Erwan Biland, Université Laval:
"Brauer blocks and Morita equivalences"
We set up the general frame of modular representation theory. We define the
Brauer blocks of a group algebra over a field of positive characteristic, with some
examples. Then we explain why classification of such blocks should be done up to an
equivalence of module categories (Morita, derived, stable equivalence), and illustrate
with our recent work.
Don Blasius, UCLA:
"Recent progress concerning existence of
automorphic forms of prescribed local type"
Reinier Bröker, Brown University:
"Abelian surfaces with extra endomorphims"
For elliptic curves, the modular polynomial Φp(X,Y) parametrizes elliptic
curves together with a p-isogeny. The polynomial Φp(X,X) parametrizes
elliptic curves together with an endomorphism of degree p. Kronecker discovered
already that the irreducible factors of Φp(X,X) are Hilbert class
polynomials.
In this talk we will consider abelian surfaces with extra endomorphisms. We
will show which factors occur when you factor the 2-dimensional analogue
of the modular polymial Φp(X,X). In the case p = 2, everything can be
explicitly computed and we will give a complete classification of abelian
surfaces admitting a (2,2)-endomorphism.
Michael Bush, Smith College:
"Heuristics for p-class towers of imaginary quadratic fields"
I will describe some recent joint work with Nigel Boston and Farshid Hajir in
which we give a conjectural description of the frequency with which certain finite
p-groups arise as the Galois groups associated to p-class towers over imaginary
quadratic fields where p is an odd prime. Our conjecture can be viewed as a non-abelian
generalization of the Cohen-Lenstra heuristics for class groups of such fields.
Hugo Chapdelaine, Université Laval:
"Computation of Galois groups via permutation group theory"
In this talk we will present a method to compute the
Galois group of certain polynomials defined over Q by combining
the theory of local fields and the theory of permutation groups.
Gautam Chinta, City College of New York:
"Recent results in Multiple Dirichlet series"
I will give a survey of some recent results in multiple Dirichlet series.
The focus of the talk will be on applications to
number theory.
John Cullinan, Bard College:
"Decomposition of the Plesken Lie algebra over finite fields"
Let G be a finite group. The Plesken Lie algebra is a
subalgebra of the complex group algebra C[G] and admits a direct-sum
decomposition into simple Lie algebras. We consider the finite field
analog of these Lie algebras and the problem of determining the simple
factors in modular characteristic.
Jean-Marie De Koninck, Université Laval:
"New methods for constructing normal numbers"
Given an integer d≥ 2, we say that an irrational number
η=0.a1a2…,
where each ai∈ {0,1,…,d-1}, is a
normal number (in base d) if the sequence
{dmη}, m=1,2,… (here {y} stands for the
fractional part of y), is uniformly distributed in the interval
[0,1[. We show how one can use the complexity of the multiplicative
structure of positive integers to construct large families of normal numbers.
Daniel Fiorilli, IAS:
"The distribution of some arithmetic sequences in arithmetic
progressions to large moduli"
As is the case for prime numbers, many arithmetic sequences behave
quite well in arithmetic progressions. However, when one looks at arithmetic
progressions of large moduli, the question becomes harder: in the case of
primes for example, this question is highly dependent on the generalized
Riemann hypothesis. Still, if we look at an average over the moduli, much can
be said. In this talk we will study such an average, and show how it can
extract information about the sequence we are studying. The examples we will
focus on are primes, integers which can be written as the sum of two squares,
representations of a fixed positive definite binary quadratic form, prime
k-tuples and integers free of small prime factors.
Amanda Folsom, Yale University:
"Mock modular forms and characters of Kac-Wakimoto"
Kac and Wakimoto recently established certain character formulas arising
from affine Lie superalgebras. In this talk, we will discuss the "mock modularity" of
these characters. In particular, we will discuss works of the author and
Bringmann-Ono, which show that these characters may be realized as parts of certain
non-holomorphic modular functions. Moreover, we show how these characters specialize
to Ramanujan's original "mock theta functions", certain combinatorial q-series. As an
application, we will then discuss joint work with Bringmann, which shows how the "mock
modularity" of these characters can be exploited to obtain improved information
regarding their asymptotic behaviors.
Avram Gottschlich, Dartmouth College:
"On positive integers n dividing the nth term of an elliptic
divisibility sequence"
Elliptic divisibility sequences are integer sequences
related to the denominator of the first coordinate of the n-fold sum
of a rational non-torsion point on an elliptic curve. Silverman and
Stange recently studied those integers n dividing Dn, where {Dn} is
an elliptic divisibility sequence. Here we discuss the distribution of
these numbers n.
