# Research: Number Theory

## SERMON 2009

SERMON is a small, friendly, and informal gathering of number theorists and combinatorialists. Faculty members, graduate students, and undergraduate students are all invited to attend, and to give talks if they wish.

The first SERMON took place at UNC Greensboro in 1988. Since then it has taken place every year in at various universities in the southeast: University of Georgia, University of South Carolina, Citadel, College of Charleston, Wake Forest, Virginia Tech, Furman, and Clemson University.

See the home page for the SouthEast Regional Meeting On Numbersfor more general information on SERMON and to subscribe to the SERMON e-mail list.

### SERMON 2009

In 2009 SERMON was held at the University of North Carolina at Greensboro Saturday April 18 and Sunday April 19, 2009. There was a party on Saturday night.

The plenary speakers at SERMON 2009 were Paul Gunnells from the University of Massachusetts, Amherst and David Ford from Concordia University, Montreal.

### Program

All talks will take place in Room 150 in the Petty Building (**29** on the campus map). The room is equipped with whiteboards, a computer, a beamer, and a document camera, as well as computer hookup.

**Friday, April 17, 2009**

17:00-20:00 | Gathering in the Old Town Draught House at 1205 Spring Garden Street (between buildings 80 and 252 on the campus map) |

**Saturday, April 18, 2009**

16:00Coffee

10:00 | Paul Gunnells: Multiple Dirichlet Series |

11:00 | Coffee |

11:30 | Jim Brown: Level lowering for half-integral weight modular forms |

12:00 | Ethan Smith: The Lang-Trotter Conjecture “on average” |

12:30 | John Webb: Partition Values and Modular L-Functions |

13:00 | Lunch |

14:30 | Max Alekseyev: On the number of two-dimensional threshold functions |

15:00 | Andrew Sills: On the Rogers-Selberg Identities and Gordon’s Theorem |

15:30 | Carlos Nicolas: k-Triangulations and k-Splitters |

16:30 | Cliff Smyth: Paths in Line Arrangements |

17:00 | Gary Walsh: A new approach to an old Diophantine problem |

18:30 | Party |

**Sunday, April 19, 2009**

9:00 | David Ford: Complexity of Ideal Factorization in an Algebraic Number Field |

10:00 | Coffee |

10:30 | Qingquan Wu: The different exponent of Artin-Schreier extension towers |

11:00 | James Carter: On the restricted Hilbert-Speiser and Leopoldt properties |

11:30 | Coffee |

12:00 | Michael Mossinghoff: The distance to an irreducible polynomial |

12:30 | Joshua Cooper: The Discrepancy of the Lexicographically Least de Bruijn Cycle |

13:00 | Igor Erovenko: Bounded generation and second bounded cohomology of wreath products |

### Abstracts of Talks

#### Max Alekseyev: On the number of two-dimensional threshold functions

A two-dimensional threshold function of $$k$$-valued logic can be viewed as coloring of the points of a $$k\times k$$ square lattice into two colors such that there exists a straight line separating points of different colors. For the number of such functions only asymptotic bounds are known. We give an exact formula for the number of two-dimensional threshold functions and derive more accurate asymptotics.

#### Jim Brown: Level lowering for half-integral weight modular forms

Let g be a half-integral weight eigenform of level Mp^{n}, gcd(p,M) = 1. Under a mild hypothesis we show there exists a half-integral weight eigenform of level M that has eigenvalues congruent to those of g modulo p. The main tools involved are the Shimura correspondence and Khare-Wintenberger’s recent proof of Serre’s conjecture. This is joint work with Yingkun Li and arose out of his summer undergraduate research project during the summer of 2008.

#### Joshua Cooper: The Discrepancy of the Lexicographically Least de Bruijn Cycle

A (binary) de Bruijn cycle of order $$n$$ is a cyclic binary word of length $$2^n$$ so that every binary word of length $$n$$ appears as a subword exactly once. There are many constructions for de Bruijn cycles, but one stands out for its extraordinary properties. The lexicographically-least de Bruijn cycle, sometimes called the Ford sequence, is also the de Bruijn cycle generated by the “greedy” algorithm, the cycle generated by a linear-shift feedback register of minimum weight, and the string of all Lyndon words of length dividing n arranged in lexicographic order. It is not hard to see that the Ford sequence is very “unbalanced” in the sense that there is an excess of 0’s near the beginning and an excess of 1’s near the end. It is possible to quantify this intuition by defining the discrepancy of a sequence to be the maximum, over all initial segments, of the difference between the number of 0’s and 1’s. We show that the discrepancy has order $$2^n \log n/n$$, answering a question of Ron Graham.

