ETHZ-UZh-Sbg Arithmetic and Geometry Research Group

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**Number Theory Seminars - Archive**

**Organizers:**

G. Wüstholz

R. Pink

Ö. Imamoglu

E. Kowalski (since HS08)

C. Fuchs (until FS12)

The seminars later than FS09 can be found here.

**Number Theory Seminar FS09:**

17.02.2009 Mini - Workshop, Day 1

(organized by G. Wüstholz and Jing Yu)

Program

Time: 14.15 - 17.15 / Place: HG G19.2

18.02.2009 Mini - Workshop, Day 2

(organized by G. Wüstholz and Jing Yu)

Program

Time: 14.15 - 17.15 / Place: HG G19.2

27.02.2009 Prof. Umberto Zannier (SNS Pisa)

Hilbert Irreducibility above algebraic groups

Abstract: We shall discuss some versions of Hilbert Irreducibility Theorem, concerning the lifting of rational points on certain commutative algebraic groups to covers of them. The proofs use a new method which works essentially by first considering maximal cyclotomic extensions and then descending the field with Chebotarev Theorem. We deal explicitly with multiplicative tori and powers of an elliptic curve (but the method in principle could work more generally). Another application is to a Bertini Theorem for multiplicative tori in place of vector spaces.

Time: 14.15 / Place: HWZ (HG G43)

06.03.2009 Prof. Klaus Schmidt (Univ. Wien/ESI)

Salem numbers, beta-shifts and quasihyperbolic toral automorphisms

Abstract: An irreducible automorphism A in GL(n,Z) of the n-dimensional torus T^n is called quasihyperbolic if it is ergodic and nonexpansive (i.e., A has an eigenvalue of absolute value 1 which is not a root of unity). This talk is about invariant sets and invariant probability measures of such automorphisms, as well as about some open questions involving beta-expansions of Salem numbers. The main results presented are joint work with Elon Lindenstrauss.

Time: 14.15 / Place: HWZ (HG G43)

13.03.2009 Dr. Alex V. Kontorovich (Brown Univ.)

Apollonian Circle Packings and Horospherical Flows on Hyperbolic 3-Manifolds

Abstract: We prove an asymptotic formula for the number of circles in an Apollonian packing of bounded curvature. Using the affine linear sieve, we give sharp upper bounds for the number of circles of prime curvature, and the number of "twin prime" tangent circles. The main ingredient of our proof is the equidistribution of long horospherical flows in the unit tangent bundle of an infinite volume hyperbolic 3-manifold, under the assumption that the Hausdorff dimension of its limit set exceeds one. This is joint work with Hee Oh.

Time: 14.15 / Place: HWZ (HG G43)

20.03.2009 Prof. Urs Hartl (WWU Münster)

Rapoport-Zink spaces and deformation spaces for local shtukas

Abstract: Bounded local shtukas are function field analogs for p-divisible groups and arise from Drinfeld modules and Anderson motives in the same way as p-divisible groups arise from abelian varietes. We describe the deformations and moduli spaces of bounded local shtukas. The latter are analogous to Rapoport-Zink spaces for p-divisible groups. The underlying schemes of these moduli spaces are affine Deligne-Lusztig varieties. This is joint work with Eva Viehmann).

Time: 14.15 / Place: HWZ (HG G43)

27.03.2009 Dr. Bernhard Köck (University of Southampton)

Real Belyi Theory

Abstract: Belyi's famous theorem states that a compact Riemann surface can be defined over a number field if and only if it admits a meromorphic function with at most three critical values. My talk will be about a generalization of this theorem to Klein surfaces, i.e. (possibly non-orientable) surfaces with boundary which carry a dianalytic structure. I will also explain some other characterizations (triangle groups, maps on surfaces, ...). The results presented are joint work with David Singerman on the one hand and with Eike Lau on the other hand.

Time: 14.15 / Place: HWZ (HG G43)

17.04.2009 No seminar.

ETH Number Theory Days at EPF Lausanne

(16.-17.04.2009)

Organizers: P. Michel and R. Pink

08.05.2009 Prof. Jianqiang Zhao (MPI Bonn)

Motivic fundamental group of G_m - mu_N and special values of multi-polylogs

Abstract: In this talk we will consider the Q-linear relations among the values of multiple polylogarithms evaluated at mu_N, where mu_N is the group of the Nth roots of unity. We show that the standard relations considered by Racinet do not provide all the possible relations in the following cases: (i) level N = 4, weight w = 3 or 4, and (ii) w = 2, 7 < N < 169, and N is a power of 2 or 3, or N has at least two prime factors. We further find some (presumably all) of the missing relations in (i) by using the octahedral symmetry of G_m - mu_4. We also prove some other results when N = p, p^2, p^3 (p prime >= 5) by using the motivic fundamental group of G_m - u_N.

Time: 14.15 / Place: HWZ (HG G43)

15.05.2009 No seminar.

Leopoldina-Symposium in Algebraic and Arithmetic Algebraic Geometry at CSF, Monte-Verità

(10.-15.05.2009)

Organizers: C. Fuchs, J. Kramer and G. Wüstholz

22.05.2009 Dr. Christopher Hall (University of Michigan)

Beyond Hilbert Irreducibility

Abstract: If f(t,x) is a polynomial in Q[t,x] which is irreducible is a polynomial with coefficients in the field of rational functions Q(t), then Hilbert's irreducibility theorem says that for 'most' positive integers t_0, the specialized polynomial f(t_0,x) is irreducible in Q[x]. We'll explain how one can use Faltings's theorems on the Mordell and Lang conjectures to say more, including cases where t_0 is allowed to vary over all elements in all finite extensions k/Q of bounded degree. In particular, we'll describe families of polynomials f(t,x) where we can give very strong finiteness statements about the Galois groups of the specialized polynomials for almost all elements of the family.

Time: 14.15 / Place: HWZ (HG G43)

07.08.2009 Prof. Yuri Zarhin (Penn State)

Endomorphisms and homomorphisms of superelliptic jacobians

Abstract: The Tate conjecture on homomorphisms of abelian varieties (now a theorem) asserts that the knowledge of the Galois action on all torsion points of an abelian variety Z allows us (in principle) to reconstruct the endomorphism algebra of Z when the ground field K is finitely generated over its prime subfield. We provide explicit examples where the knowledge of the Galois action on a (relatively) small (sub)group of torsion points allows us to recover the ring of all endomorphisms that are defined not only over K but over its algebraic closure. Applications to the computation of ranks of elliptic curves and abelian varieties in towers of function fields will be also discussed.

