Graduate Student Poster Session
Graduate students (and very recently graduated young researchers) will be presenting posters on Wednesday afternoon. Poster submissions are now closed, but you may email silvermania2015@gmail.com for late submissions.
Poster Titles and Abstracts
A Fibered Power Theorem
for Pairs of Log General Type
A theorem of
Caporaso-Harris-Mazur states that given a family of curves with
general fiber a smooth curve of genus greater than or equal to 2
(i.e. a curve of general type), some high fibered power of the family
admits a dominant rational map to a variety of general type. This was
generalized to families of surfaces of general type by Hassett, and to
families of arbitrary dimensional varieties of general type by
Abramovich. Assuming various versions of Lang's Conjecture, these
results lead to uniform bounds on the number of rational points of
varieties of general type. Using the moduli space of stable pairs, and
recent work of Kovács-Patakfalvi, we prove an analogous fibered power
theorem in the case of pairs of log general type.
Local
arithmetic for hyperelliptic curves of genus 2
In the context
of proving the 2-parity conjecture for jacobians of hyperelliptic
curves of genus 2 admitting a Richelot isogeny, we present the
computation of local arithmetic invariants of both the curve and its
jacobian. In particular, computation of invariants such as the
Tamagawa numbers, the number of real connected components and the
deficiency will be detailed.
Arithmetic of Farey-Ford Packings
The Farey sequence is a natural exhaustion of the set of rational
numbers between 0 and 1 by finite lists. Ford Circles are a natural
family of mutually tangent circles associated to Farey fractions: they
are an important object of study in the geometry of numbers and
hyperbolic geometry. We compute the distributions and moments of
certain statistics of geometric quantities such as (but not limited
to) areas, length, slopes, and angles associated to Ford circles in a
Farey-Ford Packing.
Local to
global principles for Galois representations
Let \(K\) be a
number field and let \(E/K\) be an elliptic curve whose mod \(\ell\)
Galois representation locally has image contained in a group \(G\), up
to conjugacy. We classify the possible images for the global Galois
representation in the case where \(G\) is a Cartan subgroup or the
normalizer of a Cartan subgroup. When \(K = \mathbf{Q}\), we deduce a
counterexample to the local-global principle in the case where \(G\)
is the normalizer of a split Cartan and \(\ell = 13\). In particular,
there are at least three elliptic curves (up to twist) over
\(\mathbf{Q}\) whose mod \(13\) image of Galois is locally contained
in the normalizer of a split Cartan, but whose global image is
not.
Integers that can be written
as the sum of two cubes
The Birch and Swinnerton-Dyer
conjecture predicts that we have non-torsion rational points on an
elliptic curve iff the \(L\)-function corresponding to the elliptic
curve vanishes at 1. Thus BSD predicts that a positive integer \(N\)
is the sum of two cubes if \(L(E_N, 1)=0\), where \(L(E_N, s)\) is the
\(L\)-function corresponding to the elliptic curve \(E_N: x^3+y^3=N\).
We have computed several formulas that relate \(L(E_N, 1)\) to the
trace of a modular function at a CM point. This offers a criterion for
when the integer \(N\) is the sum of two cubes. Furthermore, when
\(L(E_N, 1)\) is nonzero we get a formula for the number of elements
in the Tate-Shafarevich group.
Combinatorial
Properties of Rogers-Ramanujan-Type Identities Arising from
Hall-Littlewood Polynomials
Here we consider the \(q\)-series
coming from the Hall-Littlewood polynomials, defined by Griffin, Ono,
and Warnaar in their work on the framework of the Rogers-Ramanujan
identities. We devise a recursive method for computing the
coefficients of these series when they arise within the
Rogers-Ramanujan framework. Furthermore, we study the congruence
properties of certain quotients and products of these series,
generalizing the famous Ramanujan congruences.
Discriminants
of iterated quadratic extensions
Let \(f(x) = x^2+c \in
\mathbb Z[x]\), and let \(K\) be a number field generated by a root of
\(f^n(x)\) (assuming \(f^n(x)\) is irreducible). The purpose of this
work is to determine the multiplicities of primes dividing the
discriminant of \(K\). As a consequence of our result, we identify a
sufficient condition for \(K\) to be monogenic. Namely, \(K\) is
monogenic if \(f(0)\), \(f^2(0)\), \(f^3(0),\ldots, f^n(0)\) are all
square-free.
