# BYU Number Theory Seminar

### Fall 2017

Thursday September 14, 4 PM, 135 TMCB: Ken Ono, Emory University. Title: Can’t you just feel the moonshine? Abstract: Richard Borcherds won the Fields medal in 1998 for his proof of the Monstrous Moonshine Conjecture. Formulated in 1979 by John Conway and Simon Norton, the conjecture asserts that the representation theory of the Monster, the largest sporadic finite simple group, is dictated by a distinguished set of modular functions. This conjecture arose from astonishing coincidences noticed by finite group theorists and arithmetic geometers. Recently, mathematical physicists have revisited moonshine, and they discovered evidence of undiscovered moonshine which some believe will have applications to string theory and 3d quantum gravity. The speaker and his collaborators have been developing the mathematical facets of this theory, and have proved the conjectures which have been formulated. These results include a proof of the Umbral Moonshine Conjecture, and recently the last remaining problem raised by Conway and Norton in their groundbreaking 1979 paper. The most recent Moonshine yields unexpected applications to the arithmetic of elliptic curves thanks to theorems related to the Birch and Swinnerton-Dyer Conjecture and the Main Conjectures of Iwasawa theory for modular forms. This is joint work with John Duncan, Michael Griffin and Michael Mertens.

Friday September 15, 6:30 PM, Nelke Theatre (HFAC): Ken Ono, Emory University. Title: Why does Ramanujan, “The Man Who Knew Infinity”, matter? Abstract. This lecture is about Srinivasa Ramanujan, “The Man Who Knew Infinity”. Ramanujan was a self-trained two-time college dropout who left behind 3 notebooks filled with equations that mathematicians are still trying to figure out today. He claimed that his ideas came to him as visions from an Indian goddess. This lecture is about why Ramanujan matters. The speaker is an Associate Producer of the film "The Man Who Knew Infinity" (starring Dev Patel and Jeremy Irons) about Ramanujan. He will share several clips from the film in the lecture, and will also tell stories about the production and promotion of the film. The talk will be followed by a showing of the film.

Tuesday September 19, 10 AM, 301 TMCB: Robert Lemke Oliver, Tufts University. Title: Ranks of elliptic curves, Selmer groups, and Tate-Shafarevich groups. Abstract: The Selmer group of an elliptic curve is, in a certain not very technical sense, the set of points that "locally look like" they're on the elliptic curve. As such, it serves as an accessible algebraic proxy for the group of rational points, and many of our theorems---from the classical Mordell-Weil theorem, to the fairly recent work of Bhargava and Shankar---pass through the Selmer group. In this talk, we will present recent work of the speaker, joint with Bhargava, Klagsbrun, and Shnidman, on Selmer groups of quadratic twists of elliptic curves possessing a degree three isogeny. As applications of this work, we obtain consequences for ranks of elliptic curves and to the proportion with non-trivial elements in the associated Tate-Shafarevich group. Along the way, we will aim to make Selmer groups somewhat less mystifying.

Tuesday October 10, 10 AM, 301 TMCB: David Masser, University of Basel. Title: The unlikelihood of integrability in elementary terms. Abstract: In 1916 Hardy wrote "...no general method has been devised by which we can always tell, after a finite series of operations, whether any given integral is really elementary, or elliptic, or belongs to a higher order of transcendents." And over 100 years later nothing has changed too much. Every schoolgirl knows that ∫dx/√(x(x-λ)) is elementary; that is, it can be expressed with logarithms and exponentials. But not ∫dx/√(x(x-1)(x-λ)) unless λ= 0, 1. In 1981 (James) Davenport asserted that if f is algebraic then ∫f(x, λ) dx is elementary for at most finitely many special complex values λ (unless it is elementary for a general value of λ). I give a general discussion of the classical problem of "elementary integration" and describe some recent progress by Umberto Zannier and myself on Davenport's Assertion (this progress is conclusive if everything has coefficients which are algebraic numbers).

