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Number Theory Seminar FS08:

22.02.2008 Dr. Egon Rütsche (ETH Zurich)
Adelic Openness for Drinfeld modules in generic characteristic
Abstract: Let F be a finitely generated field of transcendence degree 1 over a finite field, and let A be the ring of all elements of F which are integral outside a fixed place. We consider a Drinfeld A-module over a function field of generic characteristic. For any prime of A, we have a continuous Galois representation on the Tate module of the Drinfeld module. If we take the product of these representations over all primes of A, we get the so-called adelic representation. If the absolute endomorphism ring of the Drinfeld module is equal to A, we prove that the image of the adelic representation is open. This is an analogue of Serre's result for elliptic curves without potential complex multiplication.
Time: 14.15 / Place: HWZ (HG G43)

29.02.2008 Prof. Damian Roessler (Jussieu, Paris)
A geometric construction of Yoshikawa's "discriminant" modular forms
Abstract: K.-I. Yoshikawa constructed a family of Siegel modular forms using spectral geometry. We give a geometric construction of these modular forms and prove that they are of arithmetic origin. This is joint work with V. Maillot.
Time: 14.15 / Place: HWZ (HG G43)

07.03.2008 David Zywina (UC Berkeley)
The image of the Galois representation associated to a random elliptic curve
Abstract: Fix a number field $k$. Associated to each elliptic curve $E$ over $k$ is a Galois representation $\rho_E : Gal(\overline{k}/k) \to GL_2(\widehat{\mathbb{Z}}),$ arising from the natural Galois action on the torsion points of $E$. In this talk we will answer (and make sense of) the following question: What is the image of $\rho_E$ for a "random'' elliptic curve $E$ over $k$?
Time: 14.15 / Place: HWZ (HG G43)

14.03.2008 Prof. Özlem Imamoglu (ETH Zurich)
ETHZ : Eta, THeta, Zeta
Abstract
Time: 14.15 / Place: HWZ (HG G43)

04.04.2008 Dr. Cormac O'Sullivan (Bronx Community College of the CUNY)
Higher-order cusp forms, inner products and Kronecker limits
Abstract: Higher-order cusp forms arose in work of Eichler, Goldfeld and Zagier in three different contexts. They generalize classical cuspforms and are also a special case of the vector-valued cusp forms of Knopp and Mason. In joint work with Imamoglu, we show that the second-order space has a natural inner product. We conjecture that this method also gives inner products for all higher-order spaces. This work relies on properties of non-holomorphic higher-order Eisenstein series. Working with Jorgenson, we identify the Kronecker limits for these series and find new analogs of the Dedekind sum.
Time: 14.15 / Place: HWZ (HG G43)

16.04.2008 PD Dr. Clemens Fuchs (ETH Zurich)
Antrittsvorlesung: Diophantische Zahlen-Welten
Abstract: Die Welt der Zahlen und ihrer Eigenschaften bergen nach wie vor viele Rätsel und Geheimnisse, selbst bei sehr einfach zu formulierenden Fragestellungen. Die ersten systematischen Untersuchungen gehen auf Diophantus von Alexandria zurück, welcher somit als Gründer der Algebra und Zahlentheorie gilt. Beispielsweise findet man folgendes Problem in seinem Buch Arithmetica (siehe [Arithmeticorum Libri Sex, cum commentariis C. G. Bacheti et observationibus D. P. de Fermat, Tolouse 1670; Lib. IV, q. XXI, p. 161]): Man finde vier (positive rationale) Zahlen mit der Eigenschaft, dass das Produkt von je zwei erhöht um eins ein Quadrat einer rationalen Zahl ist. Mit diesem Problem haben sich übrigens später auch Pierre de Fermat, der vor allem für seine Vermutungen bekannt ist, sowie Leonhard Euler besch&aouml;ftigt. In diesem Vortrag wird von den bisher bekannten Resultaten rund um dieses Problem, den Methoden zur Erlangung dieser Ergebnisse, sowie den noch offenen Fragen berichtet. Am Ende des Vortrages wird auch darauf eingegangen, wie dieser Problemkreis für sinnvolle und nicht nur für den Theoretiker interessante Einsichten genutzt werden könnte.
Time: 17.15 / Place: HG G3

