# 2013 Maine/Québec Number Theory Conference

In honor of the conference founders Claude Levesque and Chip Snyder
on the occasion of their retirements

University of Maine, Orono
October 5-6, 2013
Andrew Knightly and Benjamin Weiss, Organizers

Financial Support
The National Science Foundation
The University of Maine Office of the Vice President for Research
The University of Maine College of Liberal Arts and Sciences
University of Maine Department of Mathematics and Statistics

Full List of Participants

## Saturday, October 5, 2013

Click a name to view the abstract for the talk
 Time Speaker Title Room 8:00-8:20am Coffee/Tea, Neville Hall (NV) Lobby 8:20-8:30 am Welcoming Remarks Nigel Pitt, Chair of Mathematics and Statistics Jeffrey Hecker, Executive VP for Academic Affairs and Provost 101 NV 8:30-9:00 Radan Kučera Masaryk University, CZ On the class group of a cyclic field of odd prime power degree 101 NV 9:10-9:30 Michael Mossinghoff Davidson College Wieferich pairs and Barker sequences.   slides   paper 100 NV Siman Wong University of Massachusetts Arithmetic of octahedral sextics 101 NV 9:40-10:00 Hester Graves IDA The abc Conjecture and non-Wieferich primes in Arithmetic Progressions 100 NV Andrew Schultz Wellesley College Realizing certain $p$-groups as Galois groups  slides 101 NV 10:10-10:25 Nathan McNew Dartmouth College Sets of integers which contain no three terms in geometric progression. slides   paper 100 NV Nico Aiello University of Massachusetts Galois theory of iterated morphisms on reducible elliptic curves 101 NV 10:30-10:50 Dubi Kelmer Boston College On the Pair Correlation of Hyperbolic Angles 100 NV Dawn Nelson Bates College A Variation on Leopoldt's Conjecture: Part 2. 101 NV 11:00-11:20 David Geraghty Boston College A candidate p-adic local Langlands correspondence 100 NV Omar Kihel Brock University Denominators of algebraic numbers in a number field 101 NV 11:30-11:50 Steven J. Miller Williams College Mind the Gap: Distribution of Gaps in Generalized Zeckendorf Decompositions. slides 100 NV Alain Togbé Purdue University North Central On Diophantine equations involving normalized binomial mid-coefficients 101 NV 12:00-12:30 Abdelmalek Azizi Mohammed First University, Morocco On the capitulation problem of the $2$-ideal classes of the field $\mathbb Q(\sqrt{p_1p_2}, i)$ where $p_1\equiv p_2\equiv 1 \pmod4$ are primes. 100 NV Lunch, Memorial Union (on campus) 2:00-2:30 Eyal Goren McGill University Arithmetic intersection theory on Spin Shimura varieties 101 NV 2:40-2:55 Patrick Letendre Laval University Some results on $k$-free numbers.   slides 100 NV Joshua Zelinsky Boston University Recent result in counting Artin representations   paper 101 NV 3:00-3:15 Zeb Engberg Dartmouth College On the reciprocal sum of primes dividing Mersenne numbers. 100 NV James Ricci Wesleyan University Finiteness results for regular ternary quadratic polynomials 101 NV 3:20-3:45 Cake and tea 3:45-4:05 Moshe Adrian University of Utah Jacquet's conjecture on the local converse problem for epipelagic supercuspidal representations of GL(n,F) 100 NV Adriana Salerno Bates College Effective computations in arithmetic mirror symmetry. 101 NV 4:15-4:30 Kyle Pratt (BYU) Minh-Tam Trinh (Princeton) Zeros of Dirichlet $L$-Functions over Function Fields 100 NV Robert Grizzard University of Texas Relative Bogomolov extensions.   slides   paper 101 NV 4:35-4:55 Lei Zhang Boston College Eisenstein Series of Covers of Odd Orthogonal Groups. 100 NV Hugo Chapdelaine Laval University Cyclotomic units revisited 101 NV 5:05-5:35 Henri Darmon McGill University $p$-adic iterated integrals and algebraic points on elliptic curves over abelian extensions of $\bf Q$. 100 NV 6:45 Dinner: Anglers Seafood, 91 Coldbrook Road, Hampden. Bus leaves the hotel at 6:20 pm