Time: 14.15 / Place: HWZ (HG G43)

**Number Theory Seminar HS08:**

26.09.2008 Prof. Philippe Michel (EPFL)

The subconvexity problem for GL_2

Abstract: In this talk we will describe the general subconvexity problem for central value of L-functions. We will also explain the resolution of this problem for GL_1 and GL_2 automorphic L-functions over a general number field, and this uniformly in all parameters (the spectral, level and s-aspects). The main ingredient of the proof are -suitable representations of the central values in terms of automorphic periods which factor over local period integrals of matrix coefficients, -the spectral decomposition of such periods -the spectral gap property for GL_2 matrix coefficients -and the amplification method of Iwaniec If time permit we will also describe an application -explained to us by Andre Reznikov- of the uniformity of our bounds to the study of the restriction of Maass forms of large Laplace eigenvalue along a fixed closed geodesic. This is joint work with Akshay Venkatesh.

Time: 14.15 / Place: HWZ (HG G43)

03.10.2008 Prof. Emmanuel Kowalski (ETH Zurich)

Independence of zeros of zeta functions of algebraic curves over finite fields

Abstract: It is expected that the (positive) ordinates of zeros of Dirichlet L-functions, and in particular of the Riemann zeta function, are linearly independent over the rationals. This would have very interesting applications concerning a number of subtle issues in the distribution of prime numbers (e.g., the so-called Chebychev bias). No approach is currently known to attack this type of problems, but it is possible to consider its analogues for zeta functions over finite fields, and to prove some statements in that context that are encouraging, particularly in view of the method of proof, which combines the Random Matrix philosophy for modeling L-functions with techniques of algebraic number theory and sieve methods.

Time: 14.15 / Place: HWZ (HG G43)

10.10.2008 Prof. Alexey Parshin (Steklov Mathematical Institute)

Higher adelic groups (representations and related results)

Abstract: We will consider the following issues: 1. Adeles on an algebraic surface. 2. Unramified adelic groups, central extensions and discrete nilpotent groups. 3. Characters of representations and modular forms. 4. Relations with L-functions of surfaces.

Time: 14.15 / Place: HWZ (HG G43)

17.10.2008 Dr. Shanta Laishram (University of Waterloo / MPI Bonn)

Squares in arithmetic progression

Abstract: A result of Fermat states that there are no four squares in an arithmetic progression and Euler gave a general result that product of four terms of an arithmetic progression is never a square. Hirata-Kohno, Laishram, Shorey and Tijdeman extended Euler's result upto 109 terms. For this, we consider the Diophantine equation n(n+d)...(n+(k-1)d)=y^2 with n>=1, d>=2, k>=4 and gcd(n,d)=1. In this talk, I will give some history and discuss the above result and related results. In fact, in a joint work with Shorey, we show that the above equation has no solution when d<=10^(10) or d has at most five prime divisors.

Time: 14.15 / Place: HWZ (HG G43)

24.10.2008 Prof. Jan Hendrik Bruinier (University of Darmstadt)

Faltings heights of CM cycles and derivatives of L-functions

Abstract: We report on joint work with T. Yang. We study the Faltings height pairing of arithmetic Heegner divisors and CM cycles on Shimura varieties associated to orthogonal groups. We compute the Archimedian contribution to the height pairing and derive a conjecture relating the total pairing to the central derivative of a Rankin L-function. We briefly discuss how the conjecture can be proved in certain low dimensional cases. In particular, this leads to a new proof of the Gross-Zagier formula.

Time: 14.15 / Place: HWZ (HG G43)

31.10.2008 No seminar.

FIM-Conference Random matrices, L-functions, and primes at ETH Zurich

(27.-31.10.2008)

Organizers: E. Kowalski and A. Nikeghbali

14.11.2008 Prof. Patrick Solé (CNRS)

Codes, Invariants, and Modular Forms

Abstract: In 1972 Broue' and Enguehard derived an isomorphism between the ring of modular forms of weight multiple of 4 and the ring of polynomial invariants of a reflection group of order 192 that contains the weight enumerators of doubly even self dual codes. In the 90's there was a renaissance of that subject due to the advent of self dual codes over rings. We survey these developments and discuss recent results (in collaboration with Choie) related to modular lattices of various levels.

Time: 14.15 / Place: HWZ (HG G43)

21.11.2008 Prof. Enrico Bombieri (IAS Princeton)

Roots of Polynomials in Subgroups of F_p^* and Applications to Congruences

Abstract: The study of congruences (mod p) for sparse polynomials in one variable arises in several contexts from diophantine problems to computer science. Usual methods based on Fourier analysis (mod p) do not work well when differences between exponents have very large common factors with p-1. In this work, jointly with J. Bourgain and V.S. Konyagin, we introduce a new technique to handle the difficult case. The main new tool shows in a precise quantitative way that it is not possible to describe a 'large' subgroup of F_p^* by means of equations of 'small' degree and height; this is yet another example of the principle of 'independence' between addition and multiplication in a finite field. The reduction of the original problem about congruences to the principle alluded to above involves a study of the geometry of intersections of varieties of Fermat type, the Arithmetic Bézout Theorem, the Arithmetic Hilbert Nullstellensatz, as well as certain higher dimensional variants of the above principle of independence. The arguments involved are elementary but not entirely obvious, in the authors' opinion. After the talk coffee will be served in the Common Room of the Graduate School (HG G69). Everybody is welcome!

Time: 14.15 / Place: HG G3

28.11.2008 Prof. Frits Beukers (University of Utrecht)

Algebraic hypergeometric functions in several variables

Abstract: The problem of classifying Euler-Gauss hypergeometric functions is a classical one and basically solved by H.A. Schwarz in 1873. In the course of time there have been many generalisations of hypergeometric equations such as the higher order ones, Appell-Lauricella functions and Horn functions. All these can be fit into the elegant framework of hypergeometric functions of GKZ-type (Gel'fand, Kapranov, Zelevinski). In this lecture we address the question to recognize GKZ-hypergeometric functions which are algebraic functions of their arguments.

