PANTS Homepage Department of Mathematics and Statistics UNC Greensboro Greensboro

Palmetto Number Theory Series -- PANTS XIII


Invited Talks

Farshid Hajir: Analogies between codes, curves, graphs, number fields and 3-manifolds

I will describe a general phenomenon, that of "asymptotically good families" which arises in multiple context. For a set of interesting objects (examples: curves over finite fields, number fields, regular graphs, linear codes, or hyperbolic 3-manifolds, lattices, etc.), equipped with an appropriate notion of "type" as well as of "quality," we seek infinite families of objects of a fixed type whose quality remains high. This defines a function measuring the optimal quality of such families as a function of the type. In many contexts, zeta functions provide upper bounds for this function, and modular forms provide lower bounds. In the most favorable cases, the two bounds actually meet, as in the case of "Ramanujan graphs" for example. I will try to put all of this in a common framework, with the hope that it stimulates cross-fertilization of ideas and promotes the search for further analogies of this type in other contexts.

Harold Stark: Some Density Questions in Number Theory

Suppose S is a set of positive integers which one wishes to enumerate. The sets of interest have the property that the obstructions to being in the set S are in correspondence with the primes: For each prime p, there is a set Sp such that any integer in Sp is not in S and further S consists of all integers not in any of the Sp. In the most interesting questions of this sort, it is not even known that S is infinite, but nevertheless there are conjectures that S is infinite and the density of numbers in S is even conjectured. I will discuss one or more of these questions.

Yu Zhao: Elliptic curves over totally real fields with everywhere good reduction

The Birch and Swinnerton-Dyer conjecture asserts that for any elliptic curve E defined over a number field F, the rank r of E(F) equals ords = 1L(E/K,s). In the case r &le 1, the conjecture is still not completely solved even if we have the Gross-Zagier-Kolyvagin theorem. This talk will introduce the Darmon-Logan's conjectural construction of points on elliptic curves defined over defined over a real quadratic field and Q-curves arising from Shimura's construction.

Contributed Talks

Jaroslav Hancl: Some results concerning the factorial series

Let {an} be a sequence of integers. We say that &Sigmaan/n! is an factorial series. The speaker would like to describe the important role which factorial series play in the study of irrationality and linear independence. In addition, some unsolved problems in these theories will be included. New theorems of Tijdeman and the speaker will be presented. The results will be applied for the case of the series &Sigma[n&alpha]/n! where &alpha is a positive real number.

Marie Jameson: The Alder-Andrews Conjecture

Motivated by classical identities of Euler, Schur, and Rogers and Ramanujan, Alder investigated qd(n) and Qd(n), the number of partitions of n into d-distinct parts and into parts which are \pm 1 (mod d+3), respectively. He conjectured that qd(n) &ge Qd(n). Andrews and Yee proved the conjecture for d = 2s-1 and also for d &ge 32. We will discuss a proof of Andrews's refinement of Alder's conjecture which follows from determining effective asymptotic estimates for these partition functions (correcting and refining earlier work of Meinardus), thereby reducing the conjecture to a finite computation.

Zachary Kent: p-adic lifting of roots of Eisenstein series

For a prime p > 3, we consider j-zeros of the family of Eisenstein series whose weights are p-adically close to p - 1. In particular, the j-zeros of the weight p - 1 Eisenstein series are j-invariants of elliptic curves with supersingular reduction modulo p. We lift these j-zeros to a p-adic field, and show that when k is p-adically close to p - 1, then the j-zeros of the weight k Eisenstein series are p-adically close to supersingular j-invariants.

Michael Mossinghoff: The distance to an irreducible polynomial

More than 40 years ago, P. Turán asked if every integer polynomial is "close" to an irreducible polynomial. More precisely, he asked if there exists an absolute constant C such that for every polynomial f in Z[x] there exists an irreducible polynomial g in Z[x] with deg(g) ≤ deg(f) and L(f-g) ≤ C, where L(·) denotes the sum of the absolute values of the coefficients. This problem remains open. We show that C = 5 suffices for all polynomials with degree at most 40 by using a computational strategy, and discuss how well our results fit the predictions of a heuristic model. This is joint work with Michael Filaseta.

