In 2007 , the
Number theory/Arithmetic Geometry Seminar meets, unless indicated otherwise, on
Wednesday at 3:30 in room 304.
Other seminars in the area on related
subjects include Emory
and Clemson.
August 22: Matt Baker (Georgia Tech),
Connections between graphs and algebraic curves.
I will discuss a
"specialization map" on a regular semistable
arithmetic surface which takes divisors on the generic fiber to divisors on the
dual graph of the special fiber. I will then state the "specialization
lemma", which says that the dimension of a linear system cannot decrease
under specialization. This result has some interesting applications, including
a generalization of Ogg's results concerning Weierstrass points on modular curves and a "Brill-Noether theorem" in tropical geometry.
August 29: Matthew Smith (UGA), The classical Hardy-Littlewood
circle method.
I will give an overview of
the classical Hardy-Littlewood circle method, with
particular emphasis on its applications to Waring's
problem on representation of large natural numbers as the sum of k-th powers of integers.
This talk will serve as a prelude to a second talk next week which also
concerns a problem which can be approached via the circle method.
September
5: Matthew Smith (UGA), On
solution-free sets for simultaneous additive equations.
In this talk I will use a
combination of the classical Hardy-Littlewood circle
method and the methods developed by Gowers in his
recent proof of Szemeredi's Theorem on long
arithmetic progressions to obtain a quantitative estimate for the upper density
of a set containing no solutions to a translation and dilation invariant system
of diagonal polynomials of degrees 1, 2,..., k.
September 12: (Postponed to 10/3) Pete Clark (UGA), Existence of abelian varieties with prescribed endomorphism algebra.
This is a continuation of the 2:30 pm algebraic geometry seminar talk: namely, we seek to show that the classification of endomorphism algebras of g-dimensional abelian varieties over Q-bar is the same as it is over the complex numbers. Because this is the arithmetic geometry seminar, we will also discuss the corresponding problem over Q, which is significantly (indeed, so far as I know, prohibitively) more difficult. If possible, we may end early and solicit remarks on the density theorem of Rumely and Moret-Bailly.
September 19: Dino
Lorenzini (UGA), Riemann-Roch zeta-functions.
We
will review the definition of the zeta-function of a curve over a finite field,
and discuss some properties of an analogous 2-variable zeta-function for finite
graphs.
September 26: Mark Watkins (Bristol),
Ranks of elliptic curves.
We discuss many questions related to ranks of elliptic curves over
Q. In particular, we review conjectures and supporting data for counting the
number of rank 2 curves in a quadratic twist family. We then give a generalisation of this to obtain a heuristic for the number
of rank 2 curves with conductor up to X. We then move to the case of odd
parity, and consider the number of rank 3 curves in a quadratic twist family,
and present two or three different heuristic approaches that lead to various
conjectures. We then present data that we have obtained via calculations with Heegner points.
Thursday
September 27 (Colloquium): Mark Watkins (Bristol), Searching for
points near curves using lattice reduction (apres Elkies)
Given a plane curve $f(x,y)=0$, can we find rational numbers $x$ and $y$ with
small denominator such that $(x,y)$ is close to the
graph of $f$? An efficient method for this was introduced by Elkies about a
decade ago. We can view the method as breaking the curve into many pieces, and
then taking a linear approximation to $f$ on each. Using the famed lattice
reduction algorithm of Lenstra, Lenstra,
and Lovasz, we can find points close to each linear
approximation. One can generalise to nonplanar curves, and to varieties of higher dimension. It
turns out that embedding our curve into a higher-dimensional projective space
is often wise, as then we can take a higher-order Taylor series as an approximation. The
conditions on the curve are not very restrictive, only involving suitable
smoothness; for instance, we could find rational points close to $x^{\pi}+y^{\pi}=1$ if desired.
It was noted by Mazur that finding points close to a curve includes the
special case of finding points on it. Here we can introduce $p$-adic methods, which are numerically more robust. These have
been used by Tom Fisher to find points on a genus 1 curve represented by 54
quadrics in $P^{11}$ which correspond to points of
height more than 600 on a elliptic curve.
October 3: Pete Clark (UGA), Existence of abelian varieties with prescribed endomorphism algebra.
This is a continuation of the 2:30 pm algebraic geometry seminar talk: namely, we seek to show that the classification of endomorphism algebras of g-dimensional abelian varieties over Q-bar is the same as it is over the complex numbers. Because this is the arithmetic geometry seminar, we will also discuss the corresponding problem over Q, which is significantly (indeed, so far as I know, prohibitively) more difficult. If possible, we may end early and solicit remarks on the density theorem of Rumely and Moret-Bailly.
October 10: No meeting this week.
October 17: Patrick Corn (UGA), Tate-Shafarevich groups of genus-2 Jacobians
Let C be a genus-2 curve over a number field k, with Jacobian J. Just as for genus-1 curves, the mysterious part of the 2-Selmer group of J is the 2-torsion in Sha(J). I'll sketch two possible constructions (the second of which is new) of examples of C for which this 2-torsion is nontrivial, using Brauer-Manin obstructions on certain K3 surfaces.
October 31:
Shuhong Gao
(Clemson University), Selected problems
in Coding theory.
The
talk will be a survey of recent developments in coding theory over finite
fields as well as over real numbers. Several open problems will be presented.
Some of the problems concern deep holes and bad centers and are related to
discrete logarithm problem for finite fields, elliptic curves, NP-completeness,
etc, while other problems are related to sampling theory, image recovery and
sensor networks, linear/convex programming, etc. These problems are suitable
research topics for projects or theses for graduate and undergraduate students.
