David Penniston

University of Wisconsin–Oshkosh | UWOSH · Department of Mathematics

A partition of a positive integer n is called-regular if none of its parts is divisible by. Let b11(n) denote the number of 11-regular partitions of n. In this paper we give a complete description of the behavior of b11(n) modulo 5 when 5&n in terms of the arithmetic of the ring [-33]. This description is obtained by relating the generating functio...

Let $b_{\ell }(n)$ denote the number of $\ell$ -regular partitions of $n$ . In this paper we establish a formula for $b_{13}(3n+1)$ modulo $3$ and use this to find exact criteria for the $3$ -divisibility of $b_{13}(3n+1)$ and $b_{13}(3n)$ . We also give analogous criteria for $b_{7}(3n)$ and $b_{7}(3n+2)$ .

Let b ℓ (n) denote the number of ℓ-regular partitions of n. Recently Andrews, Hirschhorn, and Sellers proved that b 4(n) satisfies two infinite families of congruences modulo 3, and Webb established an analogous result for b 13(n). In this paper we prove similar families of congruences for b ℓ (n) for other values of ℓ.

We give exact criteria for the ℓ-divisibility of the ℓ-regular partition function b ℓ (n) for ℓ∈{5,7,11}. These criteria are found using the theory of complex multiplication. In each case the first criterion given corresponds to the Ramanujan congruence modulo ℓ for the unrestricted partition function, and the second is a condition given by J.-P. S...

Let b l (n) denote the number of l-regular partitions of n, where l is prime and 3≤l≤23. In this paper we prove results on the distribution of b l (n) modulo m for any odd integer m>1 with 3∤m if l≠3.

We prove the existence of an infinite family of non-harmonic weak Maass forms of varying weights and Laplace eigenvalues having algebraic coefficients, and show that the coefficients of these forms satisfy congruences of Ramanujan type.

We investigate arithmetic properties of the Fourier coefficients of certain harmonic weak Maass forms of weight 1/2 and 3/2. Each of the forms in question is the sum of a holomorphic function and a period integral of a theta series. In particular, for any positive integer M coprime to 6 we prove that the coefficients of the holomorphic function sat...

The function bk(n) is defined as the number of partitions of n that contain no summand divisible by k. In this paper we study the 2- divisibility of b5(n) and the 2- and 3-divisibility of b13(n). In particular, we give exact criteria for the parity of b5(2n) and b13(2n).

We present congruences for Greene's 3F2 hypergeometric functions over finite fields, which relate values of these functions to a simple polynomial in the characteristic of the field.

Let bℓ(n) denote the number of ℓ-regular partitions of n, where ℓ is a positive power of a prime p. We study in this paper the behavior of bℓ(n) modulo powers of p. In particular, we prove that for every positive integer j, bℓ(n) lies in each residue class modulo pj for infinitely many values of n.

Let Q(n) denote the number of partitions of an integer n into distinct parts. For positive integers j, the first author and B. Gordon proved that Q(n) is a multiple of 2j for every non-negative integer n outside a set with density zero. Here we show that if i≢0 (mod2j), then#{0≤n≤X:Q(n)≡i(mod2j)}≫jX/logX.In particular, Q(n) lies in every residue cl...

Let X X be a curve defined over an algebraically closed field k k with char ⁡ ( k ) = p > 0 \operatorname {char}(k)=p>0 . Assume that X / k X/k is reduced. In this paper we study the unipotent part U U of the Jacobian J X / k J_{X/k} . In particular, we prove that if p p is large in terms of the dimension of U U , then U U is isomorphic to a produc...

In this paper we study unipotent groups associated to degenerations of smooth curves. In particular, we obtain results on the exponent of these groups, and use these to get an almost complete description of the structure of unipotent groups associated to curves of genus two.

Let X be a curve defined over an algebraically closed field k with char $(k)=p>0$ . Assume that X/k is reduced. In this paper we study the unipotent part U of the Jacobian JX/k. In particular, we prove that if p is large in terms of the dimension of U, then U is isomorphic to a product of additive groups Ga.