Forthcoming events in this series


Thu, 02 May 2013

17:00 - 18:00
SR2

The p-adic monodromy group of abelian varieties over global function fields of characteristic p

Ambrus Pal
(Imperial College)
Abstract

We prove an analogue of the Tate isogeny conjecture and the
semi-simplicity conjecture for overconvergent crystalline Dieudonne modules
of abelian varieties defined over global function fields of characteristic
p, combining methods of de Jong and Faltings. As a corollary we deduce that
the monodromy groups of such overconvergent crystalline Dieudonne modules
are reductive, and after base change to the field of complex numbers they
are the same as the monodromy groups of Galois representations on the
corresponding l-adic Tate modules, for l different from p.

Thu, 02 May 2013

16:00 - 17:00
L3

Elliptic curves with rank one

Chris Skinner
(Princeton)
Abstract

I will discuss some p-adic (and mod p) criteria ensuring that an elliptic curve over the rationals has algebraic and analytic rank one, as well as some applications.

Thu, 25 Apr 2013

16:00 - 17:00
L3

Modular curves, Deligne-Lusztig curves and Serre weights

Teruyoshi Yoshida
(Cambridge)
Abstract

One of the most subtle aspects of the correspondence between automorphic and Galois representations is the weight part of Serre conjectures, namely describing the weights of modular forms corresponding to mod p congruence class of Galois representations. We propose a direct geometric approach via studying the mod p cohomology groups of certain integral models of modular or Shimura curves, involving Deligne-Lusztig curves with the action of GL(2) over finite fields. This is a joint work with James Newton.

Thu, 07 Mar 2013

16:00 - 17:00
L3

Conditional bounds for the Riemann zeta-function via Fourier analysis.

Emanuel Carneiro
(Brazil)
Abstract

In this talk I will present the best up-to-date bounds for the argument of the Riemann zeta-function on the critical line, assuming the Riemann hypothesis. The method applies to other objects related to the Riemann zeta-function and uses certain special families of functions of exponential type. This is a joint work with Vorrapan Chandee (Montreal) and Micah Milinovich (Mississipi).

Thu, 28 Feb 2013

16:00 - 17:00
L3

Probabilistic Galois Theory

Rainer Dietmann
(Royal Holloway University of London)
Abstract

Van der Waerden has shown that `almost' all monic integer

polynomials of degree n have the full symmetric group S_n as Galois group.

The strongest quantitative form of this statement known so far is due to

Gallagher, who made use of the Large Sieve.

In this talk we want to explain how one can use recent

advances on bounding the number of integral points on curves and surfaces

instead of the Large Sieve to go beyond Gallagher's result.

Thu, 21 Feb 2013

16:00 - 17:00
L3

How frequently does the Hasse principle fail?

Tim Browning
(Bristol)
Abstract

Counter-examples to the Hasse principle are known for many families of geometrically rational varieties. We discuss how often such failures arise for Chatelet surfaces and certain higher-dimensional hypersurfaces. This is joint work with Regis de la Breteche.

Thu, 14 Feb 2013

16:00 - 17:00
L3

Congruent Numbers

John Coates
(Cambridge)
Abstract

I will explain the beautiful generalization recently discovered by Y. Tian of Heegner's original proof of the existence of infinitely many primes of the form 8n+5, which are congruent numbers. At the end, I hope to mention some possible generalizations of his work to other elliptic curves defined over the field of rational numbers.

Thu, 07 Feb 2013

16:00 - 17:00
L3

C-groups

Kevin Buzzard
(Imperial College London)
Abstract

Toby Gee and I have proposed the definition of a "C-group", an extension of Langlands' notion of an L-group, and argue that for an arithmetic version of Langlands' philosophy such a notion is useful for controlling twists properly. I will give an introduction to this business, and some motivation. I'll start at the beginning by explaining what an L-group is a la Langlands, but if anyone is interested in doing some background preparation for the talk, they might want to find out for themselves what an L-group (a Langlands dual group) is e.g. by looking it up on Wikipedia!

Thu, 31 Jan 2013

16:00 - 17:00
L3

Classicality for overconvergent eigenforms on some Shimura varieties.

Christian Johansson
(Imperial College London)
Abstract

A well known theorem of Coleman states that an overconvergent modular eigenform of weight k>1 and slope less than k-1 is classical. This theorem was later reproved and generalized using a geometric method very different from Coleman's cohomological approach. In this talk I will explain how one might go about generalizing the cohomological method to some higher-dimensional Shimura varieties.

Thu, 24 Jan 2013

16:00 - 17:00
L3

p-adic functoriality for inner forms of unitary groups.