### Winter 2017

Tuesday January 10, 10 AM, 301 TMCB: Liangyi Zhao, University of New South Wales. Title: Elliptic Curves in Isogeny Classes. Abstract: We show that the distribution of elliptic curves in isogeny classes of curves with a given value of the Frobenius trace t becomes close to uniform even when t is averaged over very short intervals inside the Hasse-Weil interval. This is joint work with I. E. Shparlinski.

Thursday March 16, 10 AM, 301 TMCB: Nathan Green, Texas A&M University. Title: Special Values of L-functions over Drinfeld Modules. Abstract: We work out an explicit theory for the shtuka function of rank 1 sign-normalized Drinfeld modules over the function field of an elliptic curve. Using these explicit formulas, we obtain a product formula for the fundamental period of the exponential function associated to the Drinfeld module. We also find identities for deformations of reciprocal sums and as a result prove special value formulas for L-series over the function field.

Tuesday March 28, 10 AM, 301 TMCB: Kyle Pratt, UIUC. Title: A lower bound for the least prime in an arithmetic progression. Abstract: Fix $k$ a positive integer, and let $\ell$ be coprime to $k$. Let $p(k,\ell)$ denote the smallest prime in the residue class $\ell \pmod{k}$, and set $P(k)$ to be the maximum of $p(k,\ell)$ over residue classes $\ell$. In this talk I will describe joint work with Junxian Li and George Shakan, in which we showed that for almost every $k$ one has $P(k) \gg \phi(k) \log k \log_2 k \log_4 k/\log_3 k$. This improves an earlier bound of Pomerance, and answers a question of Ford, Green, Konyagin, Maynard, and Tao. I will discuss some ideas used in the proof, which has roots in recent work on large gaps between primes. I will also discuss some heuristics about the size of $P(k)$.

Tuesday April 4, 10 AM, 301 TMCB: Simon Myerson, University College London. Title: Diophantine inequalities in many variables. Abstract: Consider the equation f = 0, where f is a polynomial of degree d, with integral coefficients, in n variables. If n is large and f is in a suitable sense nonsingular, then one can apply the circle method to estimate the number of solutions in integers of bounded size. If we allow f to have real coefficients, one might expect a parallel result counting solutions of the inequality |f| < 1. We state a theorem of this type, and outline the new ingredients required for the proof.

### Fall 2016

Tuesday December 6: John Voight, Dartmouth. Title: A heuristic for boundedness of ranks of elliptic curves. Abstract: We describe a heuristic that suggests that ranks of elliptic curves over the rational numbers are bounded. This is joint work with Jennifer Park, Bjorn Poonen, and Melanie Matchett Wood.

### Fall 2015

Tuesday, August 25, 10 AM, 301 TMCB: Sudhir Ghorpade, Indian Institute of Technology-Bombay. Title: Linear sections of Veronese varieties and equations over finite fields. Abstract: Let d, m, r be positive integers and let F be a finite field. Consider the Veronese variety given by the image of the degree d Veronese embedding of projective m-space in a larger projective space. We consider the problem of determining the maximum number of F-rational points of a section of this Veronese variety by a linear subspace of fixed codimension r in the ambient projective space can have. In more concrete terms, this corresponds to determining the maximum number of common zeros in the projective m-space over F for a system of r linearly independent homogeneous polynomials of degree d in m+1 variables with coefficients in F. There is an elaborate conjecture of Tsfasman and Boguslavsky that predicts this maximum number when d is not too large in comparison to the cardinality of F. Special cases of the conjecture (viz., r =1 and 2) are known to be true, thanks to the results of Serre (1991) and Boguslavsky (1997), but the general case has been open for quite some time. We will give a motivated account of the above problem and some of the known results as well as an affine analogue. We will then describe some recent developments that has led to significant new results concerning the general case. This is based mainly on a joint work with Mrinmoy Datta.