Robert Lemke Oliver: Almost-primes represented by irreducible polynomials

Let G(x) be an irreducible polynomial with integer coefficients. It is conjectured that the set { n &isin N: G(n) is prime } is infinite for most G(x). If Pr denotes the set of squarefree positive integers with at most r prime factors, we consider the set { n &isin N : G(n) &isin Pr } with the goal of showing that it is infinite for a suitable choice of r. Considerable work has been done on this problem, with the most notable results being due to Iwaniec, Buh{v{s}}tab, and Richert. Here we show that if deg(G(x)) = 2, then we may take r = 2. For those G(x) with deg(G(x)) &ge 3, we establish conditions on G(x) which allow us to conclude that there is a suitable choice of
r &le deg(G(x)).

Jeremy Rouse: t-core partitions and Stanton's conjecture

t-core partitions arise naturally in the representation theory of the symmetric group. We will use a variety of tools from the theory of modular forms (the circle method, Deligne's bound, and L-functions) to study asymptotics for the number of t-core partitions of n. As a consequence, we will prove a number of cases of a conjecture of Stanton.

Ruth Stoehr: Benford's Law for coefficients of modular forms and partition functions

Here we prove that Benford's law holds for coefficients of an in finite class of modular forms. Expanding the work of Bringmann and Ono on exact formulas for harmonic Maass forms, we derive the necessary asymptotics. This implies that the unrestricted partition function p(n), as well as other natural partition functions, satisfy Benford's law.

Filip Saidak: Monotonicity of Riemann zeta and other prime-related functions

We prove that for t > 8 and &sigma < 1/2, we have
Re(&phi'(s)/&phi(s)) < Re(&zeta'(s)/&zeta(s)) < Re(&xi'(s)/&xi(s)),
where &phi, &zeta, and &xi are three prime-related complex-valued functions of Euler and Riemann. This is joint work with Y. Matiyasevich and P. Zvengrowski.

Barry Smith: The Brumer-Stark Conjecture for Some Cyclic Extensions

Recent progress on conjectures concerning special values of equivariant L-functions uses Iwasawa theory. I will discuss an alternative approach to the Brumer-Stark conjecture for degree 2p extensions of number fields using a conjectural explicit expression for the equivariant L-function value. This approach avoids some of the drawbacks arising from Iwasawa-theoretic proofs and allows verification of some new cases of the conjecture.

Brett Tangedal: Computing Stark units p-adically via a formula of Gross

We will show how the formulas mentioned in Paul Young's talk may be applied to computationally explore certain p-adic conjectures analogous to Stark's original conjecture. This talk represents joint work with Paul Young.

Dan Yasaki: Computing Stark units using Shintani domains

Let F be a number field, and let E be the ray class field of modulus m. A Stark unit is a special element n in E such that
&zetam(&sigma, 0) = -(1/wE) log|&sigma(&eta)| for all &sigma &isin H. One goal is to "get inside the absolute value'' to give an analytic expression for &eta. Shintani describes a method of computing &zetam in terms of a certain polytopal cone decomposition of a real vector space, and the decomposition into the cones sheds light into this problem. I will report on some computations over imaginary quadratic fields done this summer with B. Tangedal introducing me to these topics.

Paul Young: On p-adic multiple zeta and log gamma functions

We define p-adic multiple zeta and log gamma functions using multiple Volkenborn integrals, and develop some of their properties. Although our functions are close analogues of classical Barnes multiple zeta and log gamma functions and have many properties similar to them, we find that our p-adic analogues also satisfy reflection functional equations which have no analogues to the complex case. We conclude with a Laurent series expansion of the p-adic multiple log gamma function for
(p-adically) large x which agrees exactly with Barnes's asymptotic expansion for the (complex) multiple log gamma function, with the fortunate exception that the error term vanishes. Indeed, it was the possibility of such an expansion which served as the motivation for our functions, since we can use these expansions computationally to p-adically investigate conjectures of Gross, Kashio, and Yoshida over totally real number fields. This talk represents joint work with Brett Tangedal.

 Organizers: Sebastian Pauli Filip Saidak Brett Tangedal Dan Yasaki