## Sunday, October 6, 2013

 Time Speaker Title Room 8:15-8:40am Coffee/Tea, Neville Lobby 8:40-9:30 Barry Mazur Harvard University Arithmetic of elliptic curves over families of number fields.   notes 101 NV 9:40-10:00 John Cullinan Bard College Arithmetic properties of the Legendre polynomials.   slides 100 NV Alvaro Lozano-Robledo University of Connecticut Uniform boundedness in terms of ramification.   slides   paper 101 NV 10:10-10:25 Abbey Bourdon Wesleyan University Rationality of $\ell$-torsion Points on CM Elliptic Curves 101 NV 10:30-10:45 Alan Chang Princeton University Newman's conjecture in Various Settings 100 NV Jonah Leshin Brown University A Bound for the Failure of Noether's Problem.   paper 101 NV 10:50-11:10 Zachary Scherr University of Pennsylvania Integer Polynomial Pell Identities 100 NV Daniel Vallieres Binghamton University Non-totally real cubic number fields and Cousin groups.   slides 101 NV 11:20-11:40 Justin Sukiennik Colby College Bounds on Height Functions slides 100 NV Jonathan Bayless Husson University New bounds and computations on prime-indexed primes 101 NV 11:50-12:20 Manfred Kolster McMaster University The Quillen-Lichtenbaum Conjecture 101 NV

## Abstracts (by last name)

Moshe Adrian, University of Utah: Jacquet's conjecture on the local converse problem for epipelagic supercuspidal representations of GL(n,F)
Let F be a non-archimedean local field of characteristic zero. For any irreducible admissible generic representation of GL(n,F), a family of twisted local gamma factors can be defined using Rankin-Selberg convolution or the Langlands-Shahidi method. Jacquet has formulated a conjecture on precisely which family of twisted local gamma factors can uniquely determine an irreducible admissible generic representation of GL(n,F). In joint work with Baiying Liu, we prove that Jacquet's conjecture is true for epipelagic supercuspidal representations of GL(n,F), supplementing recent results of Jiang, Nien, and Stevens.

Nico Aiello, University of Massachusetts: Galois theory of iterated morphisms on reducible elliptic curves
Let $F$ be a number field and let $A$ be an abelian algebraic group defined over $F$. For a prime $\ell$ and a point $\alpha \in A(F)$, the tower of extensions $F([\ell^n]^{-1}(\alpha))$ contains all of the $\ell$-power torsion points of $A$ along with a Kummer-type extension. The action of the absolute Galois group, $G_{\bar{\mathbb{Q}}/F}$ on this tower encodes information regarding the density of primes $\mathcal{P}$ in the ring of integers of $F$ for which the order of $\alpha$ mod $\mathcal{P}$ is prime to $\ell$. For $A=E$ an elliptic curve, Jones and Rouse have determined necessary and sufficient conditions for the Galois action on the tower $F([\ell^n]^{-1}(\alpha))$ to be as large as possible and under these conditions the associated density has been computed. In this talk, we will consider elliptic curves for which the Galois action is not as large as possible. In particular, we will study the Galois theory of the tower of extensions corresponding to a reducible elliptic curve $E/\mathbb{Q}$ with an $\ell$-torsion point defined over $\mathbb{Q}$ and calculate the encoded density.

Abdelmalek Azizi, Mohammed First University, Morocco: On the capitulation problem of the $2$-ideal classes of the field $\mathbb Q(\sqrt{p_1p_2}, i)$ where $p_1\equiv p_2\equiv 1 \pmod4$ are primes.
In this paper, we study the capitulation problem of the 2-ideal classes of the field $\mathbb k=\mathbb Q(\sqrt{p_1p_2}, i)$, where $\mathbf{C}l_2(\mathbb k)$, the 2-class group of $\mathbb k$, is of type $(2, 2)$ or $(2, 4)$ or $(2, 2, 2)$.

Jonathan Bayless, Husson University: New bounds and computations on prime-indexed primes
If the prime numbers are listed in increasing order, then the prime-index primes are those which occur in a prime-numbered position in the list. In 2009, Barnett and Broughan established a prime-index prime number theorem analogous to the standard prime number theorem. We improve and generalize their result with explicit bounds, bound the sum of reciprocals of prime-index primes, and present empirical results on prime-index prime versions of the twin prime conjecture and Goldbach's conjecture. This is joint work with Dominic Klyve and Tomàs Oliveira e Silva.