Tuesday, September 1, 10 AM, 301 TMCB: Roger Baker, BYU. Title: Bounded intervals containing many primes. Abstract: The seminar has already heard from two visitors, James Maynard and Tristan Freiberg, about applications of Maynard's version of the Goldston-Pintz-Yildirim sieve. The method still has a lot of mileage left. Here we look at the lim inf of the diameter of a set of m primes as the left hand prime tends to infinity. We give a bound O(exp(Cm)) where the value of C is less than the one developed in the two well known Polymath papers on the subject. This is joint work with another visitor earlier this year, Alastair Irving.

Thursday, October 15, 4 PM, 292 TMCB (note the nontraditional time and location): Larry Rolen, Penn State. Title: Indefinite theta functions, higher depth mock modular forms, and quantum modular forms. Abstract: 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 $\sigma$ and $\sigma^*$ of Andrews, Dyson, and Hickerson in a natural framework.

### Winter 2015

Tuesday, April 7, 10 AM, 301 TMCB: Ram Murty, Queen's University. Title: Consecutive squarefull numbers. Abstract: A number n is called squarefull if for every prime p dividing n, we have p^2 also dividing n. Erdos conjectured that the number of pairs of consecutive squarefull numbers (n, n+1) with n < N is at most (log N)^A for some A >0. This conjecture is still open. We will show that the abc conjecture implies this number is at most N^e for any e>0. We will also discuss a related conjecture of Ankeny, Artin and Chowla on fundamental units of certain real quadratic fields and discuss its connection with the Erdos conjecture. This is joint work with Kevser Aktas.

Tuesday, April 14, 10 AM, 301 TMCB: Avner Ash, Boston College. Title: Explicit p-adic deformations of arithmetic homology. Abstract: The homology of congruence subgroups of GL(3,Z) with arbitrary coefficients can be described in very concrete terms. We use this when the coefficients are infinite-dimensional p-adic modules of distributions, where the resulting homology classes can be thought of as p-adic deformations of classical homological automorphic forms. I will describe recent work with David Pollack in which we describe a way to approximate this homology ( on a computer) in the non-ordinary case.

Tuesday, April 28, 10 AM, 301 TMCB: Alastair Irving, U. of Montreal. Title: Cubic polynomials represented by norm forms. Abstract: We discuss how analytic techniques, specifically sieves,can be used to study a problem in arithmetic geometry. For norm forms N from certain number fields and irreducible cubic polynomials f, we will establish the Hasse principle for the variety defined by N(x) = f(t).

Tuesday, May 19, 10 AM, 301 TMCB: Tristan Freiberg, U. of Missouri. Title: Limit points of normalized prime gaps.

### Fall 2014

Thursday, November 20, 10 AM, 301 TMCB: Michael Griffin, Emory University.

Title: On the distribution of Moonshine (and other theorems at the interface of number theory and representation theory).

Abstract: Monstrous Moonshine asserts that the coefficients of the modular j-function are given in terms of ''dimensions'' of virtual character for the Monster, the largest of the sporadic simple groups. There are 194 irreducible representations of the Monster, and it has been a longstanding open problem to determine their distribution in Moonshine. Witten and others have demonstrated deep connections between Monstrous Moonshine and quantum physics. The distributions of the Monster representations in Moonshine can be interpreted as the distributions of black hole states in 3 dimensional quantum gravity. In joint work with Ono and Duncan, we obtain exact formulas for these distributions.

Moonshine type-phenomena have been observed for other finite simple groups besides the Monster. The Umbral Moonshine conjectures of Cheng, Duncan, and Harvey asserts that the Moonshine extends to 24 isomorphism classes of even unimodular positive-definite rank 24 lattices. Monstrous Moonshine can be regarded as the case of the Leech lattice. In 2013, Gannon proved the case for the Mathieu group M24. We offer a method of proof for the remaining 22 cases.

In a related direction, we will also briefly discus recent work on the Rogers-Ramanujan identities which express two infinite product modular forms as number theoretic q-series. These identities give rise to the Rogers-Ramanujan continued fraction, whose values at CM points are algebraic integral units. In recent joint work with Ono, and Warnaar, we obtain a comprehensive framework of identities for infinite product modular forms and characterizes those integral units that arise from this theory.

