Illinois Number Theory Seminar : Fall 2015

Patrick Allen (UIUC)

Finiteness of unramified deformation rings

Class field theory allows one to precisely understand ramification in abelian extensions of number fields. A consequence is that infinite pro-p abelian extensions of a number field are infinitely ramified above p. Boston conjectured a nonabelian analogue of this fact, predicting that certain pro-p representations that are unramified at p act via a finite quotient, and this conjecture strengthens the unramified version of the Fontaine-Mazur conjecture. We show in many cases that one can deduce Boston's conjecture from the unramified Fontaine-Mazur conjecture, which allows us to deduce (unconditionally) Boston's conjecture in many two-dimensional cases. This is joint work with F. Calegari.

Micah Milinovich (Univ. Mississippi)

 Fourier Analysis and the zeros of the Riemann zeta-function

I will show how the classical Beurling-Selberg extremal problem in harmonic analysis arises naturally when studying the vertical distribution of the zeros of the Riemann zeta-function and other L-functions. Using this relationship, along with techniques from Fourier analysis and reproducing kernel Hilbert spaces, we can prove the sharpest known bounds for the number of zeros in an interval on the critical line and we can also study the pair correlation of zeros. Our results on pair correlation extend earlier work of P. X. Gallagher and give some evidence for the well-known conjecture of H. L. Montgomery. This talk is based on a series of joint works with E. Carneiro, V. Chandee, and F. Littmann.

Dirk Zeindler (Lancaster Univ.)

 The order of large random permutations with cycle weights

The order of a permutation of objects is the smallest integer such that the -th iterate of gives the identity. A remarkable result about the order of a uniformly chosen permutation is due to Erdős–Turán who proved in 1965 that  satisfies a central limit theorem. We show that the Erdős–Turán Law can be extended to random permutations chosen according to the so-called generalized Ewens measure and to a generalized weighted measure with polynomially growing cycle weights. Furthermore, we establish for the generalized Ewens measure a local limit theorem as well as, under some extra moment condition, a precise large deviation estimate and also show that the expectation of the logarithm of the order has a remarkable connection with the Riemann hypothesis. In addition, we provide a precise large deviation estimate for random permutations with polynomial growing cycle weights.

Rachel Davis (Purdue)

 Origami Galois representations

Let be an elliptic curve over . Let be an étale cover, ramified only above one point. The pair is called an origami. The name comes from a picture that I will draw during the talk. We study the relationship between the pre-images of rational point on under the map and Galois representations. This is joint work with Edray Herber Goins.

Adam Harper (Cambridge)

Exponential sums over smooth numbers

A number is said to be -smooth if all of its prime factors are at most . Exponential sums over the -smooth numbers less than have been widely investigated, but existing results were weak or too small compared with . For example, if is a power of then existing results were insufficient to study ternary additive problems involving smooth numbers, except by assuming conjectures like the Generalised Riemann Hypothesis.

I will try to describe my work on bounding mean values of exponential sums over smooth numbers, which allows an unconditional treatment of ternary additive problems even with a (large) power of . There are connections with restriction theory and additive combinatorics.

József Balogh (UIUC)

On some problems of Cameron and Erdős

Cameron and Erdős proposed several of enumeration problems in additive combinatorics, for example what is the number of sum-free sets in [n], or what is the number of sets in [n] which do not contain k-term arithmetic progression. I plan to survey the recent progress on these type of questions, focusing on the applications of a recent powerful tool, the hyper graph container lemma of Balogh-Morris-Samotij, and of Saxton-Thomason. It is joint work with H. Liu, M. Sharifzadeh, and A. Treglown.

Nicholas Robles (UIUC)

Mollifications of the Riemann zeta-function and families of -functions

We explain how by twisting the a mean value integral of the Riemann zeta-function by a suitable Dirichlet polynomial we can generate interesting results about the proportion of non-trivial zeros on the critical line. By using sieve results of Conrey, Iwaniec and Soundararajan, this mollification process can yield better results (over 50%) for certain averages of -functions up to degree 3. This is joint work with Dirk Zeindler, Arindam Roy and Alexandru Zaharescu.

Nickolas Andersen (UIUC)

Kloosterman sums and Maass cusp forms of half integral weight for the modular group 

We estimate the sums , where are Kloosterman sums of half-integral weight on the modular group associated to the multiplier system for the Dedekind eta function. Our estimates are uniform in and in analogy with Sarnak and Tsimerman's improvement of Kuznetsov's bound for the ordinary Kloosterman sums. We approach the problem via an analogue of Kuznetsov's trace formula; among other things this requires us to develop mean value estimates for coefficients of Maass cusp forms of weight and  uniform estimates for -Bessel integral transforms. As an application, we obtain an improved estimate for the classical problem of estimating the size of the error term in Rademacher's formula for the partition function .