#### James Carter: On the restricted Hilbert-Speiser and Leopoldt properties

Let $$G$$ be a finite abelian group. A number field $$K$$ is called a Hilbert-Speiser field of type $$G$$ if for every tame $$G$$-Galois extension $$L/K$$, the ring of integers $$\mathcal{O}_L$$ is free as an $$\mathcal{O}_K[G]$$-module. If $$\mathcal{O}_L$$ is free over the associated order $$\mathcal{A}_{L/K}$$ for every $$G$$-Galois extension $$L/K$$, then $$K$$ is called a Leopoldt field of type $$G$$. It is well-known (and easy to see) that if $$K$$ is Leopoldt of type $$G$$, then $$K$$ is Hilbert-Speiser of type $$G$$. The converse does not hold in general. However, we show that a modified version does hold for many number fields $$K$$ (in particular, for $$K/\mathbb{Q}$$ Galois) when $$G=C_{p}$$ has prime order. Finally, we show that even the modified converse is false in general, and we give examples which show that the modified converse can hold while the original does not.

#### Igor Erovenko: Bounded generation and second bounded cohomology of wreath products

We show that the (standard restricted) wreath product of groups is boundedly generated if and only if the bottom group is boundedly generated and the top group is finite. We also establish a criterion for triviality of the singular part of second bounded cohomology of wreath products. (Joint work with Nikolay Nikolov and B. Sury.)

#### David Ford: Complexity of Ideal Factorization in an Algebraic Number Field

Given

- $$F(x)$$, an irreducible monic polynomial in $$\Zx$$,
- $$p$$, a rational prime,
- $$K$$, the extension of $$\Q$$ generated by a root of $$F$$,
- $$\cO$$, the ring of integers of $$K$$,

the Montes algorithm (1999) computes $$\ldss{e_1}{e_r}$$ and $$\ldss{f_1}{f_r}$$ such that \[ p\msp\cO = \mfp_{1}^{e_1} \cdots \mfp_{r}^{e_r} \] is the complete factorization of $$p\msp\cO$$ as a product of prime ideals in $$\cO$$, with $$\ldss{f_1}{f_r}$$ the respective degrees of $$\ldss{\mfp_1}{\mfp_r}$$. \newpar Examples of very high degree are known for which the Montes algorithm terminates quickly, but a complexity estimate for the performance of the algorithm in the general case has not been made. \newpar Variants of the Zassenhaus “Round Four” algorithm can provide this factorization, via the construction of an integral basis for $$K$$. Pauli (2001) has given a complexity estimate for the “two-element” variation. \newpar The Montes algorithm avoids the construction of an integral basis, so there is reason to suspect that its complexity is lower than that of Round Four. \newpar We analyze a simplified variant of the Montes algorithm, which determines whether $$F(x)$$ is reducible in $$\Zpx$$. We assume (without proof) that this is the worst case for the Montes algorithm, since in the irreducible case the algorithm would construct the most “levels”. \newpar We show that the bit-complexity of the modified Montes algorithm is \[ O\bigp{\,(\deg F)^{\ep{3}}\,v_p(\disc F)^{\ep{2}}\,}. \]

#### Paul Gunnells: Multiple Dirichlet Series

Multiple Dirichlet series are generalizations of L-functions that involve several complex variables. While the functional equation of a usual L-series is an involution s -> 1-s, a multiple Dirichlet series satisfies a group of functional equations that intermixes all the variables. Siegel constructed the first example of such a series in 1956 by taking the Mellin transform of a half-integral weight Eisenstein series. In recent years multiple Dirichlet series have been intensively studied in a variety of contexts and with many applications in mind. In this talk we will present an overview of this subject. We show how these series provide natural tools to address questions in analytic number theory. We then describe a construction of a class of multiple Dirichlet series attached to Dynkin diagrams, where the resulting group of functional equations is the associated Weyl group. These series are expected to be Fourier-Whittaker coefficients of metaplectic Eisenstein series.

#### Michael Mossinghoff: The distance to an irreducible polynomial

Given an integer polynomial f(x), how many coefficients do you need to adjust before you are assured of finding an irreducible polynomial of the same degree or smaller? More precisely, does there exist an absolute constant C so that for every f(x) in Z[x] there exists an irreducible g(x) in Z[x] with deg(g) ≤ deg(f) and L(f-g) ≤ C, where L(h) denotes the sum of the absolute values of the coefficients of h(x)? This question was first posed by TurÃ¡n in 1962, and it remains unsolved. We discuss some algorithms designed to investigate this question, and report on the results of some recent computations.

#### Carlos Nicolas: k-Triangulations and k-Splitters

So far k-triangulations have been defined only for points in convex position in the plane (as maximal sets of diagonals of the n-gon without any k+1 of them mutually crossing). I will introduce the concept of a k-triangulation for sets of points in general position in the d-dimensional Euclidean space. Roughly speaking, a k-triangulation is a way to break down the (k-1)-splitters of the entire set into a sum of splitters for blocks of a certain size. The definition agrees with the usual triangulations of points when k=1 and also with the k-triangulations of the n-gon. I will discuss the problem of constructing k-triangulations in two dimensions using a continuous motion argument.