Geoffrey Iyer, University of Michigan:
"Constructing generalized More Sums Than Differences sets"
(Joint work with Oleg Lazarev, Steven J. Miller, and Liyang Zhang.)
A More Sums Than Differences (MSTD, or sum-dominant) set is a finite
set A of integers such that |A+A|<|A-A|. Though it was
believed that the percentage of subsets of {0,…,n} that are
sum-dominant tends to zero, in 2006 Martin and O'Bryant
proved that a positive percentage are sum-dominant. We generalize their
result to the many different ways of taking sums and differences of a
set. We prove that
|ε1A+…+εkA| >
|δ1A+…+δkA|
a positive percent of the time for all
nontrivial choices of εj and δj in {-1,1}. Previous
approaches proved the existence of infinitely many such sets given the
existence of one; however, no method existed to construct such a set.
We develop a new, explicit construction for one such set, and then
extend to a positive percentage of sets. We extend these results
further, finding sets that exhibit different behavior as more
sums/differences are taken. For example, notation as above we prove that for any m,
|ε1A + … + εkA|
- |δ1A + … + δkA| = m
a positive percentage of the time. We find the
limiting behavior of kA=A+…+A for an arbitrary set A as
k→∞ and an upper bound of k for such behavior to settle
down. Finally, we say A is k-generational sum-dominant if A,
A+A, …, kA are all sum-dominant. Numerical searches were
unable to find even a 2-generational set (heuristics indicate the
probability is at most 10-9, and almost surely significantly
less). We prove the surprising result that for any k a positive
percentage of sets are k-generational, and no set can be k-generational for all k.
Rafe Jones, College of the Holy Cross:
"The genus of y2 = fn(x)"
We classify the polynomials f such that the genus of the hyperelliptic
curve y2 = fn(x) remains bounded as n grows.
We use this to establish a
converse to a finite-ramification result of Hajir-Aitken-Maire in the case of
quadratic polynomials.
Jonah Leshin, Brown University:
"Root Discriminants and Class Field Towers"
We consider the asymptotic behavior of the root discriminant of a
certain set of number fields and give a finiteness result about this set. I will
motivate this result with the connection between root discriminants and class
field towers. Time permitting, I will discuss some open problems about root
discriminants and class field towers.
Claude Levesque, Université Laval:
"Some elliptic curves over Q which are as impressive as the banyans of Hawai`i"
We will consider certain elliptic curves over Q and we will investigate their
ranks.
Alon Levy, Brown University:
"Bounding the Height of a Postcritically Finite Map"
(Joint work with Patrick Ingram and Rafe Jones.)
Postcritically finite (PCF) maps - that is, rational maps whose critical points all have
finite forward orbit - have special arithmetic and geometric properties, and should be
regarded as a special case, much like elliptic curves with complex multiplication. Over
C, we know from Thurston's rigidity theorem that away from the Lattes curve, there are
countably many PCF maps, so the PCF maps are in a precise sense sparse over C. In this
work, we prove a sparseness result over number fields: the height of a PCF map is
bounded in terms of the degree of the map, so that away from the Lattes curve there are
only finitely many of a given degree defined over a given number field. The proof method
has some other interesting corollaries, for example about attracting cycles over
non-archimedean fields.
Ben Linowitz, Dartmouth College:
"A newform theory for Hilbert Eisenstein series"
In his thesis, Weisinger developed a newform theory for
Eisenstein series. This theory was later generalized to the Hilbert
modular setting by Wiles. We extend the theory of newforms for Hilbert
Eisenstein series. In particular we will show that Hilbert Eisenstein
newforms are uniquely determined by their Hecke eigenvalues for any
set of primes having Dirichlet density greater than 1/2. Additionally,
we will provide a number of applications of this newform theory. This
is joint work with Tim Atwill.
Alvaro Lozano-Robledo, University of Connecticut:
"Bounds on the torsion subgroup of an elliptic curve"
Let E/Q be an elliptic curve and let K be a
number field. In this talk we will discuss several bounds on the size
of the torsion subgroup of E(K) that depend on the ramification
indices of K/Q.