Time: 14.15 / Place: HWZ (HG G43)

05.12.2008 Prof. Gunther Cornelissen (University of Utrecht)

Toroidal automorphic forms and the Riemann hypothesis

Abstract: In the 1970's, Don Zagier introduced toroidal automorphic forms to study the zeros of zeta functions. An automorphic form on GL(2) is toroidal if all its right translates integrate to zero over all nonsplit tori in GL(2). In the upper half plane, this corresponds to summing over CM-points (for negative discriminant), or integrating along geodesics (for positive discriminant). An Eisenstein series is toroidal if its weight is a zero of the zeta function of the corresponding field. We compute the space of such forms for the function field of three elliptic curves over finite fields. The method is elementary: we reduce the vanishing of toroidal integrals to an infinite system of linear equations on some graph. We deduce an "automorphic'' proof for the Riemann hypothesis for the zeta function of those curves. Joint work with Oliver Lorscheid.

Time: 14.15 / Place: HWZ (HG G43)

05.12.2008 Dr. Fabien Pazuki (University of Paris 7)

A uniform bound on rational points for families of genus 2 curves over number fields

Abstract: Let k be a number field. We give in this talk an explicit uniform bound on the number of k-rational points on (families of) genus 2 curves over k. The bound depends only on the degree of k and the ranks of the Mordell-Weil groups of the jacobians. This result is a consequence of a lower bound on the canonical height for abelian varieties of dimension 2 that we will discuss in details.

Time: 15.45 / Place: HWZ (HG G43)

12.12.2008 Dr. Anna Cadoret (University of Bordeaux 1)

A uniform open image theorem for p-adic representations of etale fundamental groups of curves

Abstract

Time: 14.15 / Place: HWZ (HG G43)

**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)

**Number Theory Seminar HS07:**

28.09.2007 Dr. Jonathan Hanke (MPI Bonn / Duke Univ.)

The 290-Theorem and Representing Numbers by Quadratic Forms

Abstract: This talk will describe several finiteness theorems for quadratic forms, and progress on the question: "Which positive definite integer-valued quadratic forms represent all positive integers?". The answer to this question depends on settling the related question "Which integers are represented by a given quadratic form?" for finitely many forms. The answer to this question can involve both arithmetic and analytic techniques, though only recently has the analytic approach become practical. We will describe the theory of quadratic forms as it relates to answering these questions, its connections with the theory of modular forms, and give an idea of how one can obtain explicit bounds to describe which numbers are represented by a given quadratic form.

Time: 14.15 / Place: HWZ (HG G43)

05.10.2007 Dr. Lenny Taelman (Univ. Leiden)

Number fields and function fields: special values of L-functions

Abstract

Time: 14.15 / Place: HWZ (HG G43)

12.10.2007 Dr. Ulrich Derenthal (Univ. Zurich)

Universal torsors and rational points on del Pezzo surfaces

Abstract: Del Pezzo surfaces are rational surfaces that often contain infinitely many rational points. A conjecture of Manin predicts the distribution of rational points on these surfaces. Using universal torsors, Manin's conjecture has been proved for several smooth and singular del Pezzo surfaces. This talk will give an introduction to the universal torsor approach and report on current results.

Time: 14.15 / Place: HWZ (HG G43)

19.10.2007 Prof. Wadim Zudilin (MPI Bonn / Moscow State Univ.)

Ramanujan's formulae for 1/pi and their generalisations

Abstract: In 1914 S. Ramanujan recorded a list of 17 series for 1/pi, which produces rapidly converging (rational) approximations to pi. I will survey the methods of proofs of Ramanujan's formulae and indicate recently discovered generalisations, some of which are not yet proved. Surprisingly, the story of these formulas is related to Apery's constant, zeta(3), and in my talk I plan to show that both Ramanujan's and Apery's discoveries have several common grounds.

Time: 14.15 / Place: HWZ (HG G43)

26.10.2007 Dr. Gael Remond (Univ. Grenoble I / CNRS)

Around the Zilber-Pink conjecture

Abstract: What I call the Zilber-Pink conjecture is the intersection of a conjecture of R. Pink (2005) on mixed Shimura varieties and a conjecture of B. Zilber (2002) on semi-abelian varieties. Some cases were considered earlier by E. Bombieri, D. Masser and U. Zannier (1999, 2003) who also formulated general conjectures (2006) and S. Zhang (unpublished). The conjecture is as follows : if we take an irreducible subvariety X of a semi-abelian variety A and intersect it with the union of all semi-abelian subvarieties B of A of codimension greater than the dimension of X, then the intersection is not Zariski-dense in X unless X is contained in a proper algebraic subgroup of A. I discuss the known results with emphasis on the case where A is an abelian variety.

Time: 14.15 / Place: HWZ (HG G43)

09.11.2007 Dr. Kathrin Bringmann (Univ. of Minnesota)

Hypergeometric series, automorphic forms, and mock theta function

Abstract

Time: 14.15 / Place: HWZ (HG G43)

16.11.2007 Prof. Torsten Wedhorn (Univ. Paderborn)

Supersingular Rapoport-Zink-spaces

Abstract

Time: 14.15 / Place: HWZ (HG G43)

20.11.2007 Dr. Cristiana Bertolin (Univ. Regensburg)

The Schanuel conjecture

Abstract: We investigate the Schanuel Conjecture looking at its implications from transcendental number theory to algebraic geometry.

Time: 13.15 / Place: HWZ (HG G43)

23.11.2007 Kilian Kilger (Univ. Heidelberg)

Periods of cusp forms of level two and an application

Abstract: In this talk I show how to find an explicit basis for the dual space of S_k(2) (cusp forms of weight k and level 2) using periods in a very elementary manner. Together with work of Imamoglu and Kohnen this gives bases for S_k(2) in terms of products of Eisenstein series and has applications in elementary number theory.

Time: 14.15 / Place: HWZ (HG G43)

23.11.2007 Prof. Jay Jorgenson (CCNY)

Relating L-functions for Maass forms and holomorphic forms

Abstract: In previous joint work with Jurg Kramer (Berlin), we developed analytic relations involving an orthonormal basis of holomorphic 1-forms and the hyperbolic heat kernel, all data associated to any finite volume hyperbolic Riemann surface. In this talk, I will describe how from these relations one can prove new relations amongst L-functions.