Abbey Bourdon, Wesleyan University: Rationality of $\ell$-torsion Points on CM Elliptic Curves
Let $E$ be an elliptic curve defined over a number field $F$, and for a prime $\ell$ let $F(E[\ell])$ be the field extension of $F$ generated by the $\ell$-torsion points of $E$. It is a consequence of the Weil pairing that $F(E[\ell])$ contains the $\ell$th roots of unity, $\mu_{\ell}$, which leads us to the following question: When does $E$ have a non-trivial $\ell$-torsion point defined over $F(\mu_{\ell})$? We will show that if we consider all elliptic curves with complex multiplication defined over number fields of a fixed degree, there are only finitely primes $\ell$ for which this can occur. This result adds to the unconditionally known cases of a finiteness conjecture on abelian varieties made by Rasmussen and Tamagawa.

Alan Chang, Princeton University: Newman's conjecture in Various Settings
Polya introduced a deformation of the Riemann zeta function $\zeta(s)$, and De Bruijn and Newman found a real constant $\Lambda$ which encodes the movement of the zeros of $\zeta(s)$ under the deformation. The Riemann hypothesis (RH) is equivalent to $\Lambda \le 0$. Newman made the conjecture that $\Lambda \ge 0$ along with the remark that "the new conjecture is a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so." Newman's conjecture is still unsolved, and previous work could only handle the Riemann zeta function and quadratic Dirichlet $L$-functions, obtaining lower bounds very close to zero (for example, for $\zeta(s)$ the bound is at least $-1.14541 \cdot 10^{-11}$, and for quadratic Dirichlet $L$-functions it is at least $-1.17 \cdot 10^{-7}$). We generalize the techniques to apply to a wider class of $L$-functions, including automorphic $L$-functions as well as function field $L$-functions. We further determine the limit of these techniques by studying linear combinations of $L$-functions, and prove that these methods are insufficient to handle cases where nontrivial zeros are known to appear off the critical line. Each type of family'' of function field quadratic $L$-functions gives a different version of Newman's conjecture. These variations have connections to other fields, including random matrix theory and the Sato--Tate conjecture. In particular, the recent proof of Sato--Tate for elliptic curves over totally real fields allows us to prove a version of Newman's conjecture involving fixed $D \in \mathbb Z[T]$ of degree $3$.

Hugo Chapdelaine, Université Laval: Cyclotomic units revisited
We will give an arithmetic interpretation of cyclotomic units and explain how these can be conceived as limits of certain elliptic units. The main motivation behind this reinterpretation being the Stark conjecture.

John Cullinan, Bard College: Arithmetic properties of the Legendre polynomials.
The Legendre polynomials (which were introduced in 1785) play an important role in analysis, physics and number theory, yet their algebraic properties are not well-understood. Stieltjes conjectured in 1890 how they factor over the rational numbers, yet very little progress has been made on this conjecture. In this talk we will exhibit new cases of irreducibility of the Legendre polynomials as well as give evidence for a conjecture on their Galois groups. This is joint work with Farshid Hajir.

Henri Darmon, McGill University: $p$-adic iterated integrals and algebraic points on elliptic curves over abelian extensions of $\bf Q$.
If $E$ is an elliptic curve over ${\bf Q}$, and $L$ is a non-quadratic abelian extension of ${\bf Q}$, it is expected that the difference between the rank of $E({\bf Q})$ and $E(L)$ grows very sporadically and is most often equal to zero. For instance, David, Fearnley and Kisilevsky have conjectured that this difference is equal to zero for all but finitely many $L$ of a given prime degree $\ge 7$. On the other hand, they have tabulated extensive lists of pairs $(E,L)$ where $L$ is a cyclic cubic extension and the rank of $E(L)$ jumps by at least two. In this talk I will describe a conjectural $p$-adic analytic expression allowing the calculation of these abelian points when they exist, involving certain $p$-adic iterated integrals attached to a cusp form of weight two and a pair of weight one Eisenstein series, and describe a setting where the conjecture has been tested numerically. This is joint work with Alan Lauder and Victor Rotger.

Zeb Engberg, Dartmouth College: On the reciprocal sum of primes dividing Mersenne numbers.
Let $f(n) = \sum_{p\mid 2^n-1} \tfrac 1p$. Erdos proved that $f(n) < \log\log\log(n) + C$ for some constant $C$. Apart from the exact value of $C$, it is easy to show that this result is best possible. Although it would be more interesting to understand the maximal order of $\sum_{p\mid 2^n-1} 1$, the function $f(n)$ is more tractable, albeit still difficult. In this talk, we consider Erdos's question on the exact value of the constant $C$, as well as functions which generalize $f(n)$.

