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Number Theory Seminar (past)

Thu, 16/05
16:00 		Romyar Sharifi (Arizona) Number Theory Seminar Add to calendar L3
Refining the Iwasawa main conjecture
I will discuss conjectures relating cup products of cyclotomic units and modular symbols modulo an Eisenstein ideal. In particular, I wish to explain how these conjectures may be viewed as providing a refinement of the Iwasawa main conjecture. T. Fukaya and K. Kato have proven these conjectures under certain hypotheses, and I will mention a few key ingredients. I hope to briefly mention joint work with Fukaya and Kato on variants.
Thu, 09/05
16:00 		Kevin Hughes (Edinburgh) Number Theory Seminar Add to calendar L3
Arithmetic restriction theory and Waring's problem
We will discuss arithmetic restriction phenomena and its relation to Waring's problem, focusing on how recent work of Wooley implies certain restriction bounds.
Thu, 02/05
17:00 		Ambrus Pal (London) Number Theory Seminar Add to calendar SR2
The p-adic monodromy group of abelian varieties over global function fields of characteristic p

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, 02/05
16:00 		Chris Skinner (Princeton) Number Theory Seminar Add to calendar L3
Elliptic curves with rank one
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, 25/04
16:00 		Teruyoshi Yoshida (Cambridge) Number Theory Seminar Add to calendar L3
Modular curves, Deligne-Lusztig curves and Serre weights
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, 07/03
16:00 		Emanuel Carneiro (IMPA) Number Theory Seminar Add to calendar L3
Conditional bounds for the Riemann zeta-function via Fourier analysis.
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, 28/02
16:00 		Rainer Dietmann (Royal Holloway University of London) Number Theory Seminar Add to calendar L3
Probabilistic Galois Theory
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, 21/02
16:00 		Tim Browning (Bristol) Number Theory Seminar Add to calendar L3
How frequently does the Hasse principle fail?

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, 14/02
16:00 		John Coates (Cambridge) Number Theory Seminar Add to calendar L3
Congruent Numbers
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, 07/02
16:00 		Kevin Buzzard (Imperial College London) Number Theory Seminar Add to calendar L3
C-groups

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, 31/01
16:00 		Christian Johansson (Imperial College London) Number Theory Seminar Add to calendar L3
Classicality for overconvergent eigenforms on some Shimura varieties.
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, 24/01
16:00 		Judith Ludwig (Imperial College London) Number Theory Seminar Add to calendar L3
p-adic functoriality for inner forms of unitary groups.

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, 17/01
16:00 		Rachel Newton (Leiden University) Number Theory Seminar Add to calendar L3
Computing the local Cassels-Tate pairing.
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, 29/11/2012
16:00 		Oscar Marmon (Goettingen) Number Theory Seminar Add to calendar L3
Sums and differences of four k-th powers
Thu, 15/11/2012
16:00 		Soma Purkait (Warwick) Number Theory Seminar Add to calendar L3
Shimura Decomposition and Tunnell-like formulae.

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, 08/11/2012
16:00 		Alan Haynes (Bristol) Number Theory Seminar Add to calendar L3
Dynamical approaches to the Littlewood conjecture and its variants.
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, 25/10/2012
16:00 		Bianca Viray (Brown) Number Theory Seminar Add to calendar L3
Vertical Brauer groups and degree 4 del Pezzo surfaces.
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 Várilly-Alvarado.
Thu, 18/10/2012
16:00 		Daniel Loughran (Paris VII) Number Theory Seminar Add to calendar L3
Rational points of bounded height over number fields.
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'.

For any corrections/updates to these listings, please contact seminars [-at-] maths [dot] ox [dot] ac [dot] uk.

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