Hypergeometric Point Counts for Dwork K3-Surfaces
In 1995, Koike showed that the trace of Frobenius for elliptic
curves in the Legendre family can be expressed in terms of Greene's finite
field hypergeometric series. Further connections between hypergeometric
series and algebraic varieties have been studied since then, though the
focus has largely been on elliptic curves and Calabi-Yau threefolds. We
extend this work by showing that the number of points on the family of Dwork
K3-surfaces over finite fields can be expressed in terms of Greene's finite
field hypergeometric series.
A Brauer-Siegel theorem for elliptic curves over
function fields
The Brauer-Siegel theorem bounds the product of
the regulator and the class number of a number field in terms of its
discriminant. It can be seen as an measure of the "arithmetic
complexity" of this number field. Now consider an elliptic curve
\(E\) defined over a global field, assuming its Tate-Shafarevich group
is finite, one can form the product of the order of this group and the
Néron-Tate regulator of \(E\). Heuristically, this product
measures the complexity of computing the Mordell-Weil group of
\(E\). This prompts the question of bouding this quantity in terms of
simpler invariants of \(E\), e.g. its height. Unfortunately, it seems
unlikely that a perfect analogue of the classical Brauer-Siegel
theorem holds in this setting. This poster describes my
investigations on the behaviour of this "Brauer-Siegel ratio" for a
family of twists of an elliptic curve over a function field in
positive characteristic.
Embedding Curves into Surfaces Over Finite
Fields
When can a curve, possibly singular, be embedded into
some smooth surface, defined over the same field? This was answered
elegantly by Altman and Kleiman in the 1970s for any infinite perfect
field. We extend their result to curves defined over finite fields.
The methods involved are quite different; the key tool used is
Poonen's geometric closed-point sieve.
p-adic equidistribution of Hecke points
Let \(x\) be a point in the modular curve of level one \(X_0(1)\) over
the complex numbers such that \(x\) is not a cusp. A well known result
states that the Hecke points \(T_n(x)\) associated to \(x\) become
equidistributed, with respect to the hyperbolic measure, as \(n\) goes
to infinity. In the case of the modular curve over \(\mathbb{C}_p\)
the situation is rather different. In most cases we have
equidistribution of \(T_n(x)\) to the canonical point of a certain
Berkovich space associated to \(X_0(1)\). The type of reduction of
\(x\) mod \(p\) plays an important role in this setting. This is joint
work with Ricardo Menares and Juan Rivera Letelier.
On the arithmetic
restricted volumes and arithmetic base loci
We study
fundamental properties of the arithmetic restricted volumes and the
arithmetic multiplicities of the adelically metrized line bundles. The
arithmetic restricted volumes have the concavity property and
characterize the arithmetic augmented base loci as the null loci. We
also show a generalized Fujita approximation for the arithmetic
restricted volumes.
A Logarithmic
Equidistribution Result in Non-Archimedean Dynamics
Let \(K\)
be a complete, algebraically closed, non-Archimedean valued field, and
let \(\phi\) be a rational map of degree \(d\gt1\) defined over
\(K\). Rumely has defined a discrete probability measure, called the
crucial measure, supported on the interior of the Berkovich line over
\(K\) which carries information about the reduction of \(\phi\). In
this poster we show that the crucial measures attached to the iterates
of \(\phi\) converge weakly to the equilibrium measure \(\mu_\phi\)
when integrated against functions with logarithmic singularities.
Perfect Powers in products of Elliptic Divisibility
Sequences
In a joint work with L. Hajdu and M. Szikszai, we
show that there are only finitely many explicitly computable solutions
for the equation given by the products of terms of Elliptic
Divisibility Sequences with indices in arithmetic progression being a
perfect power. Our method also gives a way to solve such equations
completely and we show it by some examples.
J-Stability in
non-Archimedean dynamics
Roughly speaking, \(J\)-stability
means that the dynamical systems on the Julia sets of two given
rational maps are dynamically equivalent if those two rational maps
are close enough. This notion was introduced by R. Mane, P. Sad, and
D. Sullivan in complex dynamics. They also proved a \(J\)-stability
theorem, which states that a rational map is \(J\)-stable if it has a
neighborhood in the set of rational maps on which the number of
attracting cycles is constant. In this talk, we will see an analogue
of R. Mane, P. Sad, and D. Sullivan's \(J\)-stability theorem of
immediately expanding rational maps in non-Archimedean dynamics.