Time: 15.45 / Place: HWZ (HG G43)

30.11.2007 Prof. Pietro Corvaja (Univ. Udine)

Integral points on rational surfaces and applications

Abstract: In recent works with Umberto Zannier, we developed a new method to study integral points on quasi projective varieties, proving in particular the degeneracy of integral points on affine surfaces whose divisor at infinity is sufficiently reducible. In this talk, I shall present these new results, which constitute particular cases of Vojta's conjecture for open varieties; I shall also show some concrete applications to explicit diophantine equations and to problems of divisibility between values of polynomials at integral points.

Time: 14.15 / Place: HWZ (HG G43)

07.12.2007 Dr. Aaron Levin (Brown Univ. / SNS Pisa)

Ideal Class Groups and Rational Torsion in Jacobians of Curves

Abstract: We study the problem of constructing and enumerating, for any integers m,n>1, number fields of degree n whose ideal class groups have "large" m-rank. Our technique, which is new, relies on the Hilbert Irreducibility Theorem and finding certain curves whose Jacobians have a large rational torsion subgroup. Using this technique we improve on results of Nakano, Bilu-Luca, and others.

Time: 14.15 / Place: HWZ (HG G43)

14.12.2007 Martin Widmer (Univ. Basel)

Counting algebraic points of fixed degree and bounded height

Abstract

Time: 14.15 / Place: HWZ (HG G43)

**Number Theory Seminar SS07:**

23.03.2007 Dr. Evelina Viada-Aehle (Univ. of Fribourg Suisse)

About algebraic points on a curve embedded in a product of elliptic curves

Abstract

Time: 14.15 / Place: HWZ (HG G43)

30.03.2007 Prof. Yann Bugeaud (Univ. of Strasbourg)

On the decimal expansion of algebraic numbers

Abstract

Time: 14.15 / Place: HWZ (HG G43)

30.03.2007 Prof. Michel Waldschmidt (Univ. Paris VI)

Density of rational points on Abelian varieties

Abstract

Time: 15.45 / Place: HWZ (HG G43)

20.04.2007 Dr. Bernhard Heim (MPI Bonn)

Special values of automorphic L-functions

Abstract

Time: 14.15 / Place: HWZ (HG G43)

27.04.2007 Dr. Chia-Fu Yu (Academia Sinica/MPI Bonn)

Irreducibility and p-adic monodromies on the Siegel moduli spaces

Abstract

Time: 14.15 / Place: HWZ (HG G43)

04.05.2007 Philipp Habegger (Univ. Basel)

Relations on a power of an elliptic curve

Abstract

Time: 14.15 / Place: HWZ (HG G43)

01.06.2007 Dr. Tim Browning (Univ. of Bristol)

Integral and rational points on cubic

Abstract

Time: 14.15 / Place: HWZ (HG G43)

08.06.2007 Nicolas Stalder (ETH Zurich)

Algebraic Monodromy Groups of A-Motives

Abstract

Time: 14.15 / Place: HWZ (HG G43)

15.06.2007 Dr. Jens Funke (New Mexico State Univ.)

Special cohomology classes arising from the Weil representation

Abstract

Time: 14.15 / Place: HWZ (HG G43)

19.06.2007 Dr. Clemens Fuchs (ETH Zurich)

Schlechte und gute Nachrichten für Herrn Hilbert

Abstract

Time: 14.15 / Place: HWZ (HG G43)

**Number Theory Seminar WS06/07:**

23.10.2006 Thomas J. Engelsma (Operational Techniques, Inc., Marshall, Michigan, USA)

How Many Primes Can Exist Among 3159 Consecutive Integers

Abstract

Time: 14.15 / Place: HWZ (HG G43)

03.11.2006 Dr. Clemens Fuchs (ETH Zürich)

Polynomial-exponential equations and the Subspace Theorem

Abstract

Time: 14.15 / Place: HWZ (HG G43)

10.11.2006 HD Dr. Walter Gubler (Univ. Dortmund)

Tropische analytische Geometrie und die Bogomolovvermutung

Abstract

Time: 14.15 / Place: HWZ (HG G43)

17.11.2006 Prof. David Masser (Univ. Basel)

Multiplicative dependence on algebraic varieties

Abstract

Time: 14.15 / Place: HWZ (HG G43)

24.11.2006 Prof. Yuri Tschinkel (Georg-August-Univ. Göttingen, NYU)

Arithmetic over function fields of curves

Abstract

Time: 14.15 / Place: HWZ (HG G43)

27.11.2006 Prof. Winfried Kohnen (Univ. Heidelberg)

Sign changes of Fourier coefficients and Hecke eigenvalues of cusp forms

Abstract

Time: 14.15 / Place: HWZ (HG G43)

01.12.2006 Dr. Ulrich Görtz (Univ. Bonn)

Affine flag varieties and bad reduction of Shimura varieties

Abstract

Time: 14.15 / Place: HWZ (HG G43)

08.12.2006 Dr. Riad Masri (MPI Bonn)

Equidistribution, Subconvexity, and Nonvanishing of Hecke L-series

Abstract

Time: 14.15 / Place: HWZ (HG G43)

15.12.2006 Dr. Amanda Folsom (MPI Bonn)

Modular units, Selberg q-difference equations, and the cuspidal divisor class group

Abstract

Time: 14.15 / Place: HWZ (HG G43)

05.01.2007 Prof. Kang Zuo (Johannes Gutenberg-Univ. Mainz)

Special subvarieties of moduli stacks (joint work with Eckart Viehweg)

Abstract

Time: 14.15 / Place: HWZ (HG G43)

12.01.2007 Prof. Stefan Müller-Stach (Johannes Gutenberg-Univ. Mainz)

Algebraic cycles and their cohomology classes

Abstract

Time: 14.15 / Place: HWZ (HG G43)

19.01.2007 Dr. Kathrin Bringmann (Univ. of Wisconsin)

Freeman Dyson's "Challenge for the Future": The mock theta functions

Abstract

Time: 14.15 / Place: HWZ (HG G43)

26.01.2007 Prof. Robert Tichy (TU Graz)

Applications of Diophantine Equations to Combinatorial Problems

Abstract

Time: 14.15 / Place: HWZ (HG G43)

Back to the main page.

