The NT seminar takes place on Tuesday at 4pm in MC229 (McCrea 229).

Here is a campus plan.

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.

**Sara ****Checcoli** (Grenoble)

**"On certain
arithmetic properties of Mahler functions"**

February 28 **AurĂ©lien****
****Galateau** (BesanĂ§on)
!!!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.

Spring 2016

Autumn 2015

Spring 2015

Autumn 2014