Forthcoming events in this series
Analytic p-adic L-functions
Abstract
I'll sketch a construction which associates a canonical p-adic L-function with a 'non-critically refined' cohomological cuspidal automorphic representation of GL(2) over an arbitrary number field F, generalizing and unifying previous results of many authors. These p-adic L-functions have good interpolation and growth properties, and they vary analytically over eigenvarieties. When F=Q this reduces to a construction of Pollack and Stevens. I'll also explain where this fits in the general picture of Iwasawa theory, and I'll point towards the iceberg of which this construction is the tip.
Construction of p-adic L-functions for unitary groups
Abstract
This is a report on joint work (still in progress) with Ellen Eischen, Jian-Shu Li,
and Chris Skinner. I will describe the general structure of our construction of p-adic L-functions
attached to families of ordinary holomorphic modular forms on Shimura varieties attached to
unitary groups. The complex L-function is studied by means of the doubling method;
its p-adic interpolation applies adelic representation theory to Ellen Eischen's Eisenstein
measure.
Construction of 2-adic integral canonical models of Hodge-type Shimura varieties
Abstract
We extend Kisin's construction of integral canonical models of Hodge-type Shimura
varieties to p=2, using Dieudonné display theory.
Effective Ratner Theorem for $ASL(2, R)$ and the gaps of the sequence $\sqrt n$ modulo 1
Abstract
Let $G=SL(2,\R)\ltimes R^2$ and $\Gamma=SL(2,Z)\ltimes Z^2$. Building on recent work of Strombergsson we prove a rate of equidistribution for the orbits of a certain 1-dimensional unipotent flow of $\Gamma\G$, which projects to a closed horocycle in the unit tangent bundle to the modular surface. We use this to answer a question of Elkies and McMullen by making effective the convergence of the gap distribution of $\sqrt n$ mod 1.
Arithmetic of abelian varieties over function fields and an application to anabelian geometry.
Abstract
We investigate certain (hopefully new) arithmetic aspects of abelian varieties defined over function fields of curves over finitely generated fields. One of the key ingredients in our investigation is a new specialisation theorem a la N\'eron for the first Galois cohomology group with values in the Tate module, which generalises N\'eron specialisation theorem for rational points. Also, among other things, we introduce a discrete version of Selmer groups, which are finitely generated abelian groups. We also discuss an application of our investigation to anabelian geometry (joint work with Akio Tamagawa).
Isogeny classes of abelian varieties and weakly special subvarieties
Abstract
For Logic Seminar: Note change of time and place.
From quadratic polynomials and continued fractions to modular forms
Abstract
For Logic Seminar: Note change of time and location!
Covering systems of congruences
Abstract
A distinct covering system of congruences is a collection
\[
(a_i \bmod m_i), \qquad 1\ \textless\ m_1\ \textless\ m_2\ \textless\ \ldots\ \textless\ m_k
\]
whose union is the integers. Erd\"os asked whether there are covering systems for which $m_1$ is arbitrarily large. I will describe my negative answer to this problem, which involves the Lov\'{a}sz Local Lemma and the theory of smooth numbers.
Hartmanis-Stearns conjecture and Mahler's method
Abstract
Note: Change of time and (for Logic) place! Joint with Number Theory (double header)
An Euler system of diagonal cycles and the Birch and Swinnerton-Dyer conjecture for non-abelian twists of elliptic curves.
Abstract
The goal of this lecture is describing recent joint work with Henri Darmon, in which we construct an Euler system of twisted Gross-Kudla diagonal cycles that allows us to prove, among other results, the following statement (under a mild non-vanishing hypothesis that we shall make explicit):
Let $E/\mathbb{Q}$ be an elliptic curve and $K=\mathbb{Q}(\sqrt{D})$ be a real quadratic field. Let $\psi: \mathrm{Gal}(H/K) \rightarrow \mathbb{C}^\times$ be an anticyclotomic character. If $L(E/K,\psi,1)\ne 0$ then the $\psi$-isotypic component of the Mordell-Weil group $E(H)$ is trivial.
Such a result was known to be a consequence of the conjectures on Stark-Heegner points that Darmon formulated at the turn of the century. While these conjectures still remain highly open, our proof is unconditional and makes no use of this theory.
Modular forms, Eisenstein series and the ternary divisor function
Abstract
After a short survey of the notion of level of distribution for
arithmetic functions, and its importance in analytic number theory, we
will explain how our recent studies of twists of Fourier coefficients of
modular forms (and especially Eisenstein series) by "trace functions"
lead to an improvement of the results of Friedlander-Iwaniec and
Heath-Brown for the ternary divisor function in arithmetic progressions
to prime moduli.
This is joint work with É. Fouvry and Ph. Michel.
Elliptic Curves over Real Quadratic Fields are Modular.
Abstract
We combine recent breakthroughs in modularity lifting with a
3-5-7 modularity switching argument to deduce modularity of elliptic curves over real
quadratic fields. We
discuss the implications for the Fermat equation. In particular we
show that if d is congruent
to 3 modulo 8, or congruent to 6 or 10 modulo 16, and $K=Q(\sqrt{d})$
then there is an
effectively computable constant B depending on K, such that if p>B is prime,
and $a^p+b^p+c^p=0$ with a,b,c in K, then abc=0. This is based on joint work with Nuno Freitas (Bayreuth) and Bao Le Hung (Harvard).
16:00
Special numbers and special functions related to Ramanujan's mock modular forms
Abstract
This lecture will cover two recent works on the mock modular
forms of Ramanujan.
I. Solution of Ramanujan's original conjectures about these functions.
(Joint work with Folsom and Rhoades)
II. A new theorem that mock modular forms are "generating functions" for
central L-values and derivatives of quadratic twist L-functions.
(Joint work with Alfes, Griffin, Rolen).
Manin's conjecture for certain smooth hypersurfaces in biprojective space
Abstract
So far, the circle method has been a very useful tool to prove
many cases of Manin's conjecture. Work of B. Birch back in 1961 establishes
this for smooth complete intersections in projective space as soon as the
number of variables is large enough depending on the degree and number of
equations. In this talk we are interested in subvarieties of biprojective
space. There is not much known so far, unless the underlying polynomials are
of bidegree (1,1). In this talk we present recent work which combines the
circle method with the generalised hyperbola method developed by V. Blomer
and J. Bruedern. This allows us to verify Manin's conjecture for certain
smooth hypersurfaces in biprojective space of general bidegree.
On translation invariant quadratic forms
Abstract
Solutions to translation invariant linear forms in dense sets (for example: k-term arithmetic progressions), have been studied extensively in additive combinatorics and number theory. Finding solutions to translation invariant quadratic forms is a natural generalization and at the same time a simple instance of the hard general problem of solving diophantine equations in unstructured sets. In this talk I will explain how to modify the classical circle method approach to obtain quantitative results for quadratic forms with at least 17 variables.
Some structure of character sums
Abstract
I'll discuss questions about the structure of long sums of
Dirichlet characters --- that is, sums of length comparable to the modulus.
For example: How often do character sums get large? Where do character sums
get large? What do character sums "look like" when then get large? This will
include some combination of theorems and experimental data.