### Spring/Summer 2014

Tuesday, May 27, 10 AM, 301 TMCB: Bill Banks, University of Missouri. Title: Convolutions with probability distributions, zeros of L-functions, and the least quadratic nonresidue. Abstract: For a given odd prime p, we say that an integer n is a quadratic nonresidue mod p if n is not a square in the finite field with p elements. The best known bound on the size of the least positive quadratic nonresidue was given by Burgess in 1957. Heath-Brown explored the possibility that Burgess' bound was best possible, showing that this assumption implies the existence of so-called Siegel zeros of quadratic L-functions, a situation that is incompatible with the Extended Riemann Hypothesis. In this talk, I will describe some recent work with K. Makarov in which we have used results about convolutions of compact probability distributions to generalize the work of Heath-Brown.

Thursday, June 5, 10 AM, 301 TMCB: James Maynard, University of Montreal. Title: Small gaps between primes, and beyond. Abstract: It is believed that there should be infinitely many pairs of primes which differ by 2; this is the famous twin prime conjecture. More generally, it is believed that for every positive integer m there should be infinitely many sets of m primes, with each set contained in an interval of size roughly m log m. Although proving these conjectures seems to be beyond our current techniques, recent progress has enabled us to obtain some partial results. We will introduce a refinement of the 'GPY sieve method' for studying these problems. This refinement will allow us to show (amongst other things) that $\liminf_n(p_{n+m}-p_n)<\infty$ for any integer m, and so there are infinitely many bounded length intervals containing m primes. We also discuss some generalizations of this result.

Tuesday, June 17, 10 AM, 301 TMCB: Paul Pollack, University of Georgia. Title: Three themes in elementary number theory. Abstract: I will discuss three topics in elementary and analytic number theory. The first part of the talk concerns the infinitude of the primes; this part is something of an advertisement for a recent theorem of Andrew Booker connected with Euclid's classical proof. The second theme is the distribution of amicable pairs--pairs of numbers m and n, each equal to the sum of the other's proper divisors--and certain variants on these which have been studied recently. In the last segment of the talk, we discuss some problems connected with the distribution of values of the Euler function and the sum-of-divisors function.

### Winter 2014: 10:00-10:50 AM Tuesdays, 301 TMCB

January 28: Nickolas Andersen, University of Illinois at Urbana-Champaign. Title: A mock modular form for the partition function. Abstract: It is well-known that the values of the partition function p(n) appear as the coefficients of a modular form. In my talk, I will show how we can construct a mock modular form whose "shadow" encodes the values of the partition function, and whose coefficients are given by inner products of the partition generating function and other modular forms of weight -1/2. On the way, we will encounter Rademacher's exact formula for p(n). I will also show how this mock modular form is one member of an infinite basis for the space of mock modular forms of weight 5/2 on the full modular group. This is joint work with Scott Ahlgren.

March 4: Roger Baker, BYU. Title: Small gaps between the primes in a Beatty sequence. Abstract: We report on joint work with Lee Zhao showing that for every irrational a, and every m =1,2,... the Beatty sequence [an] (n = 1,2,...) contains infinitely many sets of m primes, each set being contained in an interval of length C. Here C depends in a messy but explicit way on a and m. Such a result would have been completely out of reach before the appearance of a paper by James Maynard in the arxiv last November. Maynard finds sets of m primes in the set of all positive integers and gets a corresponding value of C that is, roughly speaking, of order exp(4m).

March 11: Andreas Weingartner, Southern Utah University. Title: On the ratio of consecutive divisors. Abstract: We present a new asymptotic result for the number of integers less than x whose maximum ratio of consecutive divisors is at most t.

March 18: David Cardon, BYU. Title: Complex Zero Strip Decreasing Operators. Abstract: We study a class of linear operators that act on certain real entire functions whose zeros belong to the strip -r<Im(z)<r and map them to functions having zeros in a smaller strip. This is motivated by interest in the Riemann Xi-function which is a real entire function with zeros in the strip -1/2<Im(z)<1/2, and whose zeros conjecturally satisfy Im(z)=0.