G. Wüstholz

R. Pink

Ö. Imamoglu

E. Kowalski (since HS08)

C. Fuchs (until FS12)

The seminars later than FS09 can be found here.

17.02.2009 Mini - Workshop, Day 1

(organized by G. Wüstholz and Jing Yu)

Program

Time: 14.15 - 17.15 / Place: HG G19.2

18.02.2009 Mini - Workshop, Day 2

(organized by G. Wüstholz and Jing Yu)

Program

Time: 14.15 - 17.15 / Place: HG G19.2

27.02.2009 Prof. Umberto Zannier (SNS Pisa)

Hilbert Irreducibility above algebraic groups

Abstract: We shall discuss some versions of Hilbert Irreducibility Theorem, concerning the lifting of rational points on certain commutative algebraic groups to covers of them. The proofs use a new method which works essentially by first considering maximal cyclotomic extensions and then descending the field with Chebotarev Theorem. We deal explicitly with multiplicative tori and powers of an elliptic curve (but the method in principle could work more generally). Another application is to a Bertini Theorem for multiplicative tori in place of vector spaces.

Time: 14.15 / Place: HWZ (HG G43)

06.03.2009 Prof. Klaus Schmidt (Univ. Wien/ESI)

Salem numbers, beta-shifts and quasihyperbolic toral automorphisms

Abstract: An irreducible automorphism A in GL(n,Z) of the n-dimensional torus T^n is called quasihyperbolic if it is ergodic and nonexpansive (i.e., A has an eigenvalue of absolute value 1 which is not a root of unity). This talk is about invariant sets and invariant probability measures of such automorphisms, as well as about some open questions involving beta-expansions of Salem numbers. The main results presented are joint work with Elon Lindenstrauss.

Time: 14.15 / Place: HWZ (HG G43)

13.03.2009 Dr. Alex V. Kontorovich (Brown Univ.)

Apollonian Circle Packings and Horospherical Flows on Hyperbolic 3-Manifolds

Abstract: We prove an asymptotic formula for the number of circles in an Apollonian packing of bounded curvature. Using the affine linear sieve, we give sharp upper bounds for the number of circles of prime curvature, and the number of "twin prime" tangent circles. The main ingredient of our proof is the equidistribution of long horospherical flows in the unit tangent bundle of an infinite volume hyperbolic 3-manifold, under the assumption that the Hausdorff dimension of its limit set exceeds one. This is joint work with Hee Oh.

Time: 14.15 / Place: HWZ (HG G43)

20.03.2009 Prof. Urs Hartl (WWU Münster)

Rapoport-Zink spaces and deformation spaces for local shtukas

Abstract: Bounded local shtukas are function field analogs for p-divisible groups and arise from Drinfeld modules and Anderson motives in the same way as p-divisible groups arise from abelian varietes. We describe the deformations and moduli spaces of bounded local shtukas. The latter are analogous to Rapoport-Zink spaces for p-divisible groups. The underlying schemes of these moduli spaces are affine Deligne-Lusztig varieties. This is joint work with Eva Viehmann).

Time: 14.15 / Place: HWZ (HG G43)

27.03.2009 Dr. Bernhard Köck (University of Southampton)

Real Belyi Theory

Abstract: Belyi's famous theorem states that a compact Riemann surface can be defined over a number field if and only if it admits a meromorphic function with at most three critical values. My talk will be about a generalization of this theorem to Klein surfaces, i.e. (possibly non-orientable) surfaces with boundary which carry a dianalytic structure. I will also explain some other characterizations (triangle groups, maps on surfaces, ...). The results presented are joint work with David Singerman on the one hand and with Eike Lau on the other hand.

Time: 14.15 / Place: HWZ (HG G43)

17.04.2009 No seminar.

ETH Number Theory Days at EPF Lausanne

(16.-17.04.2009)

Organizers: P. Michel and R. Pink

08.05.2009 Prof. Jianqiang Zhao (MPI Bonn)

Motivic fundamental group of G_m - mu_N and special values of multi-polylogs

Abstract: In this talk we will consider the Q-linear relations among the values of multiple polylogarithms evaluated at mu_N, where mu_N is the group of the Nth roots of unity. We show that the standard relations considered by Racinet do not provide all the possible relations in the following cases: (i) level N = 4, weight w = 3 or 4, and (ii) w = 2, 7 < N < 169, and N is a power of 2 or 3, or N has at least two prime factors. We further find some (presumably all) of the missing relations in (i) by using the octahedral symmetry of G_m - mu_4. We also prove some other results when N = p, p^2, p^3 (p prime >= 5) by using the motivic fundamental group of G_m - u_N.

Time: 14.15 / Place: HWZ (HG G43)

15.05.2009 No seminar.

Leopoldina-Symposium in Algebraic and Arithmetic Algebraic Geometry at CSF, Monte-Verità

(10.-15.05.2009)

Organizers: C. Fuchs, J. Kramer and G. Wüstholz

22.05.2009 Dr. Christopher Hall (University of Michigan)

Beyond Hilbert Irreducibility

Abstract: If f(t,x) is a polynomial in Q[t,x] which is irreducible is a polynomial with coefficients in the field of rational functions Q(t), then Hilbert's irreducibility theorem says that for 'most' positive integers t_0, the specialized polynomial f(t_0,x) is irreducible in Q[x]. We'll explain how one can use Faltings's theorems on the Mordell and Lang conjectures to say more, including cases where t_0 is allowed to vary over all elements in all finite extensions k/Q of bounded degree. In particular, we'll describe families of polynomials f(t,x) where we can give very strong finiteness statements about the Galois groups of the specialized polynomials for almost all elements of the family.

Time: 14.15 / Place: HWZ (HG G43)

07.08.2009 Prof. Yuri Zarhin (Penn State)

Endomorphisms and homomorphisms of superelliptic jacobians

Abstract: The Tate conjecture on homomorphisms of abelian varieties (now a theorem) asserts that the knowledge of the Galois action on all torsion points of an abelian variety Z allows us (in principle) to reconstruct the endomorphism algebra of Z when the ground field K is finitely generated over its prime subfield. We provide explicit examples where the knowledge of the Galois action on a (relatively) small (sub)group of torsion points allows us to recover the ring of all endomorphisms that are defined not only over K but over its algebraic closure. Applications to the computation of ranks of elliptic curves and abelian varieties in towers of function fields will be also discussed.

