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Number Theory 2012-13

Organiser: David Loeffler

Seminars are held on Mondays at 15:00 in Room B3.03 – Mathematics Institute (unless stated otherwise)

2012-13 Term 3

2012-13 Term 2

2012-13 Term 1


Amnon Besser (15/10/2012), p-adic heights, p-adic sigma functions, and integral points on hyperelliptic curves: In this talk, based on joint work with J. Balakrishnan, I would like to advertise a new algorithm for finding integral points on hyperelliptic curves, which is based on p-adic height pairings.

There are many equivalent methods for defining p-adic height pairings. In particular, for elliptic curves over a number field one may use formulas of Mazur and Tate that rely on the p-adic sigma function, or one may use the method of Coleman and Gross, that uses Coleman integration theory to describe the local height pairing at primes above p. These two are known to be equal but the proof is rather indirect.

We use p-adic Arakelov theory to prove that the two constructions are the same. We further show that the Mazur-Tate height can be computed by extending the log of the p-adic sigma function as a Coleman integral.

In the elliptic curves case this gives an easy proof of a recent theorem of Minhyong Kim. For hyperelliptic curves, it gives a new algorithm for finding integral points when the Mordell-Weil rank equals the genus.

Bianca Viray (22/10/2012), 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 [BBFL], for computing all nonconstant classes in the Brauer group of X. This is joint work with Anthony V'arilly-Alvarado.

Christian Johansson (29/10/2012), Classicality for small slope overconvergent eigenforms on some quaternionic Shimura varieties: A theorem of Coleman states that an overconvergent modular eigenform of weight k > 1 and slope < k-1 is classical. This theorem was later reproved and generalized using a geometric method very different from Coleman's original cohomological approach. In this talk I will explain how one might go about generalizing the cohomological method to some higher-dimensional Shimura varieties.

Jens Funke (5/11/2012), Spectacle cycles and modular forms of half-integral weight: The classical Shintani lift is the adjoint of the Shimura correspondence. It realizes periods of even weight cusp forms as Fourier coefficients of a half-integral modular form. In this talk we revisit the Shintani lift from a (co)homological perspective. In particular, we extend the lift to Eisenstein series and give a geometric interpretation of this extension. This is joint work with John Millson.

Lilian Pierce (12/11/2012), Simultaneous prime values of pairs of quadratic forms: The circle method of Hardy and Littlewood is a key tool of analytic number theory. In one of its classic applications, it is used to provide an asymptotic for the number of representations of a fixed integer by a fixed homogeneous form. But it can also be used to prove results of a different flavor, such as showing that almost every number (in a certain sense) has at least one representation by the form. In joint work with Roger Heath-Brown, we have recently considered a 2-dimensional version of such a problem. Given two quadratic forms with integer coefficients, we ask whether almost every integer (in a certain sense) must be simultaneously represented by the forms. Under a modest geometric assumption, we are able to prove such a result if the forms have at least 5 variables. In particular, we show that any two such quadratic forms must simultaneously attain prime values infinitely often. In this seminar, we will review the circle method and investigate how such "almost all" results may be proved.

Haluk Sengun (19/11/2012), Cohomology of Bianchi groups and arithmetic: Bianchi groups are groups of the form SL(2,R) where R is the ring of an imaginary quadratic field. They arise naturally in the study of hyperbolic 3-manifolds and of certain generalizations of the classical modular forms (called Bianchi modular forms) for which they assume the role of the classical modular group SL(2,Z).

After giving the necessary background, I will start with a discussion of the problem of understanding the behavior of the dimensions of the cohomology of Bianchi groups and their congruence subgroups. Next, I will focus on the amount of the torsion that one encounters in the cohomology . Finally, I will discuss the arithmetic significance of these torsion classes.

Tony Scholl (26/11/2012), Congruences and modular forms: I will discuss new congruences for weakly modular forms (forms which are merely meromorphic at infinity) and their relation with de Rham cohomology. Although these were first observed for noncongruence subgroups, there are new phenomena even for level one classical modular forms. This is joint work with Matija Kazalicki (Zagreb).

Jeroen Sijsling (3/12/2012), Galois descent for hyperelliptic curves: Let k be a field, and let C be a curve over the algebraic closure of k of genus g with automorphism group G that is isomorphic with all its conjugates over k. Is it then possible to find a curve C0 defined over k that is isomorphic with C (over the algebraic closure), and if one is given such a C for which the response is affirmative, can one construct C0 and the isomorphism with C explicitly?

It is well-known that if g = 0, g = 1, or G is trivial, one can always descend. For g = 2, the obstruction to descent was described by Mestre, who has also given algorithms to descend explicitly if the obstruction vanishes.