David Geraghty, Boston College: A candidate p-adic local Langlands correspondence
I will discuss a construction which uses the patching method of Taylor--Wiles and Kisin to construct a candidate p-adic local Langlands correspondence. More precisely, for F a p-adic field, we associate a unitary Banach space representation of $GL_n(F)$ to any continuous n-dimensional p-adic representation of the absolute Galois group of F. We use this to prove many cases of the Breuil--Schneider conjecture. This is a based on a joint work with Caraiani, Emerton, Gee, Paskunas and Shin.

Eyal Goren, McGill University: Arithmetic intersection theory on Spin Shimura varieties
Claude Levesque and I share a passion for units. Much of my research over the years has been devoted to the problem of constructing units in abelian extensions of CM fields, with the hope of relating them to Stark's conjecture. I will quickly sketch a trajectory from this question, which is far from settled even in the simplest cases past imaginary quadratic fields, to questions about arithmetic intersection theory on Spin Shimura varieties and report on a recent proof of the Bruinier-Yang conjecture (with Andreatta, Howard and Madapusi-Pera).

Robert Grizzard, University of Texas: Relative Bogomolov extensions.
A field $K$ of algebraic numbers is said to satisfy the Bogomolov property if the absolute logarithmic height of non-torsion points of $K^\times$ is bounded away from 0. This can be generalized by defining a relative extension $L/K$ to be Bogomolov if the height of points of $L^\times \setminus K^\times$ is bounded away from 0. We'll briefly discuss why one would care about such things, mention an example or two, and state some results giving criteria for this condition to be satisfied.

Dubi Kelmer, Boston College: On the Pair Correlation of Hyperbolic Angles
The talk will concern the fine scale statistics of the set angles of geodesic rays corresponding to lattice points in a growing hyperbolic ball. I will discuss joint work with Kontorovich proving a conjecture of Boca Popa and Zaharescu on the pair correlation density for these hyperbolic angles.

Omar Kihel, Brock University: Denominators of algebraic numbers in a number field
Let $\gamma$ be an algebraic number over $\mathbb Q$ of degree $n\geq 1$ and $f(x)=a_nx^n+\cdots+ a_0\in\mathbb{Z}[x]$ its minimal polynomial of $\gamma$ over $\mathbb Z$. The leading coefficient $a_n$ of $f(x)$ will be denoted by $c(\gamma)$. The smallest positive integer $d$ such that $d.\gamma$ is an algebraic integer is called the denominator of $\gamma$ and will be denoted by $d(\gamma)$.
Arno et al proved that the density of the set of the algebraic numbers $\gamma$ such that $c(\gamma)=d(\gamma)$ is equal to $1/\zeta(3)=0,8319\cdots$.
Among other things, we fix a number field $K$, a prime $p$, a positive integer $k$ and we study the set of values of ${v}_p(c(\gamma))$, when $\gamma$ runs in the set of the primitive elements of $K$ over $\mathbb Q$, such that and ${v}_p(d(\gamma))=k$.

Manfred Kolster, McMaster University: The Quillen-Lichtenbaum Conjecture
It seems to be folklore (among the experts) that the Quillen-Lichtenbaum Conjecture is a consequence of the Bloch-Kato Conjecture, which has been proven by Voevodsky. We explain the connections between the two conjectures and try to throw some light on the proof.