Polynomial
analogues of Zaremba's and McMullen's conjectures
Let \(L\) be
the field of Laurent series over a field \(K\); as shown by Artin, an
analogue of the theory of continued fractions can be developed over
\(L\). We will consider the analogues of Zaremba’s and McMullen’s
conjectures. The former states that for any polynomial \(f\) there
exists a polynomial \(g\), relatively prime to \(f\), such that all
the partial quotients of \(f/g\) have degree 1. The second one states
that for any polynomial \(D\) whose square root is well defined there
exists \(r\) in \(K(T,\sqrt D)\) such that all the partial quotients
of \(r\) have degree 1. We have that Zaremba’s conjecture holds if
\(K\) is infinite and that McMullen’s holds if \(K\) is uncountable
and if \(K\) is the algebraic closure of \(\mathbb{Q}\). Moreover, if
Zaremba’s conjecture holds over finite fields, then McMullen’s
conjecture holds over finite fields and over \(\mathbb{Q}\).
Diophantine
Sets in Global Fields
We give a new proof that the set of
non-squares in a global field are diophantine, which was originally
shown by Poonen. We also show that the set of elements which are not
norms from a quadratic extension are diophantine. This is based on
work by Koenigsmann and Park.
p-adic Hubbard trees
In complex dynamics, Hubbard trees offer a combinatorial
description of the dynamics of post-critically finite (PCF) polynomials.
What are the analogous objects in a non-Archimedean setting: what is a
p-adic Hubbard tree? We begin to explore this question by studying the
critical orbit trees associated to quadratic maps \(f_c(z)=z^2+c\), with
\(c\in\mathbb{Z}_p\) (for \(p>2\)).
Algebraic analog of p-adic factorization
The
classical main conjecture in Iwasawa theory (formulated over a regular
local ring) predicts that the characteristic ideal of a Selmer group
is generated by a \(p\)-adic \(L\)-function. The main conjecture of
Iwasawa theory formulated by Ralph Greenberg predicts a precise
relationship between Selmer groups and \(p\)-adic \(L\)-functions but
generalized to arbitrary normal domains that appear as deformation
rings. We wish to exhibit algebraic analog of a phenomenon involving
factorization of \(p\)-adic \(L\)-functions that occurs on the
analytic side. As an example, results of Dasgupta involve factoring a
certain three-variable \(p\)-adic \(L\)-function (constructed by Hida)
into a product of two \(p\)-adic \(L\)-functions (constructed by
Coates-Schmidt and Kubota-Leopoldt respectively) along a certain
plane. We wish to exhibit the corresponding result involving Selmer
groups that is consistent with the main conjecture.
Some
surfaces with non-polyhedral nef cones
We present a criterion
for the cone of curves of a complex, smooth, projective surface to be
non-polyhedral. In particular, we use this to prove that the cone of
curves of \(C \times C\) where \(C\) is a curve of genus \(\ge2\) is
not polyhedral. The proof technique uses a (modified) construction of
nef classes due to Vojta.
Strengthening NTRU against
message recovery attacks
There are two basic attacks on the
NTRU cryptosystem: 1. ciphertext decryption attack (using lattice
reduction to recover the plaintext from the ciphertext), and 2. key
recovery attack (using lattice reduction to recover the secret key
from the public key). In the most basic form, the complexity of the
second attack is about the square of the first attack. We propose a
twist in NTRU to increase the complexity of the message recovery
attacks to be the same as that of the key recovery attack.
Commuting
Endomorphisms of the \(p\)-adic Unit Disk
When can a pair of
endomorphisms of \(\mathbf{Z}_p[[X]]/\mathbf{Z}_p\) commute?
Approaching this problem from the vantage point of dynamics on the
\(p\)-adic unit disk, Lubin proved that whenever a non-invertible
endomorphism \(f\) commutes with a non-torsion automorphism \(u\), the
pair \(f\) and \(u\) exhibit many of the same properties as
endomorphisms of a formal group over \(\mathbf{Z}_p.\) Because of
this, he posited that for such a pair of endomorphisms to exist, there
in fact had to be a formal group 'somehow in the background.' My
poster discusses how some of the dynamical systems of Lubin occur
naturally as the restriction of the Galois action on certain Fontaine
period rings and how one can use this observation to construct, in
some cases, the formal groups conjectured by Lubin.
