Abstracts


Talks:

  1. Hendrik Lenstra (Universiteit Leiden):

    The normal basis theorem and the combinatorial Nullstellensatz
    Noga Alon's combinatorial Nullstellensatz (1999) has had many applications of a combinatorial nature. It can also be used to give new proofs of certain results in algebra, the normal basis theorem from Galois theory being a typical example. These proofs depend on a special case of the combinatorial Nullstellensatz that can be phrased in a coordinate-free manner.

  2. Piotr Maciak (École Polytechnique Fédérale de Lausanne):

    Bounds for the Euclidean minima of algebraic number and function fields.
    The Euclidean division is a basic tool when dealing with the ordinary integers. It does not extend to rings of integers of algebraic number fields in general. The Euclidean minimum is a numerical invariant which measures the "deviation" from the Euclidean property, Its study is a classical topic in algebraic number theory, going back to Minkowski. The first part of the talk is based on the joint work with Eva Bayer and will be dedicated to some recent results concerning abelian fields of prime power conductor. In the second part of the talk (joint work with Marina Monsurrò and Leonardo Zapponi), we will also define Euclidean minima for function fields and give some bounds for this invariant. We furthermore show that the results are analogous to those obtained in the number field case.

  3. Pietro Mercuri (Universitigrave; di Roma 'La Sapienza'):

    Computing equations of Modular curves associated to the normalizers of non split Cartan subgroups.
    Serre's uniformity conjecture has raised interest in the modular curves associated to normalizers of non-split Cartan subgroups and their rational points. I will show how to compute explicit equations defining these kind of modular curves. This can be done by calculating Fourier expansions of suitable cusp forms and computing the canonical embedding. The curves are then realized as intersections of quadrics in projective space.

  4. Francesco Pappalardi (Università di Roma 3):

    Averaging with primitive roots
    After reviewing classical results on average versions of Artin Conjecture for primitive roots mainly due to Goldfeld, Stephens and Warlimont, we will adapt the known techniques of exponential sums estimates due to Burgess to several other problems. Typical examples are: the r-dimensional Artin Conjecture, the Schinzel Wojcik problem, the simultaneous primitive roots problem and more. This is joint ongoing work with several authors.

  5. Marusia Rebolledo (Université Blaise Pascal Clermont-Ferrand 2):

    Modular curves associated to non split Cartan subgroups and their normalizer
    I will present a recent work in collaboration with Christian Wuthrich where we give a new description of non-split cartan modular curves as moduli spaces, namely classifying elliptic curves endowed with a level structure that we call a necklace. I will show how this description allows to recover some classical results (on elliptic points, degenerate maps, Hecke operators etc) as well as it gives a more explicit and geometric vision of a theorem of Chen.

  6. Lajos Rónyai (Budapest University of Technology and Economics (BUTE)):

    Applications and extensions of the combinatorial Nullstellensatz
    Noga Alon's Combinatorial Nullstellensatz (CN) is a specialized, precise version of the Hilbertsche Nullstellensatz for the ideal of all polynomials vanishing on a discrete box S. It implies a simple, and amazingly widely applicable nonvanishing theorem (NvT), a sufficient condition which guarantees, that a polynomial is not identically zero on S. The NvT has dozens of applications in several branches of mathematics, including combinatorics, number theory and geometry. In the first part of the talk we present proofs of these two theorems, with some interesting applications, mostly from number theory. In the second part some generalizations are considered, including versions of the CN and NvT over general rings, and for multisets. We discuss also possible extensions to finite point sets S, other than discrete boxes.



  7. Peter Stevenhagen (Universiteit Leiden):

    Prime densities for GL1 and GL2

  8. Christian Wuthrich (University of Nottingham):

    Integrality of modular symbols for elliptic curves
    Modular symbols are certain integrals of modular forms along paths from cusp to cusp in the upper half plane. It is known that they are rational multiples of periods. I would like to discuss first a criterion for when they are an integral multiple in case the modular form corresponds to an elliptic curve over Q. As an application one can show that certain very complicated "zeta elements" by Kato are integral, too. This has direct application to the Birch and Swinnerton-Dyer conjecture.