In this paper we consider a level l Appell function, and find, for all odd l, a partial differential equation it satisfies. For l = 3 this recovers the Rank-Crank PDE, found by Atkin and Garvan, and for l = 5 we get a similar PDE found by Garvan.

In this paper we will use experimental and computational methods to find modular forms for non-congruence subgroups, and the modular forms for congruence subgroups that they are associated with via the Atkin–Swinnerton-Dyer correspondence. We also prove a generalization of a criterion due to Ligozat for an eta-quotient to be a modular function.

Hassan Kahal ([email protected]) Ahmed Aburima ([email protected]) Tamas Ungvari ([email protected]) Alan S Rigby ([email protected]) Anne M Coady ([email protected]) Rebecca V Vince ([email protected]) Ramzi A Ajjan ([email protected]) Eric S Kilpatrick ([email protected]) Khalid M Naseem ([email protected]) Stephen L Atkin ([email protected]...

In this paper we show how Rank-Crank type PDE’s (first found by Atkin and Garvan) occur naturally in the framework of non-holomorphic Jacobiforms and find an infinite family of such differential equations. As an application we show an infinite family of congruences for odd Durfee symbols, a partition statistic introduced by George Andrews.

Schoof's algorithm is used to nd a secure elliptic curve for cryptosystems, as it can compute the number of rational points on a randomly selected elliptic curve de ned over a nite eld. By realizing e cient combination of several improvements, such as Atkin-Elkies's method, the isogeny cycles method, and trial search by match-and-sort techniques, we can count the number of rational points on an...

Final report prepared for the National Institute of Corrections February 15, 2011 Submitted by Dr. Gaylene Armstrong College of Criminal Justice, Sam Houston State University Research Director, Correctional Management Institute of Texas with Doug Dretke Executive Director, Correctional Management Institute of Texas Cassandra Atkin Research Assistant, Correctional Management Institute of Texas

In 1988 P. Erdös asked if the prime divisors of xn−1 for all n = 1, 2, . . . determine the given integer x; the problem was affirmatively answered by Corrales-Rodorigáñez and R. Schoof [2] in 1997 together with its elliptic version. Analogously, K. Yamanoi [14] proved in 2004 that the support of the pull-backed divisor f∗D of an ample divisor on an abelian variety A by an algebraically non-dege...

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 ...