Judith Ludwig
(Imperial College London)
Abstract

In this talk I will explain a notion of p-adic functoriality for inner forms of definite unitary groups. Roughly speaking, this is a morphism between so-called eigenvarieties,  which are certain rigid analytic spaces parameterizing p-adic families  of automorphic forms. We will then study certain properties of classical Langlands functoriality that allow us to prove p-adic functoriality in some "stable" cases.

Thu, 17 Jan 2013

16:00 - 17:00
L3

Computing the local Cassels-Tate pairing.

Rachel Newton
(Leiden University)
Abstract

Let K be a number field and E/K be an elliptic curve. Multiplication by n induces a map from the n^2-Selmer group of E/K to the n-Selmer group. The image of this map contains the image of E(K) in the n-Selmer group and is often smaller. Thus, computing the image of the n^2-Selmer group under multiplication by n can give a tighter bound on the rank of E/K. The Cassels-Tate pairing is a pairing on the n-Selmer group whose kernel is equal to the image of the n^2-Selmer group under multiplication by n. For n=2, Cassels gave an explicit description of the Cassels-Tate pairing as a sum of local pairings and computed the local pairing in terms of the Hilbert symbol. In this talk, I will give a formula for the local Cassels-Tate pairing for n=3 and describe an algorithm to compute it for n an odd prime. This is joint work with Tom Fisher.

Thu, 15 Nov 2012

16:00 - 17:00
L3

Shimura Decomposition and Tunnell-like formulae.

Soma Purkait
(Warwick)
Abstract

Let k be an odd integer and N be a positive integer divisible by 4. Let g be a newform of weight k - 1, level dividing N/2 and trivial character. We give an explicit algorithm for computing the space of cusp forms of weight k/2 that are 'Shimura-equivalent' to g. Applying Waldspurger's theorem to this space allows us to express the critical values of the L-functions of twists of g in terms of the coefficients of modular forms of half-integral weight. Following Tunnell, this often allows us to give a criterion for the n-th twist of an elliptic curve to have positive rank in terms of the number of representations of certain integers by certain ternary quadratic forms.

Thu, 08 Nov 2012

16:00 - 17:00
L3

Dynamical approaches to the Littlewood conjecture and its variants.

Alan Haynes
(Bristol)
Abstract

We will discuss the Littlewood conjecture from Diophantine approximation, and recent variants of the conjecture in which one of the real components is replaced by a p-adic absolute value (or more generally a "pseudo-absolute value''). The Littlewood conjecture has a dynamical formulation in terms of orbits of the action of the diagonal subgroup on SL_3(R)/SL_3(Z). It turns out that the mixed version of the conjecture has a similar formulation in terms of homogeneous dynamics, as well as meaningful connections to several other dynamical systems. This allows us to apply tools from combinatorics and ergodic theory, as well as estimates for linear forms in logarithms, to obtain new results.

Thu, 25 Oct 2012

16:00 - 17:00
L3

Vertical Brauer groups and degree 4 del Pezzo surfaces.

Bianca Viray
(Brown)
Abstract

In this talk, I will show that Brauer classes of a locally solvable degree 4 del Pezzo surface X are vertical, that is, that every Brauer class is obtained by pullback from an element of Br k(P^1) for some rational map f : X ----> P^1. As a consequence, we see that a Brauer class does not obstruct the existence of a rational point if and only if there exists a fiber of f that is locally solvable. The proof is constructive and gives a simple and practical algorithm, distinct from that in [Bright,Bruin,Flynn,Logan (2007)], for computing all nonconstant classes in the Brauer group of X. This is joint work with Anthony V\'arilly-Alvarado.

Thu, 18 Oct 2012

16:00 - 17:00
L3

Rational points of bounded height over number fields.

Daniel Loughran
(Paris VII)
Abstract

Given a variety X over a number field, one is interested in the collection X(F) of rational points on X. Weil defined a variety X' (the restriction of scalars of X) defined over the rational numbers whose set of rational points is naturally equal to X(F). In this talk, I will compare the number of rational points of bounded height on X with those on X'.

Fri, 28 Sep 2012

15:05 - 15:45
L1

Efficient computation of Rankin $p$-adic L-functions

Alan Lauder
(Oxford)
Abstract

I will present an efficient algorithm for computing certain special values of Rankin triple product $p$-adic L-functions and give an application of this to the explicit construction of rational points on elliptic curves.