Radan Kučera, Masarykova Univerzita (CZ): On the class group of a cyclic field of odd prime power degree
Let $p$ be an odd prime and $K/\mathbb Q$ be a Galois extension of degree $\ell=p^k$ whose Galois group $G=\operatorname{Gal}(K/\mathbb Q)$ is cyclic. Let $\operatorname{cl}_K$ be the ideal class group of $K$ and $h_K=|\operatorname{cl}_K|$ be the class number of $K$.
Let $p_1,\dots,p_s$ be the primes which ramify in $K/\mathbb Q$, let $e_j$ be the ramification index of $p_j$ and $g_j$ be the number of prime ideals of $K$ dividing $p_j$. We assume that $s>1$ and that the primes $p_1,\dots,p_s$ are ordered in such a way that $\ell=e_1\ge e_2\ge\dots\ge e_s\ge p$.
Let $C_K$ be the Sinnott group of circular units of $K$, which is a subgroup of the group $E_K$ of all units of $K$ of finite index defined by explicit generators. Sinnott's index formula for our field $K$ gives that the index $[E_K:C_K]=2^{\ell-1}\cdot h_K\cdot e_2^{-1}$.
The aim of this talk is to show that, if $s>2$, we can enlarge the Sinnott group $C_K$ by other explicit generators to a subgroup $\overline{C}_K$ of $E_K$ having smaller index $[E_K:\overline{C}_K]=2^{\ell-1}\cdot h_K\cdot p^n\cdot\prod_{j=1}^s e_j^{-g_j}$, where $n=\sum_{i=1}^k\max\{g_j\mid e_j\ge p^i\}$. This formula gives that $h_K$ is divisible by $p^{-n}\cdot\prod_{j=1}^s e_j^{g_j}$, which is stronger than the usual divisibility result obtained by genus theory if and only if there are at least two ramified primes $p_j$ having $g_j>1$. Moreover, assuming that $p$ does not ramify in $K/\mathbb Q$, by a modification of Thaine-Rubin machinery we can show that if $\alpha\in\mathbb Z[G]$ annihilates the $p$-Sylow subgroup of the quotient $E_K/\overline{C}_K$ then $(1-\sigma^{\ell/\bar e})\cdot\alpha$ annihilates the $p$-Sylow subgroup of the class group $\operatorname{cl}_K$, where $\sigma$ is a generator of the Galois group $G$ and $\bar e=\max\{e_j\mid g_j=\max_{i=1,\dots,s}g_i\}$, the maximal ramification index among the most decomposed ramified primes. (Joint work with Cornelius Greither.)

Jonah Leshin, Brown University: A Bound for the Failure of Noether's Problem.
Let $G$ be a finite group acting on affine $n$-space over a field $K$. Noether's problem asks whether the quotient $A^n/G$ of affine space by this group action is again affine (also called rational). In general, the rationality of $A^n/G$ depends on $G$ as well as the field $K$. When $G$ is abelian, we generalize a purely number theoretic criterion of Lenstra for the rationality of $A^n/G$ to a bound for the extent to which $A^n/G$ may fail to be rational.

Patrick Letendre, Laval University: Some results on $k$-free numbers.
We obtain asymptotic and upper bounds for $\displaystyle \mathcal{T}(h,M):=\sum_{n=0}^{M}|E(n,h)|^2\quad\mbox{and}\quad\mathcal{W}(q,N) :=\sum_{a=1}^{q}|E(a,q,N)|^2$ where $\displaystyle E(n,h)=\sum_{a=1}^{h}\mu_k(nh+a)-\frac{h}{\zeta(k)}\quad\mbox{and}\quad E(a,q,N)=\sum_{\substack{n=1\\n\equiv a\mod q}}^{N}\mu_k(n)-g_k(a,q)N.$ We also present some applications.

Alvaro Lozano-Robledo, University of Connecticut: Uniform boundedness in terms of ramification
Let $d\geq 1$ be fixed. Let $F$ be a number field of degree $d$, and let $E/F$ be an elliptic curve. Let $E(F)_\text{tors}$ be the torsion subgroup of $E(F)$. In 1996, Merel proved the uniform boundedness conjecture, i.e., there is a constant $B(d)$, which depends on $d$ but not on the chosen field $F$ or on the curve $E/F$, such that the size of $E(F)_\text{tors}$ is bounded by $B(d)$. Moreover, Merel gave a bound (exponential in $d$) for the largest prime that may be a divisor of the order of $E(F)_\text{tors}$. In 1996, Parent proved a bound (also exponential in $d$) for the largest $p$-power order of a torsion point that may appear in $E(F)_\text{tors}$. It has been conjectured, however, that there is a bound for the size of $E(F)_\text{tors}$ that is polynomial in $d$. In this talk we discuss that under certain hypotheses there is a linear bound for the largest $p$-power order of a torsion point defined over $F$, which in fact is linear in the maximum ramification index of a prime ideal of the ring of integers $F$ over $(p)$.

Barry Mazur, Harvard University: Arithmetic of elliptic curves over families of number fields.
I will discuss what Zev Klagsbrun, Karl Rubin and I call the disparity of an elliptic curve $E$ over a given number field $K$, this being the ratio of even versus odd rank $2$-Selmer groups of twists of $E$ by quadratic characters of $K$. I will also discuss on-going joint work with Maarten Derickx and Sheldon Kamienny on computations regarding what we call Basic Brill-Noether modular varieties which classify rational families of elliptic curves having torsion over fields of degree $d$, where $d$ is the smallest degree in which there are such rational families.