Ken McMurdy, Ramapo College:
"Stable Models and Up Slope Calculations"
In joint work with LJP Kilford, we recently computed the slopes of the U7
operator acting on overconvergent modular forms on X1(49) with an infinite
family of weight-characters. Instead of restricting to an affinoid subspace of the
modular curve (as is the usual method), our key step was to lift the forms up
onto the stable reduction, thereby arguing mod p independence via Riemann-Roch.
In this talk we begin to explore the extent to which this might be possible
in general. I.e., we seek a family of models for X0(pn) whose parameters (1)
provide a Banach basis for the overconvergent forms and (2) adequately
describe certain components in the stable reduction.
Steven Miller, Williams College:
"Finite conductor models for zeros near the central point of elliptic curve
L-functions" (slides)
Random Matrix Theory has successfully modeled the behavior of zeros of elliptic curve
L-functions in the limit of large conductors. In this talk we explore the behavior of
zeros near the central point for one-parameter families of elliptic curves with rank
over Q(T) and small conductors. Zeros of L-functions are conjectured to be simple except
possibly at the central point for deep arithmetic reasons; these families provide a
fascinating laboratory to explore the effect of multiple zeros on nearby zeros. Though
theory suggests the family zeros (those we believe exist due to the Birch and
Swinnerton-Dyer Conjecture) should not interact with the remaining zeros, numerical
calculations show this is not the case; however, the discrepency is likely due to small
conductors, and unlike excess rank is observed to noticeably decrease as we increase the
conductors. We shall mix theory and experiment and see some surprisingly results, which
leads us to conjecture that a discretized Jacobi ensemble correctly models the small
conductor behavior.
Dawn Nelson, Bates College:
"Variation on Leopoldt's Conjecture"
Leopoldt's Conjecture is a statement about the relationship between the global and local
units of a number field. Approximately the conjecture states that the
Zp-rank of the
diagonal embedding of the global units into the product of all local units equals
the Z-rank of the global units. The variation that we consider in this talk addresses
the question: Can we say anything about the Zp-rank
of the diagonal embedding of the
global units into the product of some local units? The answer is yes. Moreover we
can give a value for the Zp-rank
(of the diagonal embedding of the global units into
the product of some local units) in terms of the Z-rank
of the global units and
a property of the the local units included in the product.
Keith Ouellette, UCLA:
"On the Fourier Inversion Theorem for PGL(2, Qp)"
We translate our proof of the Fourier inversion theorem for SL(2,R) to the
setting of p-adic groups. We will present the key techniques used in the proof in the
case of spherical unramified principal series for PGL(2, Qp).
Carl Pomerance, Dartmouth College:
"A problem of Arnold on the average multiplicative order"
For n an odd natural number, let l2(n) denote
the multiplicative order of 2 in the unit group mod n. In a
recent paper, V. I. Arnold conjectured that on average l2(n)
behaves like a constant times n/log n. Kurlberg and I show,
under assumption of the Generalized RH, that one needs a correction
factor of about (log n)B/logloglog n, for a certain
explicit constant B. This can be generalized to lg(n),
where |g|>1 is an integer and n is coprime to g. We also
compute the average where n is restricted to prime numbers.
Juan Rivera-Letelier,
Brown University / Pontificia Universidad Católica de Chile:
"Ergodic theory of p-adic rational maps"
The topological entropy is one of the most important invariants of a
topological dynamical system. Since the late 1970s it is known that the topological
entropy of a rational map acting on the Riemann sphere is equal to the logarithm of
its degree. However, this is not true for a p-adic rational map acting on
Berkovich's projective line: the topological entropy could be zero and it is difficult
to compute in general. We show a rigidity result for a p-adic rational map whose
equidistribution measure does not charge the wildly ramified locus: if the topological
entropy is not equal to the logarithm of tis degree (as in the complex case), then the
rational map possesses a smooth invariant metric.
This is a work in progress with Charles FAVRE.
Adriana Salerno, Bates College:
"The Dwork family and hypergeometric functions"
In his work studying the Zeta functions of families of
hypersurfaces, Dwork came upon a one-parameter family of hypersurfaces (now
known as the Dwork family). These examples were not only useful to Dwork in
his study of his deformation theory for computing Zeta functions of
families, but they have also proven to be extremely useful to physicists
working in mirror
symmetry. A startling result is that these families are very closely linked
to hypergeometric functions. This phenomenon was carefully studied by Dwork
and Candelas, de la Ossa, and Rodriguez-Villegas in a few special cases.
Dwork, Candelas, et.al. observed that, for these families, the differential
equation associated to the Gauss-Manin connection is in fact hypergeometric.
