Royal Holloway - Number Theory Seminar

The NT seminar takes place on Tuesday at 4pm in MC229 (McCrea 229).
Here is a campus plan.

Spring Term 2017

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.

February 21 !!Bourne Lecture Theatre 1!! Sara Checcoli (Grenoble)

"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