Nathan McNew, Dartmouth College: Sets of integers which contain no three terms in geometric progression.
The problem of looking for subsets of the natural numbers which contain no 3-term arithmetic progressions has a rich history. Roth's theorem famously shows that any such subset cannot have positive upper density. In contrast, Rankin in 1960 suggested looking at subsets without geometric progressions, and constructed such a subset with asymptotic density about 0.719. More recently, several authors have found upper bounds for the upper density of such sets. We significantly improve upon these upper bounds, and demonstrate a method of constructing sets with a greater upper density than Rankin's set. This construction is optimal in the sense that this method gives a way of effectively computing the greatest possible upper density of a geometric-progression-free set. Finally, we show that geometric progressions mod N behave more like Roth's theorem in that one cannot take any fixed positive proportion of the integers modulo a sufficiently large value of N while avoiding geometric progressions.

Steven Miller, Williams College: Mind the Gap: Distribution of Gaps in Generalized Zeckendorf Decompositions.
Zeckendorf proved that any integer can be decomposed into a unique sum of non-adjacent Fibonacci numbers, $F_n$. Using continued fractions, Lekkerkerker showed that the average number of summands in a decomposition of an integer in $[F_n, F_{n+1})$ is essentially $n/(\phi^2 +1)$, where $\phi$ is the golden ratio. Miller-Wang generalized this by adopting a combinatorial perspective, proving that for any positive linear recurrence of the form $A_n = c_1 A_{n-1} + c_2 A_{n-2} + \cdots + c_L A_{n+1-L}$, the number of summands in decompositions for integers in $[A_n, A_{n+1})$ converges to a Gaussian distribution as $n \to \infty$. We prove that the probability of a gap larger than the recurrence length converges to decaying geometrically, with decay rate equal to the largest eigenvalue of the characteristic polynomial of the recurrence, and that the distribution of the smaller gaps depends on the coefficients of the recurrence (which we analyze through the combinatorial perspective). These results hold both for the average over all $m \in [A_n, A_{n_1})$, as well as holding almost surely for the gap measure associated to individual $m$.

Michael Mossinghoff, Davidson College: Wieferich pairs and Barker sequences.
A Barker sequence is a finite sequence of integers, each $\pm1$, whose off-peak aperiodic autocorrelations are all at most $1$ in absolute value. Very few Barker sequences are known, and it has long been conjectured that no additional ones exist. Many arithmetic restrictions have been established that severely limit the allowable lengths of Barker sequences, so severely that no permissible lengths were even known. These restrictions involve certain Wieferich prime pairs, which are pairs $(q, p)$ with the property that $q^{p-1} \equiv 1$ mod $p^2$. We identify the smallest plausible value for the length of a new Barker sequence, and we compute a number of permissible lengths up to a sizable bound. This is joint work with Peter Borwein.

Dawn Nelson, Bates College: A Variation on Leopoldt's Conjecture: Part 2.
What is the relationship between the (global) units of a number field and the (local) units of the related local fields? Leopoldt conjectured an answer. Informally his conjecture states that the $\mathbb{Z}_p$-rank of the diagonal embedding of the global units into the product of all local units equals the $\mathbb{Z}$-rank of the global units. I consider the variation: Can we say anything about the $\mathbb{Z}_p$-rank of the diagonal embedding of the global units into the product of some local units? The answer is yes. In particular, in the case of an Abelian extension I use ideas from linear algebra, the theory of linear representations, and Galois theory to give a precise formula for the $\mathbb{Z}_p$-rank (of the diagonal embedding of the global units into the product of some local units) in terms of the $\mathbb{Z}$-rank of the global units and a property of the the local units included in the product.

Kyle Pratt, Brigham Young University, and Minh-Tam Trinh, Princeton University: Zeros of Dirichlet $L$-Functions over Function Fields
We consider the distribution of zeros in the family of Dirichlet $L$-functions over $\mathbb{F}_q(T)$ of fixed irreducible conductor $Q$, as $\deg Q \to \infty$. Our approach allows us to compare the strength of results obtained in three separate settings, called the global, mesoscopic, and local regimes. In the first two, fluctuations in the equidistribution of the zeros are Gaussian, as we would predict if the matrices that correspond to the $L$-functions in our family were equidistributed in the unitary group. In the third, we only obtain agreement with unitary predictions for test functions whose Fourier transforms have support in the range $[-2, +2]$. Finally, we state some natural arithmetic hypotheses that improve the strength of our results. Our work extends and brings together work of Hughes-Rudnick, Faifman-Rudnick, Xiong, and Fiorilli-Miller.

