Forthcoming events in this series
Smooth numbers in arithmetic progressions
Abstract
A number is said to be $y$-smooth if all of its prime factors are
at most $y$. A lot of work has been done to establish the (equi)distribution
of smooth numbers in arithmetic progressions, on various ranges of $x$,$y$
and $q$ (the common difference of the progression). In this talk I will
explain some recent results on this problem. One ingredient is the use of a
majorant principle for trigonometric sums to carefully analyse a certain
contour integral.
Degree 1 L-functions and the Discrete Fourier Transform
Abstract
I will review the basic properties of the DFT and describe how these can be exploited to efficiently compute degree 1 L-functions.
New perspectives on the Breuil-Mézard conjecture
Abstract
I will discuss joint work with Matthew Emerton on geometric
approaches to the Breuil-Mézard conjecture, generalising a geometric
approach of Breuil and Mézard. I will discuss a proof of the geometric
version of the original conjecture, as well as work in progress on a
geometric version of the conjecture which does not make use of a fixed
residual representation.
Sharpening `Manin-Mumford' for certain algebraic groups of dimension 2
Abstract
(Joint work with P. Corvaja and D.
Masser.)
The topic of the talk arises from the
Manin-Mumford conjecture and its extensions, where we shall
focus on the case of (complex connected) commutative
algebraic groups $G$ of dimension $2$. The `Manin-Mumford'
context in these cases predicts finiteness for the set of
torsion points in an algebraic curve inside $G$, unless the
curve is of `special' type, i.e. a translate of an algebraic
subgroup of $G$.
In the talk we shall consider not merely the set of torsion
points, but its topological closure in $G$ (which turns out
to be also the maximal compact subgroup). In the case of
abelian varieties this closure is the whole space, but this is
not so for other $G$; actually, we shall prove that in certain
cases (where a natural dimensional condition is fulfilled) the
intersection of this larger set with a non-special curve
must still be a finite set.
We shall conclude by stating in brief some extensions of
this problem to higher dimensions.
A hyperbolic Ax-Lindemann theorem in the cocompact case
Abstract
This is a joint work with Emmanuel Ullmo.
This work is motivated by J.Pila's strategy to prove the Andre-Oort conjecture. One ingredient in the strategy is the following
conjecture:
Let S be a Shimura variety uniformised by a symmetric space X.
Let V be an algebraic subvariety of S. Maximal algebraic subvarieties of the preimage of V in X are precisely the
components of the preimages of weakly special subvarieties contained in V.
We will explain the proof of this conjecture in the case where S is compact.
Lower bounds for CM points and torsion in class groups
Abstract
Let $x$ be a CM point in the moduli space $\mathcal{A}_g(\mathbb{C})$ of principally
polarized complex abelian varieties of genus $g$, corresponding to an
Abelian variety $A$ with complex multiplication by a ring $R$. Edixhoven
conjectured that the size of the Galois orbit of x should grow at least
like a power of the discriminant ${\rm Disc}(R)$ of $R$. For $g=1$, this reduces to the
classical Brauer-Siegel theorem. A positive answer to this conjecture
would be very useful in proving the Andr\'e-Oort conjecture unconditionally.
We will present a proof of the conjectured lower bounds in some special
cases, including $g\le 6$. Along the way we derive transfer principles for
torsion in class groups of different fields which may be interesting in
their own right.
16:00
Nodal length fluctuations for arithmetic random waves
Abstract
Using the spectral multiplicities of the standard torus, we
endow the Laplace eigenspaces with Gaussian probability measures.
This induces a notion of random Gaussian eigenfunctions
on the torus ("arithmetic random waves''.) We study the
distribution of the nodal length of random Laplace eigenfunctions for high
eigenvalues,and our primary result is that the asymptotics for the variance is
non-universal, and is intimately related to the arithmetic of
lattice points lying on a circle with radius corresponding to the
energy. This work is joint with Manjunath Krishnapur and Par Kurlberg
Linear Combinations of L-functions
Abstract
If two L-functions are added together, the Euler product is destroyed.
Thus the linear combination is not an L-function, and hence we should
not expect a Riemann Hypothesis for it. This is indeed the case: Not
all the zeros of linear combinations of L-functions lie on the
critical line.
However, if the two L-functions have the same functional equation then
almost all the zeros do lie on the critical line. This is not seen
when they have different functional equations.
We will discuss these results (which are due to Bombieri and Hejhal)
during the talk, and demonstrate them using characteristic polynomials
of random unitary matrices, where similar phenomena are observed. If
the two matrices have the same determinant, almost all the zeros of
linear combinations of characteristic polynomials lie on the unit
circle, whereas if they have different determinants all the zeros lie
off the unit circle.
16:00
Unlikely intersections for algebraic curves.