Conductors and
discriminants for hyperelliptic curves
The Ogg-Saito formula
relates the conductor of the minimal proper regular model of an
elliptic curve over a local field (which is a certain numerical
invariant that can be computed from the Galois action on étale
cohomology groups) with the valuation of the discriminant of a minimal
Weierstrass equation for the curve. The definition of the conductor
naturally extends to models of curves of higher genus as well. Deligne
defined a certain discriminant attached to regular models of curves,
and Saito showed that the conductor equals the discriminant in this
setting as well. Upper bounds on the conductor of a regular model can
be used to give upper bounds on the number of components of the
special fiber of the model, and this has applications to the study of
rational points (Chabauty's method). We show that Deligne's
discriminant for the minimal proper regular model of a hyperelliptic
curve with rational Weierstrass points, over a local field with
perfect residue field (and residue characteristic either zero or large
enough compared to the genus) is bounded above by the valuation of the
discriminant of an integral Weierstrass equation for the curve.
On the
distribution of zeros of the derivatives of the Riemann zeta function
and Dirichlet \(L\)-functions
The number of zeros and the
distribution of the real part of non-real zeros of the derivatives of
the Riemann zeta function have been investigated by Berndt, Levinson,
Montgomery, and Akatsuka. Berndt, Levinson, and Montgomery
investigated the general case, meanwhile Akatsuka gave sharper
estimates for the first derivative of the Riemann zeta function under
the truth of the Riemann hypothesis. Many properties of the zeros of
the derivatives of the Dirichlet \(L\)-functions associated with
primitive Dirichlet characters were studied by Yildirim. Among,
Yildirim also studied the number of zeros. In this talk, we introduce
a generalization of the results of Akatsuka to the \(k\)-th derivative
(for positive integer \(k\)) of the Riemann zeta function. We also
give a sharper estimate to the result of Yildirim on the number of
zeros for the first derivative of the Dirichlet \(L\)-functions
associated with primitive Dirichlet characters under the assumption of
the generalized Riemann hypothesis.
The Brauer
Manin Obstruction for Cubic Surfaces
"The Hasse principle asks
whether solutions to an equation in a local field extend to those in a
global field. This does not always happen, with the Brauer-Manin
obstruction being a common explanation. It is conjectured by
Colliot-Thelene and Sansuc that for cubic surfaces the Brauer-Manin
obstruction explains every instance where the Hasse principle fails.
In 1973, Birch and Swinnerton-Dyer gave some of the first examples of
the failure of the Hasse princple for cubic surfaces
\[m\operatorname{Norm}_{L/\mathbb{Q}}(ax+by+\phi z+\psi w) =
(cx+dy)\operatorname{Norm}_{k/\mathbb{Q}}(x+\theta y),\] for given
fields \(L\) and \(k\) over \(\mathbb{Q}\) and fixed constants \(m,\
a,\ b,\ c,\ d,\ \phi,\ \psi,\) and \(\theta\). They make a rough
number theoretic argument for the Brauer Manin obstruction in the case
that the Hasse principle fails, focusing on the particular fields and
constants. We make use of advancements in geometry and class field
theory, taking a geometric look at this object and utilizing the
correspondence between the Brauer group and the Picard group of a
surface. "
A function-field
analogue of Conway's topograph
Conway's topograph is a visual
method to display values of an integral binary quadratic form over
\(\mathbb{Z}\). This method leads to a simple and elegant method of
classifying all integral binary quadratic forms and answering some
basic questions about them. Let \(\mathbb{F}_{q}\) be a finite field
with odd characteristic, \(A = \mathbb{F}_{q}[T]\), and \(\widehat{K}
= \mathbb{F}_{q}((T^{-1}))\). We develop an analogue of Conway's
topograph for binary quadratic forms over \(A\) by exploiting the
connection between Conway's topograph and hyperbolic geometry.
Following Paulin, we use the Bruhat–Tits tree
\(\mathcal{T}_{q+1}\) of \(\operatorname{SL}_{2}(\widehat{K})\) as our
function-field analogue of the hyperbolic plane. After we recast the
underlying infrastructure of Conway's topograph in terms of
constructions on \(\mathcal{T}_{q+1}\), we formulate and prove an
analogue of Conway's climbing lemma. We then show that just as in the
classical setting, there is a unique "well" (respectively, "river") on
the topograph of any definite (respectively, indefinite) binary
quadratic form over \(A\).