James Ricci, Wesleyan University: Finiteness results for regular ternary quadratic polynomials
Any quadratic polynomial can be written in the form $f(x) = Q(x) + l(x) + c$ where $Q$ is a quadratic form, $l$ is a linear form, and $c$ is a constant; it is called regular if it represents all the integers which are represented locally by the polynomial itself over $\mathbb{Z}_p$ for all primes $p$. Given a positive definite $Q$, we can associate certain types of quadratic polynomials to a coset of a $\mathbb{Z}$-lattice in order to view quadratic polynomials through the geometric perspective of quadratic spaces and lattices. In this talk we will define an invariant called the conductor, a notion of a semi-equivalence class of a regular quadratic polynomial and present our result: Given a fixed conductor, there are finitely many semi-equivalence classes of primitive regular integral quadratic polynomials in three variables.

Adriana Salerno, Bates College: Effective computations in arithmetic mirror symmetry.
In this talk, I will talk about computational approaches to the problem of arithmetic mirror symmetry. One of the biggest questions facing string theorists is the one of mirror symmetry. In arithmetic mirror symmetry, we approach the conjecture from a number theoretic point of view, namely by computing Zeta functions of mirror pairs. I will define all of these terms and then explain our work through a couple of examples of families of K3 surfaces. This is joint work with Xenia de la Ossa, Charles Doran, Tyler Kelly, Stephen Sperber, and Ursula Whitcher.

Zachary Scherr, University of Pennsylvania: Integer Polynomial Pell Identities
Let $d$ be a non-square positive integer. It is well known that the Pell equation $x^2-dy^2=1$ has infinitely many non-zero integer solutions $x$ and $y$. Euler, in the 1760s, noticed that if $d$ is of the form $n^2+1$ for some integer $n$ then $$(2n^2+1)^2-(n^2+1)(2n)^2=1$$ gives a solution to the Pell equation. Motivated by Euler's example, one looks at a polynomial analogue of the Pell equation over the ring $\mathbb{Z}[x]$. In this talk we will survey some known results about when quadratic polynomials over $\mathbb{Z}$ admit solutions to the Pell equation, and then present new results giving a complete classification of when quartic polynomials admit solutions. Our methods draw on ideas from the theory of ellptic curves, specifically Mazur's theorem and explicit parametrizations of modular curves.

Andrew Schultz, Wellesley College: Realizing certain $p$-groups as Galois groups
We build on the classical parameterizing spaces for elementary $p$-abelian extensions to give a module-theoretic description for the appearance of a large family of $p$-groups. We then use this to generalize some of the known results concerning the appearance of the non-abelian groups of order $p^3$ as Galois groups, including automatic realization results as well as some realization multiplicity calculations.

Justin Sukiennik, Colby College: Bounds on Height Functions
We look at strict upper bounds for the difference in a point's height and its linear fractional mapping over a number field. From these results, we can demonstrate an interesting asymmetry when we interchange the difference terms. Also, the proof of the bound follows from the Artin-Whaples approximation theorem for $p$-adic metrics.

Alain Togbé, Purdue University North Central: On Diophantine equations involving normalized binomial mid-coefficients
For any nonnegative integer $n$, the normalized binomial mid-coefficient is defined by $$\mu_{n}=2^{-2n}\binom{2n}{n} = \frac{1\cdot3\cdot5\cdot\cdots\cdot (2n-1)}{2\cdot4\cdot6\cdot\cdots\cdot(2n)}.$$ Let $t$ be a real number. The power mean of order $t$ of the positive real numbers $x_{1}, \cdots, x_{n}$ is defined by $M_{t}(x_{1}, \cdots, x_{n})=\Big(\frac{1}{n}\sum_{j=1}^{n}x_{j}^{t}\Big)^{\frac{1}{t}}\quad \mbox{if } t\neq 0,$ and $M_{0}(x_{1}, \cdots, x_{n})=\lim_{t\rightarrow0}M_{t}(x_{1}, \cdots, x_{n})=\Big(\prod_{j=1}^{n}x_{j}\Big)^{\frac{1}{n}}.$ It is very interesting to study the Diophantine equation involving power means of $n$ variables $\mu_{n}$, $$M_{k}(\mu_{a_{1}},\cdots, \mu_{a_{n}})=M_{l}(\mu_{b_{1}}, \cdots, \mu_{b_{n}}), \; k, l\in {\mathbb{Z}}.$$ During this talk, we will discuss the above Diophantine equation particularly for $n=2, 3$ and other general cases. (The talk is based on a joint work with Shichun Yang and Wenquan Wu).