Time: 14.15 / Place: HWZ (HG G43)

26.09.2008 Prof. Philippe Michel (EPFL)

The subconvexity problem for GL_2

Abstract: In this talk we will describe the general subconvexity problem for central value of L-functions. We will also explain the resolution of this problem for GL_1 and GL_2 automorphic L-functions over a general number field, and this uniformly in all parameters (the spectral, level and s-aspects). The main ingredient of the proof are -suitable representations of the central values in terms of automorphic periods which factor over local period integrals of matrix coefficients, -the spectral decomposition of such periods -the spectral gap property for GL_2 matrix coefficients -and the amplification method of Iwaniec If time permit we will also describe an application -explained to us by Andre Reznikov- of the uniformity of our bounds to the study of the restriction of Maass forms of large Laplace eigenvalue along a fixed closed geodesic. This is joint work with Akshay Venkatesh.

Time: 14.15 / Place: HWZ (HG G43)

03.10.2008 Prof. Emmanuel Kowalski (ETH Zurich)

Independence of zeros of zeta functions of algebraic curves over finite fields

Abstract: It is expected that the (positive) ordinates of zeros of Dirichlet L-functions, and in particular of the Riemann zeta function, are linearly independent over the rationals. This would have very interesting applications concerning a number of subtle issues in the distribution of prime numbers (e.g., the so-called Chebychev bias). No approach is currently known to attack this type of problems, but it is possible to consider its analogues for zeta functions over finite fields, and to prove some statements in that context that are encouraging, particularly in view of the method of proof, which combines the Random Matrix philosophy for modeling L-functions with techniques of algebraic number theory and sieve methods.

Time: 14.15 / Place: HWZ (HG G43)

10.10.2008 Prof. Alexey Parshin (Steklov Mathematical Institute)

Higher adelic groups (representations and related results)

Abstract: We will consider the following issues: 1. Adeles on an algebraic surface. 2. Unramified adelic groups, central extensions and discrete nilpotent groups. 3. Characters of representations and modular forms. 4. Relations with L-functions of surfaces.

Time: 14.15 / Place: HWZ (HG G43)

17.10.2008 Dr. Shanta Laishram (University of Waterloo / MPI Bonn)

Squares in arithmetic progression

Abstract: A result of Fermat states that there are no four squares in an arithmetic progression and Euler gave a general result that product of four terms of an arithmetic progression is never a square. Hirata-Kohno, Laishram, Shorey and Tijdeman extended Euler's result upto 109 terms. For this, we consider the Diophantine equation n(n+d)...(n+(k-1)d)=y^2 with n>=1, d>=2, k>=4 and gcd(n,d)=1. In this talk, I will give some history and discuss the above result and related results. In fact, in a joint work with Shorey, we show that the above equation has no solution when d<=10^(10) or d has at most five prime divisors.

Time: 14.15 / Place: HWZ (HG G43)

24.10.2008 Prof. Jan Hendrik Bruinier (University of Darmstadt)

Faltings heights of CM cycles and derivatives of L-functions

Abstract: We report on joint work with T. Yang. We study the Faltings height pairing of arithmetic Heegner divisors and CM cycles on Shimura varieties associated to orthogonal groups. We compute the Archimedian contribution to the height pairing and derive a conjecture relating the total pairing to the central derivative of a Rankin L-function. We briefly discuss how the conjecture can be proved in certain low dimensional cases. In particular, this leads to a new proof of the Gross-Zagier formula.

Time: 14.15 / Place: HWZ (HG G43)

31.10.2008 No seminar.

FIM-Conference Random matrices, L-functions, and primes at ETH Zurich

(27.-31.10.2008)

Organizers: E. Kowalski and A. Nikeghbali

14.11.2008 Prof. Patrick Solé (CNRS)

Codes, Invariants, and Modular Forms

Abstract: In 1972 Broue' and Enguehard derived an isomorphism between the ring of modular forms of weight multiple of 4 and the ring of polynomial invariants of a reflection group of order 192 that contains the weight enumerators of doubly even self dual codes. In the 90's there was a renaissance of that subject due to the advent of self dual codes over rings. We survey these developments and discuss recent results (in collaboration with Choie) related to modular lattices of various levels.

Time: 14.15 / Place: HWZ (HG G43)

21.11.2008 Prof. Enrico Bombieri (IAS Princeton)

Roots of Polynomials in Subgroups of F_p^* and Applications to Congruences

Abstract: The study of congruences (mod p) for sparse polynomials in one variable arises in several contexts from diophantine problems to computer science. Usual methods based on Fourier analysis (mod p) do not work well when differences between exponents have very large common factors with p-1. In this work, jointly with J. Bourgain and V.S. Konyagin, we introduce a new technique to handle the difficult case. The main new tool shows in a precise quantitative way that it is not possible to describe a 'large' subgroup of F_p^* by means of equations of 'small' degree and height; this is yet another example of the principle of 'independence' between addition and multiplication in a finite field. The reduction of the original problem about congruences to the principle alluded to above involves a study of the geometry of intersections of varieties of Fermat type, the Arithmetic Bézout Theorem, the Arithmetic Hilbert Nullstellensatz, as well as certain higher dimensional variants of the above principle of independence. The arguments involved are elementary but not entirely obvious, in the authors' opinion. After the talk coffee will be served in the Common Room of the Graduate School (HG G69). Everybody is welcome!

Time: 14.15 / Place: HG G3

28.11.2008 Prof. Frits Beukers (University of Utrecht)

Algebraic hypergeometric functions in several variables

Abstract: The problem of classifying Euler-Gauss hypergeometric functions is a classical one and basically solved by H.A. Schwarz in 1873. In the course of time there have been many generalisations of hypergeometric equations such as the higher order ones, Appell-Lauricella functions and Horn functions. All these can be fit into the elegant framework of hypergeometric functions of GKZ-type (Gel'fand, Kapranov, Zelevinski). In this lecture we address the question to recognize GKZ-hypergeometric functions which are algebraic functions of their arguments.