#### Andrew Sills: On the Rogers-Selberg Identities and Gordon’s Theorem

The Rogers-Ramanujan identities are among the most famous in the theory of integer partitions. For many years, it was thought that they could not be generalized, so it came as a big surprise when Basil Gordon found an infinite family of partition identities that generalized Rogers-Ramanujan in 1961. Since the publication of Gordon’s result, it has been suspected that a certain special case of his identity should provide a combinatorial interpretation for a set of three analytic identities known as the Rogers-Selberg identities. I will discuss a bijection between two relevant classes of integer partitions that explains the connection between Gordon and Rogers-Selberg. This work appeared in JCTA 115 (2008) 67-83.

#### Ethan Smith: The Lang-Trotter Conjecture “on average”

Abstract: The Lang-Trotter Conjecture concerns a prime distribution problem related to elliptic curves. In this talk, I will describe the original conjecture of Lang and Trotter as well as a version for elliptic curves defined over number fields. I will also give a survey of several results (by various authors) which may be interpreted as saying that the conjecture holds “on average.”

#### Cliff Smyth: Paths in Line Arrangements

Consider an arrangement $$A = \{l_1, \ldots , l_n\} of $$n$$ lines in the plane. A path in $$A$$ of length $$N$$ is an alternating sequence $$P = (L_0, P_1, L_1, P_2, \ldots , P_N, L_N)$$ of lines $$L_i$$ from $$A$$ and points $$P_i$$, where each $$P_i$$ is the intersection the lines $$L_{i-1}$$ and $$L_i$$. A path is monotone if the $$x$$ coordinates of the $$P_i$$ are increasing with $$i$$. It is conjectured that the maximum possible length $$N$$ of a monotone path in any line arrangment of $$n$$ lines is $$N = o(n^2)$$. For any $$\epsilon > 0$$, we show the maximum length is $$N = \Omega(n^{2-\epsilon}).$$ This is joint work with Joszef Balogh, Oded Regev, William Steiger and Mario Szegedy.

#### Gary Walsh: A new approach to an old Diophantine problem

Our goal is to obtain an upper bound for the number of integer solutions to the equation $$x^2-dy^4=k$$, in which the upper bound depends as lightly as possible on $$d$$ and $$k$$. A result in this direction was proved recently by the speaker, where the dependence on $$k$$ was given in terms of the number of its prime factors. We refine these results for a particular case in which $$k$$ is a power of $$2$$. This is joint work with Pingzhi Yuan.

#### John Webb: Partition Values and Modular L-Functions

Using Waldspurger’s theorem, interesting congruences can be built relating values of the partition function to central critical values of particular modular L-functions. This expands upon the work of Guo and Ono from the late 1990’s.

#### Qingquan Wu: The different exponent of Artin-Schreier extension towers

Let $$K$$ be a function field over a perfect constant field of positive characteristic $$p$$, and $$L$$ the compositum field of some (degree $$p$$) Artin-Schreier extensions of $$K$$. We describe how places of $$K$$ split in this elementary abelian $$p$$-extension $$L/K$$. One consequence is that the converse of Abhyankar’s Lemma is also true here. It turns out that a place of $$K$$ totally ramifies in $$L/K$$ (is inert, splits completely, resp.) if and only if it totally ramifies (is inert, splits completely, resp.) in all of the intermediate degree $$p$$ Artin-Schreier extensions over $$K$$. Similar scenarios are rare in number field extensions. But in the function field setting, examples are given to show that all possible decompositions are indeed possible. These examples are independent of the function field structure of $$K$$ and its characteristic $$p$$. Our main result is to compute the different exponent of such extension towers. This is joint work with Renate Scheidler.

### Participants

- Max Alekseyev, University of South Carolina
- Ken Berenhaut, Wake Forest University
- Adam Boseman, UNCG
- Matt Boylan, University of South Carolina
- Jim Brown, Clemson University
- Adam Boseman, UNCG
- James Carter, College of Charleston
- Joshua Cooper, University of South Carolina
- Paul Duvall, UNCG
- Igor Erevenko, UNCG
- David Ford, Concordia University, Montreal
- Margaret Francel, Wake Forest University
- Yair Goldberg, UNCG
- Samuel Gross, University of South Carolina
- Paul Gunnells, University of Massachusetts, Amherst
- Norman Hill, UNCG
- Frederic Howard, Wake Forest University
- Danielle Moran, UNC Chapel Hill
- Michael Mossinghoff, Davidson College
- Carlos Nicolas, UNCG
- Sebastian Pauli, UNCG
- Michael Rogers, Oxford College of Emory University
- Filip Saidak, UNCG
- Andrew Sills, Georgia Southern
- Brian Sinclair, UNCG
- Ethan Smith, Clemson University
- Clifford Smyth, UNCG
- Brett Tangedal, UNCG
- Catherine Trentacoste, Clemson University
- Theresa Vaughan, UNCG
- Andrew Vincent, University of South Carolina
- Gary Walsh, University of Ottawa
- John Webb, University of South Carolina
- Qingquan Wu, University of Calgary
- Dan Yasaki, UNCG
- Paul Young, College of Charleston

### Acknowledgements

SERMON 2009 supported by the National Science Foundation under Grant No. 0921700 and by the Number Theory Foundation.

### Organizers

Founder: Theresa Vaughan; Sebastian Pauli Filip Saidak Brett Tangedal Dan Yasaki