Fri, 28 Sep 2012

14:00 - 15:00
L1

p-adic iterated integrals and rational points on elliptic curves

Henri Darmon
(McGill)
Abstract

The $p$-adic Gross-Zagier formula for diagonal cycles and the $p$-adic Beilinson formulae described in the lectures of Rotger and Bertolini respectively suggest a connection between certain  {\em $p$-adic iterated integrals} attached to modular forms and rational points on elliptic curves. I will describe an ongoing project (in collaboration with Alan Lauder and Victor Rotger) whose goal is to explore these relationships numerically, with the goal of better understanding the notion of {\em Stark-Heegner points}. It is hoped that these experiments might suggest new perspectives on Stark-Heegner points based on suitable {\em $p$-adic deformations} of the global objects--diagonal cycles, Beilinson-Kato and Beilinson-Flach elements-- described in the lectures of Rotger, Bertolini, Dasgupta, and Loeffler, following  the influential approach to $p$-adic $L$-functions pioneered by Coates-Wiles, Kato, and Perrin-Riou.

Fri, 28 Sep 2012

11:00 - 12:00
L1

Eisenstein cocycle on ${\rm GL}_n$ and computation \\ of $p$-adic L-functions of totally real fields

Pierre Charollois
(Paris VI)
Abstract

We define an integral version of Sczech cocycle on ${\rm GL}_n(\mathbf{Z})$ by raising the level at a prime $\ell$.As a result, we obtain a new construction of the $p$-adic L-functions of Barsky/Cassou-Nogu\`es/Deligne-Ribet. This cohomological construction further allows for a study of the leading term of these L-functions at $s=0$:\\1) we obtain a new proof that the order of vanishing is at least the oneconjectured by Gross. This was already known as result of Wiles.\\2) we deduce an analog of the modular symbol algorithm for ${\rm GL}_n$ from the cocyclerelation and LLL. It enables for the efficient computation of the special values of these $p$-adic L-functions.\\When combined  with a refinement of the Gross-Stark conjecture, we obtain some examples of numerical construction of $\mathfrak p$-units in class fields of totally real (cubic) fields.This is joint work with Samit Dasgupta.

Fri, 28 Sep 2012

09:30 - 10:30
L1

Euler systems for Rankin--Selberg convolutions of modular forms

David Loeffler
(Warwick)
Abstract

I will describe a construction of special cohomology classes over the cyclotomic tower for the product of the Galois representations attached to two modular forms, which $p$-adically interpolate the "Beilinson--Flach elements" of Bertolini, Darmon and Rotger. I will also describe some applications to the Iwasawa theory of modular forms over imaginary quadratic fields.

Thu, 27 Sep 2012

16:00 - 17:00
L1

Factorization of $p$-adic Rankin L-series

Samit Dasgupta
(UCSC)
Abstract


We show that the $p$-adic L-function associated to the tensor square of a $p$-ordinary eigenform factors as the product of the symmetric square $p$-adic L-function of the form with a Kubota-Leopoldt $p$-adic L-function.  Our method of proof follows that of Gross, who proved a factorization for Katz's $ p$-adic L-function for a character arising as the restriction of a Dirichlet character.  We prove certain special value formulae for classical and $p$-adicRankin L-series at non-critical points.  The formula of Bertolini, Darmon, and Rotger in the $p$-adic setting is a key element of our proof.  As demonstrated by Citro, we obtain as a corollary of our main result a proof of the exceptional zero conjecture of Greenberg for the symmetric square.

Thu, 27 Sep 2012

14:45 - 15:25
L1

$p$-adic Beilinson's formulas for Rankin $p$-adic L-functions and applications

Massimo Bertolini
(Milan)
Abstract

I will report on $p$-adic Beilinson's formulas, relating the values of certain Rankin $p$-adic L-functions outside their range of classical interpolation, to $p$-adic syntomic regulators of Beilinson-Kato and Beilinson-Flach elements. Applications to the theory of Euler systems and to the Birch and Swinnerton-Dyer conjecture will also be discussed. This is joint work with Henri Darmon and Victor Rotger.

Thu, 27 Sep 2012

14:00 - 14:40
L1

Triple product $p$-adic L-functions and diagonal cycles

Victor Rotger
(UP Catalunya)
Abstract

In this lecture I shall introduce certain generalised Gross-Kudla-Schoen diagonal cycles in the product of three Kuga-Sato varieties and a $p$-adic formula of Gross-Zagier type which relates the images of these diagonal cycles under the $p$-adic Abel-Jacobi map to special values of the $p$-adic L-function attached to the Garrett triple convolution of three  Hida families of  modular forms. This formula has applications to the Birch--Swinnerton-Dyer conjecture and the theory of Stark-Heegner points. (Joint work with Henri Darmon.)