Time: 14.15 / Place: HWZ (HG G43)

05.12.2008 Prof. Gunther Cornelissen (University of Utrecht)

Toroidal automorphic forms and the Riemann hypothesis

Abstract: In the 1970's, Don Zagier introduced toroidal automorphic forms to study the zeros of zeta functions. An automorphic form on GL(2) is toroidal if all its right translates integrate to zero over all nonsplit tori in GL(2). In the upper half plane, this corresponds to summing over CM-points (for negative discriminant), or integrating along geodesics (for positive discriminant). An Eisenstein series is toroidal if its weight is a zero of the zeta function of the corresponding field. We compute the space of such forms for the function field of three elliptic curves over finite fields. The method is elementary: we reduce the vanishing of toroidal integrals to an infinite system of linear equations on some graph. We deduce an "automorphic'' proof for the Riemann hypothesis for the zeta function of those curves. Joint work with Oliver Lorscheid.

Time: 14.15 / Place: HWZ (HG G43)

05.12.2008 Dr. Fabien Pazuki (University of Paris 7)

A uniform bound on rational points for families of genus 2 curves over number fields

Abstract: Let k be a number field. We give in this talk an explicit uniform bound on the number of k-rational points on (families of) genus 2 curves over k. The bound depends only on the degree of k and the ranks of the Mordell-Weil groups of the jacobians. This result is a consequence of a lower bound on the canonical height for abelian varieties of dimension 2 that we will discuss in details.

Time: 15.45 / Place: HWZ (HG G43)

12.12.2008 Dr. Anna Cadoret (University of Bordeaux 1)

A uniform open image theorem for p-adic representations of etale fundamental groups of curves

Abstract

Time: 14.15 / Place: HWZ (HG G43)

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)

28.09.2007 Dr. Jonathan Hanke (MPI Bonn / Duke Univ.)

The 290-Theorem and Representing Numbers by Quadratic Forms

Abstract: This talk will describe several finiteness theorems for quadratic forms, and progress on the question: "Which positive definite integer-valued quadratic forms represent all positive integers?". The answer to this question depends on settling the related question "Which integers are represented by a given quadratic form?" for finitely many forms. The answer to this question can involve both arithmetic and analytic techniques, though only recently has the analytic approach become practical. We will describe the theory of quadratic forms as it relates to answering these questions, its connections with the theory of modular forms, and give an idea of how one can obtain explicit bounds to describe which numbers are represented by a given quadratic form.

Time: 14.15 / Place: HWZ (HG G43)

05.10.2007 Dr. Lenny Taelman (Univ. Leiden)

Number fields and function fields: special values of L-functions

Abstract

Time: 14.15 / Place: HWZ (HG G43)

12.10.2007 Dr. Ulrich Derenthal (Univ. Zurich)

Universal torsors and rational points on del Pezzo surfaces

Abstract: Del Pezzo surfaces are rational surfaces that often contain infinitely many rational points. A conjecture of Manin predicts the distribution of rational points on these surfaces. Using universal torsors, Manin's conjecture has been proved for several smooth and singular del Pezzo surfaces. This talk will give an introduction to the universal torsor approach and report on current results.

Time: 14.15 / Place: HWZ (HG G43)

19.10.2007 Prof. Wadim Zudilin (MPI Bonn / Moscow State Univ.)

Ramanujan's formulae for 1/pi and their generalisations

Abstract: In 1914 S. Ramanujan recorded a list of 17 series for 1/pi, which produces rapidly converging (rational) approximations to pi. I will survey the methods of proofs of Ramanujan's formulae and indicate recently discovered generalisations, some of which are not yet proved. Surprisingly, the story of these formulas is related to Apery's constant, zeta(3), and in my talk I plan to show that both Ramanujan's and Apery's discoveries have several common grounds.

Time: 14.15 / Place: HWZ (HG G43)

26.10.2007 Dr. Gael Remond (Univ. Grenoble I / CNRS)

Around the Zilber-Pink conjecture

Abstract: What I call the Zilber-Pink conjecture is the intersection of a conjecture of R. Pink (2005) on mixed Shimura varieties and a conjecture of B. Zilber (2002) on semi-abelian varieties. Some cases were considered earlier by E. Bombieri, D. Masser and U. Zannier (1999, 2003) who also formulated general conjectures (2006) and S. Zhang (unpublished). The conjecture is as follows : if we take an irreducible subvariety X of a semi-abelian variety A and intersect it with the union of all semi-abelian subvarieties B of A of codimension greater than the dimension of X, then the intersection is not Zariski-dense in X unless X is contained in a proper algebraic subgroup of A. I discuss the known results with emphasis on the case where A is an abelian variety.

Time: 14.15 / Place: HWZ (HG G43)

09.11.2007 Dr. Kathrin Bringmann (Univ. of Minnesota)

Hypergeometric series, automorphic forms, and mock theta function

Abstract

Time: 14.15 / Place: HWZ (HG G43)

16.11.2007 Prof. Torsten Wedhorn (Univ. Paderborn)

Supersingular Rapoport-Zink-spaces

Abstract

Time: 14.15 / Place: HWZ (HG G43)

20.11.2007 Dr. Cristiana Bertolin (Univ. Regensburg)

The Schanuel conjecture

Abstract: We investigate the Schanuel Conjecture looking at its implications from transcendental number theory to algebraic geometry.

Time: 13.15 / Place: HWZ (HG G43)

23.11.2007 Kilian Kilger (Univ. Heidelberg)

Periods of cusp forms of level two and an application

Abstract: In this talk I show how to find an explicit basis for the dual space of S_k(2) (cusp forms of weight k and level 2) using periods in a very elementary manner. Together with work of Imamoglu and Kohnen this gives bases for S_k(2) in terms of products of Eisenstein series and has applications in elementary number theory.

Time: 14.15 / Place: HWZ (HG G43)

23.11.2007 Prof. Jay Jorgenson (CCNY)

Relating L-functions for Maass forms and holomorphic forms

Abstract: In previous joint work with Jurg Kramer (Berlin), we developed analytic relations involving an orthonormal basis of holomorphic 1-forms and the hyperbolic heat kernel, all data associated to any finite volume hyperbolic Riemann surface. In this talk, I will describe how from these relations one can prove new relations amongst L-functions.

