January
10 Quentin Guignard
(ENS, Paris)
"A
zetafunctionological approach to the effective Northcott
theorem"
Abstract:
Northcott's theorem asserts the finiteness of the set of
algebraic points of bounded height and degree on a projective
space. I
will survey known results on the cardinality of these finite
sets, and I
will explain how to combine the theory of height zeta functions
with the
Harder-Narasimhan-Grayson-Stuhler theory of slopes in order to
give new
results on this problem. The main novelty is a procedure
allowing to
relate the counting problems on two different projective spaces.
January 17 David Masser
(Basel)
"Relative Manin-Mumford for
commutative group varieties"
January 24
January
31 Titus Hilberdink
(Reading)
"Orders of
growth of real functions"
Abstract: In this talk we define the notion of order of a
function, which measures its growth rate
with respect to a given function. We introduce the notions of
continuity and linearity at infinity
with which we characterize order-comparability and equivalence.
Using the theory we have developed,
we apply orders of functions to give a simple and natural
criterion for the uniqueness of fractional
and continuous iterates of a function.
February 7 No seminar!
February 14 Rainer Dietmann (Royal Holloway)
"Schmidt's
subspace theorem, and systems of cubic forms"
Abstract: One of the fundamental results in Diophantine
approximation is Schmidt's subspace theorem.
Schmidt also worked on applications of the circle method to
systems of homogeneous forms, and in
particular in a series of four papers showed that any system of
$r$ rational cubic forms in at least $(10r)^5$
variables has a non-trivial rational zero. In our talk we want
to show how one can apply the subspace
theorem to obtain further progress on systems of cubic forms.
"On certain
arithmetic properties of Mahler functions"
February 28 Aurélien
Galateau (Besançon)
!!!CANCELLED!!
March 7 Fabien Pazuki (Copenhagen)
"Heights, bad
reduction and ranks of abelian varieties"
Abstract: The goal is to give new inequalities between classical
invariants of the Mordell-Weil groups of
abelian varieties. We provide an explicit comparison between the
Faltings height of an abelian variety A
defined over a number field K and the bad reduction primes of A.
Our proof is indirect, deriving the general
case from the jacobian case by a Bertini argument. As a
consequence we obtain an inequality between the
height of A and the rank of the group of rational points
A(K), with (non-optimal) explicit constants. This
approach is independent of the recent works of
Hindry-Pacheco and Wagener. If time permits, we will show
how it applies to studying a Northcott property for the
regulator of abelian varieties.
March 14 Laura Capuano (Max Planck Institute Bonn)
"Unlikely
Intersections in certain families of abelian varieties and the
polynomial Pell equation"
Abstract: In a joint work with F. Barroero (Basel), we proved
that, given n independent points on the Legendre family
of elliptic curves of equation Y^2=X(X-1)(X-c) with coordinates
algebraic over Q(c), there are at most finitely many
specializations of c such that two independent relations hold
between the n points on the specialized curve. This result
fits in the framework of the so-called Unlikely Intersections. We
will see analogues of this result in certain families of
abelian varieties and in a family of split semi-abelian varieties.
We will finally explain some applications of these results
to the study of the solvabilty of almost-Pell equations in
polynomials.
March 21 Holly Krieger (Cambridge) !!CANCELLED!!
May 26 Manfred Madritsch (Nancy) !!at 2pm in C229!!
"Forms
of differing degrees over number fields"
Abstract:
Consider a system of m forms of degree d in n variables over the
integers. The classical question on the number of integers
solutions
was solved by Birch. Using the circle method he gave an
asymptotic
formula for the number of integer solutions to this system in a
homogeneously expanding box provided n is large compared to m
and
d. An analogous result over arbitrary number fields was proved
by
Skinner, where the asymptotic formula is independent of the
degree of
the number field. In joint work with C. Frei, we combine
Skinner's
techniques with a recent generalization of Birch's theorem by
Browning
and Heath-Brown, where they allow the forms to have differing
degrees.
We discuss the main ingredients of the proof as well as
consequences
of this result to the Hasse principle, weak approximation and
Manin's
conjecture.