Abstract
In the last twelve years there has been much study of what happens when an algebraic curve in $n$-space is intersected with two multiplicative relations $x_1^{a_1} \cdots x_n^{a_n}~=~x_1^{b_1} \cdots x_n^{b_n}~=~1 \eqno(\times)$ for $(a_1, \ldots ,a_n),(b_1,\ldots, b_n)$ linearly independent in ${\bf Z}^n$. Usually the intersection with the union of all $(\times)$ is at most finite, at least in zero characteristic. In Oxford nearly three years ago I could treat a special curve in positive characteristic. Since then there have been a number of advances, even for additive relations $\alpha_1x_1+\cdots+\alpha_nx_n~=~\beta_1x_1+\cdots+\beta_nx_n~=~0 \eqno(+)$ provided some extra structure of Drinfeld type is supplied. After reviewing the zero characteristic situation, I will describe recent work, some with Dale Brownawell, for $(\times)$ and for $(+)$ with Frobenius Modules and Carlitz Modules.
Class invariants for quartic CM-fields
Abstract
I show how invariants of curves of genus 2 can be used for explicitly constructing class fields of
certain number fields of degree 4.
Iwasawa theory for modular forms
Abstract
he Iwasawa theory of elliptic curves over the rationals, and more
generally of modular forms, has mostly been studied with the
assumption that the form is "ordinary" at p -- i.e. that the Hecke
eigenvalue is a p-adic unit. When this is the case, the dual of the
p-Selmer group over the cyclotomic tower is a torsion module over the
Iwasawa algebra, and it is known in most cases (by work of Kato and
Skinner-Urban) that the characteristic ideal of this module is
generated by the p-adic L-function of the modular form.
I'll talk about the supersingular (good non-ordinary) case, where
things are slightly more complicated: the dual Selmer group has
positive rank, so its characteristic ideal is zero; and the p-adic
L-function is unbounded and hence doesn't lie in the Iwasawa algebra.
Under the rather restrictive hypothesis that the Hecke eigenvalue is
actually zero, Kobayashi, Pollack and Lei have shown how to decompose
the L-function as a linear combination of Iwasawa functions and
explicit "logarithm-like" series, and to modify the definition of the
Selmer group correspondingly, in order to formulate a main conjecture
(and prove one inclusion). I will describe joint work with Antonio Lei
and Sarah Zerbes where we extend this to general supersingular modular
forms, using methods from p-adic Hodge theory. Our work also gives
rise to new phenomena in the ordinary case: a somewhat mysterious
second Selmer group and L-function, which is related to the
"critical-slope L-function" studied by Pollack-Stevens and Bellaiche.
Ribet points on semi-abelian varieties : a nest for counterexamples
Abstract
The points in question can be found on any semi-abelian surface over an
elliptic curve with complex multiplication. We will show that they provide
counter-examples to natural expectations in a variety of fields : Galois
representations (following K. Ribet's initial study from the 80's),
Lehmer's problem on heights, and more recently, the relative analogue of
the Manin-Mumford conjecture. However, they do support Pink's general
conjecture on special subvarieties of mixed Shimura varieties.
Reductions of local Galois representations arising from Hilbert modular forms
Applications of nilsequences to number theory
Abstract
I will introduce the notion of a nilsequence, which is a kind of
"higher" analogue of the exponentials used in classical Fourier analysis. I
will summarise the current state of our understanding of these objects. Then
I will discuss a variety of applications: to solving linear equations in
primes (joint with T. Tao), to a version of Waring's problem for so-called
generalised polynomials (joint with V. Neale and Trevor Wooley) and to
solving certain pairs of diagonal quadratic equations in eight variables
(joint work with L. Matthiesen). Some of the work to be described is a
little preliminary!
Constructing Abelian Varieties over $\overline{\mbthbb{Q}}$ Not Isogenous to a Jacobian
Abstract
We discuss the following question of Nick Katz and Frans Oort: Given an
Algebraically closed field K , is there an Abelian variety over K of
dimension g which is not isogenous to a Jacobian? For K the complex
numbers
its easy to see that the answer is yes for g>3 using measure theory, but
over a countable field like $\overline{\mbthbb{Q}}$ new methods are required. Building on
work
of Chai-Oort, we show that, as expected, such Abelian varieties exist for
$K=\overline{\mbthbb{Q}}$ and g>3 . We will explain the proof as well as its connection to
the
Andre Oort conjecture.
On Nahm's conjecture
Abstract
We consider certain q-series depending on parameters (A,B,C), where A is
a positive definite r times r matrix, B is a r-vector and C is a scalar,
and ask when these q-series are modular forms. Werner Nahm (DIAS) has
formulated a partial answer to this question: he conjectured a criterion
for which A's can occur, in terms of torsion in the Bloch group. For the
case r=1, the conjecture has been show to hold by Don Zagier (MPIM and
CdF). For r=2, Masha Vlasenko (MPIM) has recently found a
counterexample. In this talk we'll discuss various aspects of Nahm's conjecture.