Roman Holowinsky (Ohio State)

New variants of the delta-method and applications

We will present simple "conductor lowering" techniques in the classical delta-method of Duke-Friedlander-Iwaniec which have proven useful in various shifted convolution sum and subconvexity problems.  We also describe new variants arising from Petersson-Kuznetsov trace formulae and shall discuss their benefits in application to analytic problems on higher rank groups. Some of the recent works of Ritabrata Munshi, Zhi Qi, and current graduate students will be highlighted.

Fan Zhou (Ohio State)

The Voronoi formula and double Dirichlet series

We present a proof of Voronoi formula for coefficients of a large class of L-functions, in the style of the classical converse theorem of Weil. Our formula applies to full-level cusp forms, Rankin-Selberg convolutions, and certain isobaric sums. Our proof is based on the functional equations of L-functions twisted by Dirichlet characters and does not directly depend on automorphy. Hence it has wider application than any previous proofs. The key ingredient is the construction of a double Dirichlet series associated with these coefficients and the structure of nonprimitive Gauss/Ramanujan sums. This is joint work with Eren Mehmet Kıral.

Larry Rolen (Penn State)

Indefinite theta functions, higher depth mock modular forms, and quantum modular forms 

In this talk, I will describe several new results concerning the modularity of indefinite theta functions. From Zwegers' thesis, we know that special types of indefinite theta functions with prescribed signatures give rise to mock modular forms, which combined with important work of Andrews and others gives one road to understanding the mock theta functions of Ramanujan. Here, we will study several important examples of more general indefinite theta series inspired by physics and geometry and describe how to study the modularity properties of more complicated objects such as these, giving a glimpse into the general structure of indefinite theta functions. We will also study another class of indefinite theta functions, and we will discuss a new family of examples which give rise to quantum modular forms, and provide a family of canonical Maass waveforms whose Fourier coefficients are described by combinatorial functions with integer coefficients, placing the famous functions and of Andrews, Dyson, and Hickerson in a natural framework.

Bruce Reznick (UIUC)

 Quotients of sums of distinct powers of three

Let $A$ denote the set of Newman polynomials, , . Which integers may be written as , where ? There are two complementary  approaches: the algebraic and the combinatorial. In the first, one can show that 4 arises only when and if 22 arises, then cannot divide . In the second, all possible 's are encoded by certain labeled closed walks in an uncomplicated digraph; in this way it can be shown that 529 and 592 can never occur. No particular prerequisites for this seminar.

Kagan Kursungoz (Sabancı)

 Andrews and Bressoud Style Identities for Partitions and Overpartitions

We propose a method to construct a variety of partition identities at once. The main applications are an all-moduli generalization of some of Andrews' results in [Andrews, Parity in partition identities. Ramanujan  Journal 23:45-90 (2010)] and Bressoud's even moduli generalization of Rogers-Ramanujan-Gordon identities, and their counterparts for overpartitions due to Lovejoy et al. and Chen et al. We obtain unusual companion identities to known theorems as well as to the new ones in the process. The novelty is that the method constructs solutions to functional equations which are satisfied by the generating functions. In contrast, the conventional approach is to show that a variant of well-known series satisfies the system of functional equations, thus reconciling two separate lines of computations.

Bob Hough (IAS)

Recent progress on extreme central values of -functions

Soundararajan's resonance method is a first moment method for demonstrating the existence of large central values of -functions within natural families. In recent work I have supplemented the resonance method to study the angle distribution of the large values found. Also, recent work of Bondarenko and Seip has introduced a new combinatorial argument which strengthens the estimate obtained for extreme values of the Riemann zeta function on the half line.  I will describe some of the new ideas in these arguments.

Brandon Levin (U Chicago)

The weight part of Serre's conjecture

Serre's modularity conjecture (now a Theorem due to Khare-Wintenberger and Kisin) states that every odd irreducible two dimensional mod p representation of the absolute Galois group of Q comes from a modular form. I will begin with an overview of the Serre's original conjecture on modular forms focusing on the weight part of the conjecture. Herzig gave a generalization of the conjecture for n-dimensional Galois representations which predicts the modularity of so-called shadow weights.  After briefly describing Herzig's conjecture, I will discuss joint work with D. Le, B. Le Hung, and S. Morra where we prove instances of this conjecture in dimension three.

Yifeng Liu (Northwestern)

Bad reduction of Hilbert modular varieties and arithmetic application

In this talk, we will study the reduction of Shimura varieties attached to certain quaternion algebras at some ramified prime. We explain how the global structure of the bad reduction is related to the level raising phenomenon for modular forms. As an application, we will use this to bound the Selmer groups of certain motives of high rank.