November 7:
Ken McMurdy
(Ramapo College), Stable Reduction of Modular Curves.
This talk will begin with an introduction
to the equivalent notions of semi-stable reduction and semi-stable covering for
an algebraic curve defined over a complete subfield of C_p.
With examples, we will show how the stable reduction and associated inertia
action can in some cases be computed quite explicitly. We then turn our
attention to our current program for computing the stable reduction of the
modular curve X_0(p^n) as a
curve over Q_p. Part of our strategy involves
explicit analysis based on approximation formulas of Gross-Hopkins. Equally
fundamental to our work, however, is moduli-theoretic
reasoning. This means that we think of points of X_0(p^n) as elliptic curves with level structure, and study p-adic properties of the associated elliptic curves. Both
aspects will be discussed in some detail, as well as the current state of our
work.
November 14, Math Club Talk, 5:00-6:00, Boyd 304:
Janice Wethington (National Security Agency), Factoring Polynomials Over
Finite Fields.
The topic of polynomials over finite fields is basic to the
study of cryptography. Certainly, we would want to know when one is irreducible
or how it might factor into irreducibles over the
field of interest. This talk starts with a short review of finite fields and a
look at Stickelberger's Theorem. Then I will give
recent results by NSA mathematicians on factoring polynomials over finite
fields. This talk is designed to be accessible by undergraduate math majors.
November 28: Charles Pooh (Wolfram Research), Holonomic sequences and automated proofs of identities.
An holonomic sequence is a sequence that satisfies a linear recurrence relation with polynomial coefficients.
Numerous combinatorial sequences such as Fibonacci, binomial sequences, hypergeometric sequences are holonomic.
We will present the theoretical framework needed to prove identities for holonomic sequences
(Cassini's identity, summation identities that involve Fibonacci, factorials, binomial coefficients, ...)
December 5 : Pete Clark (UGA), Torsion Points on elliptic curves.
It has been known for almost 80 years that for any elliptic curve E over any number field K,
the group E(K) of K-rational points is abelian and finitely generated, and thus equal to the
direct sum of a free abelian group Z^r and a finite group T, the torsion subgroup.
Of the two summands, the torsion subgroup is relatively better understood: a theorem of Mazur
describes exactly which finite groups occur as the torsion subgroup of an elliptic curve over the
rational numbers, and a theorem of Merel asserts that there is a uniform bound on the size of T
depending only on the degree of the number field K. However, these results still leave open many questions,
e.g. the complete list of torsion subgroups of elliptic curves over cubic number fields is not quite known,
and much less is known about number fields of higher degree.
In this talk I will discuss some of these open problems, state a conjecture about the asymptotic
(maximum) size of the torsion subgroup as a function of the number field degree,
and explain how the computational work in progress by this year's Number Theory VIGRE Research Group is related to this conjecture.
January 24, Colloquium, Boyd 304, 3:30, Daniel Krashen (U. Penn.), The u-invariant of fields.
The u-invariant of a field is defined to be the maximal dimension of a quadratic form which has no
nontrivial zeros. Although there are some expectations for what u-invariants should be of most "naturally occurring" fields, these
invariants are unknown in a great number of situations. For example, if F is a nonreal number field, it is known that u(F) = 4, and it
is expected that the u-invariant of the rational function field F(t) should be 8. At this point, however, there is no known bound for
u(F(t)) (and no proof it is even finite).
Important progress on this type of problem was obtained by Parimala and Suresh late last year, who showed that the u-invariant of a
rational function field F(t) is 8 when F is p-adic (p odd). In this talk I will describe joint work with David Harbater and Julia Hartmann
in which we give an independent proof and a generalization of this result using the method of "field patching."
February 27, Colloquium, Carl Pomerance (Dartmouth), Euler's function
A familiar concept in elementary number theory and algebra, Euler's function at $n$ is the order of the
unit group in the ring of integers mod $n$. It is a surprisingly rich source of interesting problems, some of them still
unsolved. For example, is it always at least 2 to 1 as a mapping from the natural numbers to themselves? What
is the computational complexity of computing Euler's function? Is there an asymptotic formula for the distribution of
its range? These, and many more problems and results (some with UGA connections) will be discussed.
February 27, Math Club Talk, 5:30-6:30, Boyd 304: Carl Pomerance (Dartmouth), The covering congruences of Paul Erdos.
Note that every integer is either even or odd. That is, the residue classes 0 mod 2
(the even numbers) and 1 mod 2 (the odd numbers) cover all of the integers.
Can this be done where the moduli are all different and larger than 1? Sure, but it's harder: try 0 mod 2, 0 mod 3, 1 mod 4, 1 mod 6, and 11
mod 12. Over 50 years ago, Paul Erdos asked if one can cover with a finite collection of residue
classes with distinct moduli, where the least modulus is arbitrarily large. He later wrote that this was perhaps his favorite problem.
It's not so difficult to find examples with least modulus 3 or 4 or so, but no one knows any examples with least modulus greater than 36.
Can you find one? This talk will give an introduction to this thorny, yet accessible research problem, discussing its origins in antiquity,
some new results, and some related problems.
March 26, 27, and 28, 2008 Cantrell Lectures: Bjorn Poonen
(Berkeley),
April 9, John Voight
(University of Vermont), Quadratic forms that represent almost the
same primes.
The quadratic forms x2 + 9y2 and x2 + 12y2 represent the same
primes, namely, those of the form p = 1 (mod 12). What other such pairs
of forms exist? We give a complete answer to this question using the
tools of class field theory, proving a conjecture of Jagy and Kaplansky.
Archived seminars: Fall 2006