We have developed a computer algorithm, implemented in Pari-GP, which can
check this result for more cases by computing the Gauss-Manin connection and
the parameters of the hypergeometric differential equation.
Andrew Schultz, Wellesley College:
"A q-analog of Fleck's congruence"
In the early 1900's, Fleck proved that alternating sums of
binomial coefficients taken across particular residue classes modulo a
prime number are highly divisible by that prime number. In this talk,
I'll discuss some recent work for analogous sums of q-binomial
coefficients, and we'll see that these give "half" of a generalization
of Fleck's result.
Joseph Silverman, Brown University:
"Arithmetic Geometry and Arithmetic Dynamics"
Arithmetic dynamics is a relatively new field in which one studies
arithmetic properties of orbits of points under iteration of an
algebraic map, while arithmetic geometry, and especially the study of
Diophantine equations, has a long history. Many of the motivating
problems in arithmetic dynamics are dynamical analogues of well-known
theorems and conjectures in arithmetic geometry. In this talk I will
discuss these connections, highlighting both similarities and
differences in the two subjects. As time permits, I will also discuss
p-adic dynamics, where the motivations and analogies come primarily
from classical complex dynamics.
Ian Sprung, Brown University:
"The Shafarevich-Tate group of an elliptic curve in towers of number
fields and its modesty"
The Shafarevich-Tate group is a fundamental invariant that arises when
computing the Mordell-Weil group of an elliptic curve. It is
conjectured to be finite, but this is only known in special cases.
Assuming finiteness, people have studied how the order grows in
towers, in particular studying the growth of its p-primary part in the
cyclotomic Zp-extension of the rationals. In this talk, we show that
the Shafarevich-Tate group grows MODESTLY: When presented with two
ways of growing, it chooses the slower way.
Jim Stankewicz, University of Georgia:
"Twists of Shimura curves"
We describe some techniques for determining if Atkin-Lehner twists of
Shimura curves (and thus Atkin-Lehner twists of classical modular curves) have
rational points.
Lucien Szpiro, City University of New York:
"Critical bad reduction for a self-map"
We will explain a new notion of bad reduction for a self map of the projective line over
a number field or a function field: critical bad reduction. It is comparable to the
usual one (dropping of the degree in reduction) but has been proved more useful for
finiteness theorems.
Lola Thompson, Dartmouth College:
"On the divisors of xn-1 in Fp[x]"
In a recent paper, we considered integers n for which the polynomial
xn-1 has a divisor in Z[x] of every degree up to n, and we gave upper and
lower bounds for their distribution. In this talk, we will discuss some new
work in which we consider those n for which the polynomial xn-1 has a
divisor in Fp[x] of every degree up to n, where p is a rational prime.
Thomas Tucker, University of Rochester:
"Towards a dynamical Manin-Mumford conjecture"
The Manin-Mumford conjecture, now a theorem of Raynaud, says
that a subvariety of an abelian variety A contains a dense set of
torsion points if and only if the subvariety is itself a torsion
translate of an abelian subvariety of A. A natural dynamical analog
asks that a subvariety contains a dense set of preperiodic points
exactly when the subvariety is itself preperiodic. We will present a
counterexample to this conjecture and discuss other possible
formulations of this problem.
Bianca Viray, Brown University:
"On a uniform bound for the number of exceptional linear subvarieties in
the dynamical Mordell--Lang conjecture"
Let F : Pn --> Pn be a morphism of degree d > 1 defined over C.
The dynamical Mordell--Lang conjecture says that the intersection
of an orbit OF(P) and a subvariety X of Pn is usually finite. We
consider the number of linear subvarieties L in Pn such that the
intersection of OF(P)and L is "larger than expected". When F is
the d'th-power map and the coordinates of P are multiplicatively
independent, we prove that there are only finitely many linear
subvarieties that are "super-spanned" by OF(P), and further that
the number of such subvarieties is bounded by a function of n,
independent of the point P or the degree d. More generally, we
show that there exists a finite subset S, whose cardinality is
bounded in terms of n, such that any n+1 points in OF(P)\S are in
linear general position in Pn.
Jonathan Webster, Bates College:
"Simple Cubic Function Fields"
In this talk, we study simple cubic fields in the function field setting,
and also generalize the notion of a set of exceptional units to cubic function fields,
namely the notion of k-exceptional units. We give a simple proof that the Galois simple
cubic function fields are the immediate analog of Shanks simplest cubic number
fields. We prove that a unit arising as a root of the polynomial is a fundamental unit.