### Fall 2013: 10:00-10:50 AM Tuesdays, 301 TMCB

Friday September 6, 4:00 PM (note nonstandard time): Riad Masri, Texas A&M University. Title: Harmonic weak Maass forms and partition ranks. Abstract: The rank of an integer partition is the largest part minus the number of parts. The notion of rank was introduced by Freeman Dyson to give a combinatorial explanation of Ramanujan's famous partition congruences. Recent developments in the theory of harmonic weak Maass forms have led to new exact formulas for partition ranks. I will explain how these formulas can be combined with analytic methods to give asymptotics for partition ranks with power saving error terms.

September 17: Roger Baker, BYU. Title: Character sums over Piatetski-Shapiro sequences. Abstract: I will describe a few problems associated with the 'somewhat random' behavior of the Piatetski-Shapiro sequences [n^c] (this is the floor function, of course). Then I will focus on one problem in particular. Can we give a nontrivial estimate when a Dirichlet character to modulus q is summed over values taken at [n^c], n ranging over [1,x]? Here c is a constant , c>1, c not an integer. At present an affirmative answer is limited to certain ranges of (log q)/log x. For c<4/3, the existence of a divisor of q in a convenient range can enable us to do better.

October 1: Lee Zhao, BYU and Nanyang Technological University. Title: Hecke Eigenvalues at Piatetski-Shapiro Primes. Abstract: Continuing the discussion on the Piatetski-Shapiro primes, I'll discuss of average of Hecke eigenvalues at these primes. Let \lambda(n) be the normalized n-th Fourier coefficient of a holomorphic cusp form for the full modular group. We show that for some constant C > 0 depending on the cusp form and every fixed c with 1 < c < 8/7, the mean value of \lambda(p) is << exp(-C ˆ{logN} ) as p runs over all (Piatetski-Shapiro) primes of the form [n^c] with for some natural number n ¾ N. This is joint work with S. Baier.

October 15: Avner Ash, Boston College. Title: Reducible Galois representations and homology of GL(n,Z). Abstract: In joint work with Darrin Doud, we have been connecting reducible 3-dimensional representations of the absolute Galois group of Q with homology Hecke eigenclasses for congruence subgroups of GL(3,Z). I will explain what we have done and how it fits into the larger picture of generalized reciprocity.

October 29: Sam Dittmer, BYU. Title: The real-analytic Eisenstein series and the existence of nonvanishing class group L-functions. Abstract: The real-analytic Eisenstein series encodes important number theoretic data. After giving some background results on Gauss's class number problem and class group L-functions, we will examine the Eisenstein series in more depth to demonstrate the existence of at least one class group L-functions that is nonvanishing at s=1/2 for all imaginary quadratic fields.

November 12: Jaap Top (U of Groningen, The Netherlands). Title: The Hasse inequality. Abstract: Several introductory texts on elliptic curves or on number theory use an elementary argument by Yu. I. Manin (1956) to prove the celebrated Hasse inequality for the number of points on an elliptic curve over a finite field. For example, this applies to books or lecture notes by Chahal, by Gelfond & Linnik, by Husemoller, by Knapp and by Lemmermeyer. The somewhat ad-hoc flavour of the argument plus some doubts expressed in the review of Manin's paper by J. W. S. Cassels concerning some details of the proof, possibly provide a reason why some other textbooks do not mention the argument. Motivated by the PhD thesis of Soomro and the master's thesis of Soeten (both June 2013) in Groningen, I will discuss some aspects and generalizations of Manin's proof.

November 26: No seminar.