Zagier's
sums of powers of quadratic polynomials when the discriminants are
negative
Zagier studied some functions defined as sums of
powers of quadratic polynomials with integer coefficients and
discovered that these functions have several surprising properties and
are related to many other subjects, including modular forms of weight
\(2k\) and special values of zeta functions. Zagier mentions that his
definition does not work well and/or becomes unnatural when \(k\) is
odd. We redefine Zagier's sums by changing the summation
condition. That allows us to consider splittings of positive
discriminants of the quadratic forms under summation into products of
two (positive or negative) discriminants. Finally, our sums, while
essentially coincide with those of Zagier in the case when \(k\) is
even, allow us to cover in a similar way the case when \(k\) is
odd.
Sums of zeroes in
the Arthur-Selberg Trace Formula
The Arthur-Selberg trace
formula for a noncompact reductive group involves logarithmic
derivatives of automorphic \(L\)-functions, which may be rewritten as
certain sums of zeroes of \(L\)-functions. This suggests studying the
distribution of zeroes using the trace formula. I will describe this
in the simplest case, which is \(\operatorname{GL}(2)\).
Generalization of
Voronoi Formula and a New Proof for \(\operatorname{GL}(3)\)
We discover new Voronoi formulae for automorphic forms on
\(\operatorname{GL}(n)\) for \(n\ge4\). There are \([n/2]\) different
Voronoi formulae on \(\operatorname{GL}(n)\), which are Poisson
summation formulae weighted by Fourier coefficients of the automorphic
form with twists by some hyper-Kloosterman sums.
A generalized
conjecture of ternary pure exponential Diophantine equation in ring of
integers
Let \(a,b,c\) be positive integers. In 1933,
K. Mahler used his \(p\)-adic analogue of the Thue-Siegel method to
prove that the ternary pure exponential Diophantine equation
\(a^{x}+b^{y}=c^{z}\) has only finitely many solutions
\((x,y,z)\). His method is ineffective in the sense that it gives no
indication on the number of possible solutions. In 1940, an effective
result for the number of the solutions was given by A. O. Gel'fond. In
1956, L. Jesmanowicz conjectured that if \( a^{2}+b^{2}=c^{2}\), then
the Diophantine equation \(a^{x}+b^{y}=c^{z}\) has only one integer
solution \((x,y,z)=(2,2,2)\). In 1994, N. Terai conjectured that
\(a^{x}+b^{y}=c^{z}\) always has at most one positive integer solution
\((x,y,z)\), especially when \(a^{p}+b^{q}=c^{r}\), \( a,b,c,p,q,r\gt
1\) are fixed integers, \(a^{x}+b^{y}=c^{z}\) has exactly one positive
integral solution \((x,y,z)=(p,q,r)\). L. J. Alex pointed that
Terai's conjecture is clearly false. For example the equation \(
2^{x}+2^{y}=2^{z}\) has infinitely many solutions of the form
\((x,y,z)=(k,k,k+1)\), \(k\gt1\). The condition \(\gcd(a,b)=1\) should
be added to the hypotheses of the conjecture. In 1999, simple
counterexamples to this statement have been found by Z.-F. Cao, who
suggested that the condition \(\max\{a,b,c\}\gt7\) should be added to
the hypotheses of Terai's conjecture. However, it turns out that this
condition is not sufficient to ensure the correctness. In 2003, a
family of counterexamples has been found by M.-H. Le, that is,
\((2^{n}-1)^{x}+2^{y}=(2^{n}+1)^{z}\) has two positive integer
solutions \((x,y,z)=(1,1,1)\) and \((2,n+1,2)\). Now the accepted
Terai conjecture is the following: for any fixed and coprime integers
\(a\) ,\(b\) and \(c\) greater than \(1\), \(a^{x}+b^{y}=c^{z}\) has
at most one solution \((x,y,z)\) with \(\min\{x,y,z\}\gt1\). In 2006,
N. Hirata-Kohno proved \(a^{x}+b^{y}=c^{z}\) has at most \(2^{32}\)
positive integer solutions. We will give a generalized conjecture on
ternary pure exponential Diophantine equation \(a^{x}+b^{y}=c^{z}\) in
ring of integers, that is to say, if we allow \(x,y,z\) to be negative
or zero or positive integers and \(a,b,c\) to be coprime or not
coprime, we will describe an upper bound for the number of solutions
of \(a^{x}+b^{y}=c^{z}\) according to discussion in some cases.