This talk will give the solution to the descent problem for general hyperelliptic curves. Building on work by Huggins, we reduce to the case where G modulo the hyperelliptic involution is cyclic. The descent obstruction then usually reduces to finding a point on a conic over k associated with C. We also indicate how to construct a model C0 of C if the obstruction vanishes. This is joint work with Reynald Lercier and Christophe Ritzenthaler.

Tom Ward (14/1/2013), Congruences between Artin-twisted L-values: Let E be an elliptic curve defined over Q. Given an Artin representation over Q (a complex Galois representation which factors through a finite extension) we may define an L-series associated to the 'twist' of E by this representation.
Such L-functions play an important role in the conjectures of non-commutative Iwasawa theory; in the particular case we will discuss, these deep conjectures imply that their special values should satisfy certain strong congruences. We will outline a method using Hilbert modular forms by which one can prove a weaker form of these congruences, based on work of Panchishkin, and also work of V. Dokchitser and Bouganis.

Stefan Keil (21/1/2013), On non-square order Tate-Shafarevich groups of non-simple abelian surfaces: For an elliptic curve (over a number field) it is known that the order of its Tate-Shafarevich group is a square, provided it is finite. In higher dimensions this no longer holds true. We will present work in progress on the classification of all occurring non-square parts of orders of Tate-Shafarevich groups of non-simple abelian surfaces over the rationals. As it seems the only possible values are k=1,2,3,5,6,7,10 and maybe 13.

Ghaith Hiary (28/1/201), Detecting square-free numbers via the explicit formula: Let d = m2 n, where n is square-free, and where m and n are unknown to us. A method to obtain a lower bound on n without attempting to factor d is presented. If d happens to be square-free, then the method might yield a sufficiently good lower bound on n so that the square-freeness of d can then be certified fast -- in particular, faster than could have been done had we immediately applied one of the other known methods that also can produce partial information about n (such as the Pollard-Strassen algorithm). The running time of the method is heuristically sub-exponential in the lower bound over a relatively wide initial range, and perhaps further. The method is based on the explicit formula for the Dirichlet L-function associated with a suitably chosen real character, and assuming the generalized Riemann hypothesis for that L-function. Several optimizations of the method will be discussed. Examples of computations using the method will be given. This is joint work with Andy Booker and Jon Keating.

Daniel Loughran (11/1/2013), Rational points of bounded height and the Weil restriction: If one is interested in studying diophantine equations over number fields, there is a clever trick due to Weil where one may move the problem from the number field setting to the usual field of rational numbers by performing a "restriction of scalars". In this talk, we consider the problem of how the height of a solution (a measure of the complexity of a solution) changes under this process, and in particular how the number of solutions of bounded height changes.

Dino Lorenzini (25/02/2013), Wild quotient singularities of surfaces and models of curves: Let K be a discrete valuation field and let X/K be a smooth projective curve. In many important instances, it is useful to have a description of the regular minimal model of X over the ring of integers O of K. Such a model is often most difficult to describe when X/K admits semi-stable reduction only after a wildly ramified extension of K. We will discuss in this talk the case where the curve X admits good ordinary reduction after a wildly ramified extension. Our method involves resolving wild quotient singularities, and we will present some general combinatorial results on the resolution of such singularities. As an application, we also obtain results on the desingularization of a normal surface Z over a field k obtained as the quotient of the product of two smooth curves by a diagonal automorphism of order equal to the characteristic of k. Again, our most complete results are when one of the two curves is ordinary.

Paul-James White (04/03/2013), Studying L-functions via the trace formula: Sarnak, following ideas of Langlands on "Beyond Endoscopy", gave a trace formula proof of the analytic continuation of the L-function associated to a holomorphic cusp form. Herman recently deduced the functional equation via the trace formula. We shall talk about a possible approach to studying base change L-functions via the trace formula.

Oliver Bräunling (11/03/2013), The many faces of the higher tame symbol: I will explain several perspectives from which one could look at the classical tame symbol and its generalizations. This encompasses determinants, geometry, class field theory and K-theory. We show how to 'factor' the tame symbol to more elementary structures, giving a unified explanation why it occurs in such different ways. (Joint work with M. Groechenig and J. Wolfson).

Ariel Pacetti (14/3/2013), Proving modularity of Galois representations: In this talk we will recall the Falting-Serre method and Livné's method to prove modularity of Galois representations. We will show how this method can be used to prove modularity of elliptic curves over number fields and also the modularity of a threefold studied by Consani and Scholten. If time allows, we will show an application of a work in progress to prove modularity of an abelian surface which is not of GL(2) type.