Daniel Vallieres, Binghamton University: Non-totally real cubic number fields and Cousin groups.
In this talk, we will explain a connection between certain complex connected abelian Lie groups and non-totally real cubic number fields. This can be viewed as an analogue of what is happening in the theory of complex multiplication between complex tori of dimension one and imaginary quadratic number fields.

Siman Wong, University of Massachusetts: Arithmetic of octahedral sextics
Given a quartic field with Galois group $S_4$, we relate its ramification to that of the non-Galois sextic subfields of its Galois closure, and we construct explicit generators of these sextic fields from that of the quartic field, and vice versa. This allows us to recover examples of $S_4$ sextic fields of Cohen and of Tate unramified outside 229, and to easily determine the tame part of the conductor of an octahedral Artin representation. We study class number divisibility arised from $S_4$-quartics whose discriminants are odd and square-free, we explicitly construct infinitely many $S_4$-quartics whose discriminants are $-1$ times square, and experimental data suggests two surprising conjectures about $S_4$-quartic fields with prime power discriminants.

Joshua Zelinsky, Boston University: Recent result in counting Artin representations
We will discuss recent results in counting ray class character as well as an elementary analog related to the Artin primitive root conjecture. The talk will focus on novel upper bounds on sums related to these two problems.

Lei Zhang, Boston College: Eisenstein Series of Covers of Odd Orthogonal Group.
We study the Whittaker coefficients of the minimal parabolic Eisenstein series on the n-fold cover of the split odd orthogonal group SO(2r+1). If the degree of the cover is odd, we establish the Beineke, Brubaker and Frechette's conjecture on the p-power coefficients to the Whittaker coefficients. In general, we use a combination of automorphic and combinatorial-representation-theoretic methods. We aslo establish a formula for Whittaker coefficients in the even degree cover case, based on crystal graphs of type C.

## List of Participants

Domenico Aiello, University of Massachusetts (grad)
Joseph Arsenault, University of Maine (grad)
Josh Audibert, University of Maine (undergraduate)
Abdelmalek Azizi, Mohammed First University, Morocco
Matthew Bates, University of Massachusetts (grad)
Jonathan Bayless, Husson University
Elliot Benjamin, CALCampus
Matthew Brenc, University of Maine (grad)
Daniel Buck, University of Maine (undergrad)
Byungchul Cha, Muhlenberg College
Hugo Chapdelaine, Laval University
Michael Chou, University of Connecticut (grad)
John Cullinan, Bard College
Harris Daniels, Amherst College
Henri Darmon, McGill University
Emma Dowling, University of Massachusetts (grad)
Zhenguang Gao, Framingham University
David Geraghty, Boston College
Eyal Goren, McGill University
Fernando Gouvêa, Colby College
Hester Graves, IDA
Bobby Grizzard, University of Texas (grad)
Samuel Gross, Bloomsburg University
Dubi Kelmer, Boston College
Omar Kihel, Brock University
Myoungil Kim, University of Connecticut
Andrew Knightly, University of Maine
Manfred Kolster, McMaster University
Mark Kozek, Whittier College
Radan Kučera, Masaryk University, Czech Republic
Jennifer Letourneau, University of Maine (grad)
Claude Levesque, Laval University
Alvaro Lozano-Robledo, University of Connecticut
Barry Mazur, Harvard University
Steven Miller, Williams College
Chung Pang Mok, McMaster University
Michael Mossinghoff, Davidson College
Dawn Nelson, Bates College
Elliot Ossanna, University of Maine (undergrad)
Dean Pelletier, University of Maine (undergrad)
Nigel Pitt, University of Maine
Sophia Potoczak, University of Maine (grad)
Kyle Pratt, Brigham Young University (undergrad)
Jonathan Sands, University of Vermont
Zachary Scherr, University of Pennsylvania
Andrew Schultz, Wellesley College
Chip Snyder, University of Maine
Justin Sukiennik, Colby College
Alain Togbé, Purdue Northern University