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Number Theory
Spring 2008 Schedule:
| TITLE: |
Integer solutions to decomposable form
inequalities |
| SPEAKER: |
Min Ru (University of Houston) |
| DATE: |
February 14, 2008 |
| TIME: |
2:00 - 3:00 p.m. |
| PLACE: |
GAB 473 |
| FOOD: |
Cookies, coffee and tea served outside the
office at 3:30 p.m. |
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Abstract:
In this talk, I will report some recent
progress on the finiteness of integer solutions to
decomposable homogeneous polynomial equations and
inqualities, using the tools in the Diophantine
approximations. |
Future
Number Theory Talks
Archive
November 8, 2007
Matt Papanikolas (Texas A&M)
Frobenius difference equations and applications
We will focus on Frobenius difference
equations over function fields of characteristic p,
whose solutions are related to periods of Drinfeld
modules. More specifically we show that the
transcendence degree of the period matrix of a Drinfeld
module is equal to the dimension of its associated
difference Galois group. We will discuss applications to
various transcendence problems over function fields.
September 27, 2008
Speaker: Aaron Levin (Brown and Scuola Normale Superiore
di Pisa)
Title: Ideal Class Groups and Rational Torsion in Jacobians
of Curves
Abstract: We study the problem of constructing and
enumerating, for any integers m, n> 1, number fields of degree n
whose ideal class groups have "large" m-rank. Our technique,
which appears to be new, relies on the Hilbert Irreducibility
Theorem and finding certain curves whose Jacobians have a large
rational torsion subgroup. Using this technique we improve on
results of Nakano, Bilu-Luca, and others.
September 13, 2007
Speaker: Dale Henderson
Title: Zagier's paper on algebraic numbers close to both 0
and 1
Abstract: A long-standing conjecture of Lehmer implies that
if K is a number field and x is not a root of unity in K\{0},
then the relative logarithmic height h_K(x) >=
log(alpha_0)=0.1623, where alpha_0 is a root of a particular
10th degree polynomial. If one considers instead the absolute
height h(x) and allows x to vary over all algebraic numbers, it
is easy to see that one can find a sequence x_n with h(x_n)->0.
However, a consequence of Zhang's equidistribution theorems for
points of small height in semi-Abelian varieties is that there
exists a positive constant C such that h(x)+h(1-x)>=C>0 for all
but the four algebraic points where either x or 1-x is zero or
both x and 1-x are roots of unity. On can view this as a
statement that these are the only four algebraic points
simultaneously "close to" 0 and 1 at all places.
In my talk, I will explain an elementary argument of Zagier
to prove this theorem and to compute the optimal constant C. My
talk will not discuss Zhang's work or semi-Abelian varieties, so
will be accessible to a general audience.
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