there is a basis whose Fourier coefficients at infinity satisfy a three-term Atkin and Swinnerton-Dyer congruence relation, which is the p-adic analogue of the threeterm recursion satisfied ...

We extend methods of Greenberg and the author to compute in the cohomology of a Shimura curve defined over a totally real field with arbitrary class number. Via the Jacquet-Langlands correspondence, we thereby compute systems of Hecke eigenvalues associated to Hilbert modular forms of arbitrary level over a totally real field of odd degree. We conclude with two examples which illustrate the eff...