In addition to computing the invariants, including a formula for the regulator, we
compute the class numbers of the Galois simple cubic function fields over F5 and F7.
Finally, as an additional application, we determine all Galois simple cubic function
fields with class number one, subject to a mild restriction.
Andrew Yang, Dartmouth College:
"Counting binary n-ic forms by Julia invariant"
The natural action of SL2(Z) on binary forms of degree n
partitions the set of these forms into equivalence classes. One can ask how many such
classes there are of "size" up to X, for a suitable notion of size. For example, in the
n = 2, 3 cases, size is commonly measured by the (absolute value of the) discriminant,
and these counting results have applications to enumeration questions in algebraic
number theory. We study this counting question for general n where size is measured by
an invariant first studied by G. Julia. This is joint work with Manjul Bhargava.
Liyang Zhang, Williams College, and Oleg Lazarev, Princeton
University:
"Low-lying zeros of cuspidal Maass forms"
(Joint work with Steven J. Miller and Nadine Amersi.)
We study the distribution of zeros near the central point of
L-functions of level 1 Maass forms; this is essentially summing a
smooth test function whose Fourier transform is compactly supported
over the scaled zeros. Using the Petersson formula, Iwaniec, Luo and
Sarnak proved that the zeros near the central point of holomorphic cusp
forms agree with the eigenvalues of orthogonal matrices for suitably
restricted test functions. We prove a similar result for Maass forms.
We derive an explicit formula (via complex analysis) relating sums of
our test function at scaled zeros to sums of the Fourier transform at
the primes weighted by the Maass forms coefficients, and use the
Kuznetsov trace formula to average over the family. There are numerous
technical obstructions in handling the terms in the trace formula,
which are surmounted through the use of smooth weight functions and
results on Kloosterman sums and Bessel and hyperbolic functions.
|
List of Participants
Domenico Aiello, University of Massachusetts (grad)
Jackie Anderson, Brown University (grad)
Jean Auger, Laval University (undergrad)
Erwan Biland, Laval University (grad)
Don Blasius, UCLA
David Bradley, University of Maine
Reinier Bröker, Brown University
Michael Bush, Smith College
Hugo Chapdelaine, Laval University
Gautam Chinta, City College of New York
Maurice-Etienne Cloutier, Laval University (grad)
John Cullinan, Bard College
Harris Daniels, University of Connecticut (grad)
Jean-Marie DeKoninck, Laval University
Nicolas Doyon, Laval University
Zeb Engberg, Dartmouth College (grad)
Daniel Fiorilli, Institute for Advanced Study, Princeton
Amanda Folsom, Yale University
Thomas Gassert, University of Massachusetts (grad)
Avram Gottschlich, Dartmouth College (grad)
Anna Haensch, Wesleyan University (grad)
Emily Igo, University of Maine (grad)
Geoffrey Iyer, University of Michigan (undergrad)
Rafe Jones, Holy Cross
Susie Kimport, Yale University (grad)
Andrew Knightly, University of Maine
Oleg Lazarev, Princeton University (undergrad)
Jonah Leshin, Brown University (grad)
Claude Levesque, Laval University
Alon Levy, Brown University
Benjamin Linowitz, Dartmouth College (grad)
Alvaro Lozano-Robledo, University of Connecticut
Mostapha Mache, Laval University (grad)
Ken McMurdy, Ramapo College
Nathan McNew, Dartmouth College (grad)
Jeffery Merckens, University of Maine (grad)
Steven Miller, Williams College
Dawn Nelson, Bates College
Keith Ouellette, UCLA
Ali Özlük, University of Maine
Carl Pomerance, Dartmouth College
Carl Ragsdale, University of Maine (grad)
James Ricci, Wesleyan University (grad)
Juan Rivera-Letelier, Pontificia Universidad Católica de Chile
Adriana Salerno, Bates College
Jonathan Sands, University of Vermont
Andrew Schultz, Wellesley College
Joseph Silverman, Brown University
Chip Snyder, University of Maine
Ian Sprung, Brown University (grad)
Jim Stankewicz, University of Georgia (grad)
Lucien Szpiro, City University of New York
Lola Thompson, Dartmouth (grad)
Adam Towsely, University of Rochester (grad)
Thomas Tucker, University of Rochester
Bianca Viray, Brown University
Jonathan Webster, Bates College
Andrew Yang, Dartmouth College
Liyang Zhang, Williams College (undergrad)
|
