In Fall 2006, 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.
We will prove the non-existence of elliptic curves of certain prime conductors
over certain quadratic number fields by using properties about ramification in
its division fields.
The period and index of a curve are two numbers which measure the failure of a
curve to have points. Lichtenbaum in the 60's proved
divisibility conditions that the genus, period, and index must satisfy. We will
show that for genus one curves over number fields containing certain roots of
unity, all possible periods and indices occur, and consequences for the size of
Tate-Shafarevich groups of elliptic curves. This is
joint work with P. Clark.
Let G be a connected linear algebraic group defined over a field and X a
principal homogeneous space under G. There are open questions concerning
whether the existence of a zero cycle of degree one implies the existence of a
rational point on X. We discuss the case of number fields where these questions
admit a positive response and explain known results in a general context.
Elkies' proof that abc implies Faltings
theorem (that there are finitely many rational points on an algebraic curve X
of genus >1) does effectively bound the height of rational points though in
terms of the height of the Belyi map (from X to P^1)
which tends to be very large. However by running through certain (carefully
selected) families of curves we can have more control over the Belyi maps, and thus obtain more useful height bounds.
We explain how certain key results in analytic number theory can be rephrased
in terms of pretentiousness, and discuss some joint results with K. Soundararajan motivated by this new concept.
The theory of Galois groups acting on p-adic vector
spaces has an analogue where they act on locally finite, rooted trees. This is
developed particularly in the case coming from iteration of a given polynomial.
Joint work with Rafe Jones (U.
Wisconsin).
At the beginning of the XX century the following question was posed by Hilbert:
is there an algorithm that can determine whether an arbitrary polynomial
equation in several variables and with integer coefficients has integer
solutions? This problem became known as "Hilbert's Tenth Problem". In
the early 1970's, Yurii Matiyasevich,
building on the work by Martin Davis, Hilary Putnam and Julia Robinson showed
that Diophantine sets and computably enumerable sets of integers were the same
and thus showed that an algorithm sought by Hilbert did not exist. Matiyasevich's result immediately raised another question
which proved to be even more vexing: is there an algorithm as described above
but for the solutions in rational numbers? This problem is unsolved to this
day. We will discuss the current state of this problem and other problems and
conjectures which came out of attempts to solve HTP for rational numbers.
We discuss extensions of Hilbert's
Tenth Problem to rings of integers and bigger subrings
of number fields. More specifically we describe the "vertical
methods" using norm equations and elliptic curves.
In 1989 and 2000, the National Council of Teachers of Mathematics (NCTM)
published standards for school mathematics (grades PreK
- 12). These standards are widely regarded as a vision for what school
mathematics should become and have influenced most states' school mathematics
standards. In September, NCTM published Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics. The speaker
was a member of the writing team. This presentation will discuss the rationale
and background for the Focal Points, what the Focal Points are, how they were
developed, and how they should be used.
On a weighted graph G, i.e. a metrized graph, the
diagonal value of the Arakelov-Green's function g_{m}(x,y) is a constant for a
certain "canonical measure" m on G. This constant is called "the
tau constant", denoted as tau(G),
introduced by M.Baker and R.Rumely,
studied in Summer 2003 REU at UGA and related to S. Zhang's work on the
reduction of curves. There are a number of ways to describe tau(G). In terms of spectral
theory, it is the trace of the inverse of the continuous Laplacian
on G. Also, it is closely related to the resistance and the voltage functions
on G when G is considered as an electrical network. Our main focus is to show
the existence of a universal positive lower bound to tau(G)
for any G with length(G)=1. We will show how tau(G) changes under various
graph operations e.g., doubling edges, deleting and contracting edges, union of
graphs along one or two points. We will establish some identities which we call
"the deletion and the contraction identities". We will show that tau(G)
>= length(G)/12*(1-4/N)^2 , where the edge connectivity N > 4. We will
show how tau(G) is related to the discrete Laplacian
and present an effective Matlab program computing tau(G). We will present several families of graphs with
equal edge lengths and having tau constants
asymptotically approaches to length(G)/108. We will
also present an application by giving the relation between tau(G) and an another
constant that was used in one of
I hope to give a conceptual explanation of the formula for the number of ways
to write a positive integer as a sum of 4 integer squares. In addition to
philosophy, there will be lots of concrete computations with something for
everyone. In particular, we will combine tools from seemingly unrelated areas
of mathematics such as - Clock/modular arithmetic with prime numbers - Infinite
sums, products, and transcendental numbers - Calculus and the geometry of
4-dimensional space to compute the number of ways of writing 1 and 2 as a sum
of four squares. This talk should assume only undergraduate mathematics, and is
meant to be a fun talk for first-year graduate students, who want to see
something interesting in the land of number theory!
This talk will describe
several finiteness theorems for quadratic forms, and progress on the question:
"Which positive definite integer-valued quadratic forms represent all
positive integers?". The answer to this question
depends on settling the related question "Which integers are represented
by a given quadratic form?" for finitely many forms. The answer to this
question can involve both arithmetic and analytic techniques, though only
recently has the analytic approach become practical. We will describe the
theory of quadratic forms as it relates to answering these questions, its
connections with the theory of modular forms, and give an idea of how one can
obtain explicit bounds to describe which numbers are represented by a given
quadratic form. http://www.sciencenews.org/articles/20060311/bob9.asp
We will discuss the proofs of some of the key theorems that we mentioned last
week. In order to give more insight, our main focus is to discuss the ideas
that are used rather than the computations. We will also present illustrative
examples. In this talk, we will assume that the audiences attended the last
week's talk or know the basic concepts related to the tau
constant tau(G) for a metrized graph G.
We shall present a few conjectures on small points of subvarieties
of group varieties and describe their application to counting problems
(uniformity in the Mordell-Lang counting problem).
Specializing to the case of elliptic curves, we shall present results in the
direction of these conjectures.
Starting with a few
historical problems on transcendental numbers, we shall explain how the tools
developed for transcendental number theory can now be used to tackle modern
problems of arithmetic geometry.
I will attempt to summarize
a research project of van Luijk and Logan whose goal
is to exhibit examples of nontrivial 2-torsion
elements of the Tate-Shafarevich
group of the Jacobian of certain genus-2 curves. The
problem is not new, and so far the results may not be either, but the method
is--such nontrivial elements will appear whenever there is a Brauer-Manin obstruction to rational points on a certain K3
Kummer surface.