Time: 15.45 / Place: HWZ (HG G43)

30.11.2007 Prof. Pietro Corvaja (Univ. Udine)

Integral points on rational surfaces and applications

Abstract: In recent works with Umberto Zannier, we developed a new method to study integral points on quasi projective varieties, proving in particular the degeneracy of integral points on affine surfaces whose divisor at infinity is sufficiently reducible. In this talk, I shall present these new results, which constitute particular cases of Vojta's conjecture for open varieties; I shall also show some concrete applications to explicit diophantine equations and to problems of divisibility between values of polynomials at integral points.

Time: 14.15 / Place: HWZ (HG G43)

07.12.2007 Dr. Aaron Levin (Brown Univ. / SNS Pisa)

Ideal Class Groups and Rational Torsion in Jacobians of Curves

Abstract: We study the problem of constructing and enumerating, for any integers m,n>1, number fields of degree n whose ideal class groups have "large" m-rank. Our technique, which is new, relies on the Hilbert Irreducibility Theorem and finding certain curves whose Jacobians have a large rational torsion subgroup. Using this technique we improve on results of Nakano, Bilu-Luca, and others.

Time: 14.15 / Place: HWZ (HG G43)

14.12.2007 Martin Widmer (Univ. Basel)

Counting algebraic points of fixed degree and bounded height

Abstract

Time: 14.15 / Place: HWZ (HG G43)

23.03.2007 Dr. Evelina Viada-Aehle (Univ. of Fribourg Suisse)

About algebraic points on a curve embedded in a product of elliptic curves

Abstract

Time: 14.15 / Place: HWZ (HG G43)

30.03.2007 Prof. Yann Bugeaud (Univ. of Strasbourg)

On the decimal expansion of algebraic numbers

Abstract

Time: 14.15 / Place: HWZ (HG G43)

30.03.2007 Prof. Michel Waldschmidt (Univ. Paris VI)

Density of rational points on Abelian varieties

Abstract

Time: 15.45 / Place: HWZ (HG G43)

20.04.2007 Dr. Bernhard Heim (MPI Bonn)

Special values of automorphic L-functions

Abstract

Time: 14.15 / Place: HWZ (HG G43)

27.04.2007 Dr. Chia-Fu Yu (Academia Sinica/MPI Bonn)

Irreducibility and p-adic monodromies on the Siegel moduli spaces

Abstract

Time: 14.15 / Place: HWZ (HG G43)

04.05.2007 Philipp Habegger (Univ. Basel)

Relations on a power of an elliptic curve

Abstract

Time: 14.15 / Place: HWZ (HG G43)

01.06.2007 Dr. Tim Browning (Univ. of Bristol)

Integral and rational points on cubic

Abstract

Time: 14.15 / Place: HWZ (HG G43)

08.06.2007 Nicolas Stalder (ETH Zurich)

Algebraic Monodromy Groups of A-Motives

Abstract

Time: 14.15 / Place: HWZ (HG G43)

15.06.2007 Dr. Jens Funke (New Mexico State Univ.)

Special cohomology classes arising from the Weil representation

Abstract

Time: 14.15 / Place: HWZ (HG G43)

19.06.2007 Dr. Clemens Fuchs (ETH Zurich)

Schlechte und gute Nachrichten für Herrn Hilbert

Abstract

Time: 14.15 / Place: HWZ (HG G43)

23.10.2006 Thomas J. Engelsma (Operational Techniques, Inc., Marshall, Michigan, USA)

How Many Primes Can Exist Among 3159 Consecutive Integers

Abstract

Time: 14.15 / Place: HWZ (HG G43)

03.11.2006 Dr. Clemens Fuchs (ETH Zürich)

Polynomial-exponential equations and the Subspace Theorem

Abstract

Time: 14.15 / Place: HWZ (HG G43)

10.11.2006 HD Dr. Walter Gubler (Univ. Dortmund)

Tropische analytische Geometrie und die Bogomolovvermutung

Abstract

Time: 14.15 / Place: HWZ (HG G43)

17.11.2006 Prof. David Masser (Univ. Basel)

Multiplicative dependence on algebraic varieties

Abstract

Time: 14.15 / Place: HWZ (HG G43)

24.11.2006 Prof. Yuri Tschinkel (Georg-August-Univ. Göttingen, NYU)

Arithmetic over function fields of curves

Abstract

Time: 14.15 / Place: HWZ (HG G43)

27.11.2006 Prof. Winfried Kohnen (Univ. Heidelberg)

Sign changes of Fourier coefficients and Hecke eigenvalues of cusp forms

Abstract

Time: 14.15 / Place: HWZ (HG G43)

01.12.2006 Dr. Ulrich Görtz (Univ. Bonn)

Affine flag varieties and bad reduction of Shimura varieties

Abstract

Time: 14.15 / Place: HWZ (HG G43)

08.12.2006 Dr. Riad Masri (MPI Bonn)

Equidistribution, Subconvexity, and Nonvanishing of Hecke L-series

Abstract

Time: 14.15 / Place: HWZ (HG G43)

15.12.2006 Dr. Amanda Folsom (MPI Bonn)

Modular units, Selberg q-difference equations, and the cuspidal divisor class group

Abstract

Time: 14.15 / Place: HWZ (HG G43)

05.01.2007 Prof. Kang Zuo (Johannes Gutenberg-Univ. Mainz)

Special subvarieties of moduli stacks (joint work with Eckart Viehweg)

Abstract

Time: 14.15 / Place: HWZ (HG G43)

12.01.2007 Prof. Stefan Müller-Stach (Johannes Gutenberg-Univ. Mainz)

Algebraic cycles and their cohomology classes

Abstract

Time: 14.15 / Place: HWZ (HG G43)

19.01.2007 Dr. Kathrin Bringmann (Univ. of Wisconsin)

Freeman Dyson's "Challenge for the Future": The mock theta functions

Abstract

Time: 14.15 / Place: HWZ (HG G43)

26.01.2007 Prof. Robert Tichy (TU Graz)

Applications of Diophantine Equations to Combinatorial Problems

Abstract

Time: 14.15 / Place: HWZ (HG G43)

Impressum 21.03.2016