December 10: Tonghai Yang, University of Wisconsin. Title: Factorization of some big numbers and Borcherds product. Abstract: It was observed 90 years ago that the famous modular  j-function has the following cool property: j(\frac{1+\sqrt{-163}2) =-e^{\sqrt{163} \pi} +744 + tiny = -2^{18}3^3 5^3 23^3 29^3 is a huge integer with very small prime factor, and so is  j(\frac{1+\sqrt{-163}2)-1728. In early 80s, Gross and Zagier proved this is a general  phenomenon---the norm of j(\frac{d_1+\sqrt d_1}}2) -j(\frac{d_2+\sqrt  d_2}2) has an explicit factorization formula with prime factors less  than or equal to 4d_1d_2. It turns out that this phenomenon is much more general and works for a whole family of modular functions called Borcherds product. In this talk, we will give an informal account of this story.

## Student Computational Number Theory Seminar

### Winter 2015: 10:00-10:50 AM Thursdays, 301 TMCB

All interested students are welcome to attend.

January 15: Darrin Doud, Modern factorization methods

January 22: Darrin Doud, Modern factorization methods

January 29: Pace Nielsen, Odd perfect numbers

February 5: Pace Nielsen, Odd perfect numbers

February 12: Grant Molnar, Hensel's lemma

February 19: Paul Jenkins, Zeros of weakly holomorphic modular forms of half integral weight

February 26: Kevin Childers

March 5:

March 12: SRC practice talks: DJ Thornton

March 19: SRC practice talks: Grant Molnar

March 26:

April 2:

April 9:

### Fall 2014: 10:00-10:50 AM Thursdays, 301 TMCB

All interested students are welcome to attend.

September 4: Organizational Meeting

September 11: Pace Nielsen, Small gaps between primes

September 18: David Cardon, Computing units in quadratic fields

September 25: Brett Clarke, Silverman/Tate chapter 1

October 2: Ben Herrera, Silverman/Tate chapter 2

October 9: Amanda Francis, Silverman/Tate chapter 3

October 16: DJ Thornton, Silverman/Tate chapter 3 examples

October 23: Brett Clarke, Silverman/Tate chapter 4

October 30: Tyler Owens, Silverman/Tate chapter 4

November 6: No seminar

November 13: Ben Herrera, Silverman/Tate chapter 5

November 20: Michael Griffin, Emory University (see above)

December 4: DJ Thornton, Silverman/Tate chapter 6

December 11: Tyler Owens

### Winter 2014: 10:00-10:50 AM Thursdays, 301 TMCB

All interested students are welcome to attend.

January 9: Organizational Meeting

January 16: Pace Nielsen, Bounded gaps between primes

January 23: DJ Thornton, Ireland/Rosen ch. 17

January 30: Kevin Childers, Ireland/Rosen ch. 17

February 6: Kyle Pratt, Open problems in number theory

February 13: Joey Adams, Ireland/Rosen ch. 18

February 20: SRC practice talks: DJ, Tyler

February 27: SRC practice talks: Chris, Ben

March 6: SRC practice talks: Andrew, Grant

March 13: SRC practice talks: Joey, Kevin

Saturday March 15: CPMS Student Research Conference

March 20: David Wagner, 290 theorem on quadratic forms

March 27: Grant Molnar, Newton polynomials

April 3: Tyler Owens, Covering systems

April 10: Wil Cocke, TBA

### Fall 2013: 10:00-10:50 AM Thursdays, 301 TMCB

September 5: Organizational Meeting

September 12: Kyle Pratt, Ireland/Rosen Chapter 6

September 19: Kyle Pratt, Ireland/Rosen Chapter 6

September 26: Kevin Childers, Serre's Conjecture

October 3: Chris Hettinger, Public Key Encryption

October 10: Andrew Haddock, Ireland/Rosen Chapter 8

October 17: Andrew Haddock, Ireland/Rosen Chapter 8

October 24: DJ Thornton, p-adic numbers

October 31: Pace Nielsen, Recent results and open problems in odd perfect numbers

November 7: Tyler Owens, Ireland/Rosen Chapter 9

November 14: Tyler Owens, Ireland/Rosen Chapter 9

November 21: Darrin Doud, Reciprocity Laws

December 5: Seminar canceled

December 12: Joey Adams, Ash/Doud/Pollack Conjecture