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
Moshe Adrian, University of Utah
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
Abbey Bourdon, Wesleyan University (grad)
David Bradley, University of Maine
Matthew Brenc, University of Maine (grad)
Daniel Buck, University of Maine (undergrad)
Byungchul Cha, Muhlenberg College
Alan Chang, Princeton University (undergrad)
Hugo Chapdelaine, Laval University
Sara Chari, Bates College (undergrad)
Michael Chou, University of Connecticut (grad)
John Cullinan, Bard College
Harris Daniels, Amherst College
Henri Darmon, McGill University
Emma Dowling, University of Massachusetts (grad)
Zeb Engberg, Dartmouth College (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
Jonah Leshin, Brown University (grad)
Patrick Letendre, Laval University (grad)
Jennifer Letourneau, University of Maine (grad)
Claude Levesque, Laval University
Alvaro Lozano-Robledo, University of Connecticut
Mostafa Mache, Laval University (grad)
Barry Mazur, Harvard University
Nathan McNew, Dartmouth College (grad)
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)
James Ricci, Wesleyan University (grad)
Azar Salami, Laval University (grad)
Adriana Salerno, Bates College
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
Minh-Tam Trinh, Princeton University (undergrad)
Daniel Vallieres, Binghamton University
Douglas Weathers, University of Maine (grad)
Benjamin Weiss, University of Maine
Siman Wong, University of Massachusetts
Joshua Zelinsky, Boston University (grad)
Wesley Zeng, University of Maine (undergrad)
Lei Zhang, Boston College
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