25.04.2008 Prof. Tarlok Shorey (Tata Institute Mumbay)
Irreducibility of some polynomials considered by Schur
Abstract: Schur considered the polynomial a_n x^n/n! + a_{n-1} x^{n-1}/(n-1)! + ... + a_1 x/1! + a_0, where a_i are integers with a_0 and a_n are either 1 or -1, and showed that it is irreducible. We shall consider more general polynomials a_n x^n/(n+a)! +a{n-1} x^{n-1}/(n-1+a)!+ +a_1 x/ (1+a)! + a_0 /a!, where a is not necessarily zero, and give several extensions of Schur's result.
Time: 14.15 / Place: HWZ (HG G43)

25.04.2008 Dr. Aurélien Galateau
The effective Bogomolov conjecture in abelian varieties
Abstract: I will speak about an explicit version of the Bogomolov problem in abelian varieties. Under a conjecture concerning ordinary primes in abelian varieties, I give a lower bound for the essential minimum of subvarieties with small codimension, which is the analog of the bound known in the toric case since the work of Amoroso and David. The key point is a p-adic lemma concerning the formal group of an abelian variety in positive caracteristic. I will finally discuss the diophantine setting and the possible extension to general codimension.
Time: 15.45 / Place: HWZ (HG G43)

02.05.2008 Antonella Perucca (Sapienza University of Rome)
The support problem for abelian varieties and related questions
Abstract: The support problem for abelian varieties originated with a question by Erdös on the integers and was solved by Larsen in 2003. After giving the history of this problem, we present some variants and some open questions.
Time: 14.15 / Place: HWZ (HG G43)

09.05.2008 Prof. Hélène Esnault (Univ. Duisburg-Essen)
Fundamental groups, motivic and étale
Abstract: We will report on joint work with Marc Levine on a description of Deligne's motivic fundamental group which draws some analogy with Grothendieck's geometric fundamental group. What then plays the role of the Galois group of the base field is the Tannaka group scheme of mixed Tate motives over a number field.
Time: 14.15 / Place: HWZ (HG G43)

16.05.2008 Prof. Per Salberger (Göteborg University)
Counting rational points on projective varieties
Abstract: We present new results on the asymptotic behaviour of the number of rational points of bounded height on projective varieties. We use thereby a refined version of the determinant method initiated by Bombieri/Pila and Heath-Brown.
Time: 14.15 / Place: HWZ (HG G43)

23.05.2008 Prof. Sergei Vostokov (St. Petersburg State Univ.)
The classical reciprocity law as an analog of an Abelian integral theorem
Abstract: Apparently Leopold Kronecker came up with the idea of an analogy between numbers and functions. David Hilbert was the first who began to study this analogy in algebraic number fields. He, in particular , observed that his reciprocity law for norm residue symbols resembles the Cauchy integral theorem. To make this analogy more transparent, we consider the classical reciprocity law for power residues. Class field theory connects the product of power residues with the product of local norm residue symbols and this relation must be an analog of integral theorem stating that the Abelian integral of a differential form on a Riemann surface equals the sum of residues of the form at the singular points. I obtain the explicit global reciprocity law for power residue and show that this explicit formula is an analog of the Abelian integral theorem.
Time: 14.15 / Place: HWZ (HG G43)

23.05.2008 Cécile Armana (Jussieu, Paris)
Rational points on Drinfeld modular curves
Abstract: Since the work of Mazur, Kamienny and Merel, we know the existence of a uniform bound for the torsion of elliptic curves over number fields. Their approach is based on the study of rational points on classical modular curves. I will discuss to what extent this method applies to Drinfeld modular curves over function fields and the uniform boundedness conjecture for torsion of rank-2 Drinfeld modules. When N is a prime polynomial in F_q[T] of degree 3, we show that the Drinfeld modular curve Y_1(N) has no rational points of small degree. For general N, we have a similar statement, under a hypothesis on the Hecke structure of Drinfeld modular forms.
Time: 15.30 / Place: HWZ (HG G43)

13.06.2008 Kate Stange (Brown University)
Elliptic Nets in Cryptography
Abstract: Elliptic divisibility sequences are integer recurrence sequences, eachof which is associated to an elliptic curve over the rationals together with a rational point on that curve. I'll give the background on these and present a higher-dimensional analogue over arbitrary fields. Suppose E is an elliptic curve over a field K, and P_1,...,P_n are points on E defined over K. To this information we associate an n-dimensional array of values of K satisfying a nonlinear recurrence relation. These are called elliptic nets. I'll talk about applications of elliptic nets to cryptography: specifically, I give a new algorithm for computing the Tate or Weil pairing for pairing-based cryptography. If time permits, I may briefly discuss recent work relating elliptic nets to the elliptic curve discrete logarithm problem.
Time: 11:00 / Place: HWZ (HG G43)