# arithmetic algebraic geometry

3-7 September 2007

# Abstracts

## Andrea Bandini

### Control theorem and L-functions for elliptic curves over function fields

Let F be a global field of characteristic p>0, F/F a Galois extension with Gal(F/F)\simeq Zp<b>N and E/F a non-isotrivial elliptic curve. Via an appropriate version of Mazur's Control Theorem (and with a mild hypothesis on SelE(F)p) we prove that for any Zpd-extensions Fd/F contained in F the p-parts SelE(Fd)p of the Selmer groups are cofinitely generated Zp[[Gal(Fd/F)]]-modules. The Fitting ideals associated to the Pontrjagin duals of the Selmer groups form an inverse system and their inverse limit defines an algebraic L-function", providing an ingredient for a function field analogue of Iwasawa's Main Conjecture for elliptic curves.

## Stefan Barańczuk

### Reduction maps for groups of Mordell-Weil type

We consider reduction maps of groups of Mordell-Weil type. Principal examples of such groups are Mordell-Weil groups of abelian varieties over number fields, odd K-groups of number fields and K-groups of curves of number fields. I will discuss some local-global principles for groups of Mordell-Weil type with respect to the reduction maps.

## Spencer Bloch

### Motives associated to graphs

Feynman amplitudes computed by physicists are periods in the sense of arithmetic algebraic geometry associate to graphs. They have a strong tendency to be multiple zeta numbers. I will talk about the construction of motives associated to graphs and how motivic cohomology is related to combinatorial invariants of the graphs.

## Jerzy Browkin

### Cyclotomic elements in K2F for number fields F

The symbol of the form {a,Φn(a)}∈ K2F, where Φn(X) is the n-th cyclotomic polynomial, and a, Φn(a)∈ F×, is called the cyclotomic element. Basing on results of Hourong Qin and Xuejun Guo we prove that for some number fields F every element in K2F is the finite or infinite product of cyclotomic elements.

## David Burns

### Main conjectures in non-commutative Iwasawa

theory and related conjectures

We review some recent results related to the main conjecture of non-commutative Iwasawa theory formulated by Coates, Fukaya, Kato, Sujatha and Venjakob.

## Fred Diamond

### Serre's conjecture and mod p Langlands correspondences

Serre's conjecture, whose proof was recently announced by Khare and Wintenberger, states that every odd, irreducible 2-dimensional mod p Galois representation arises from a modular form. This result can be viewed as a global mod p Langlands correspondence; moreover Serre's refinement predicting the weight and level of the form suggests compatibility with a hypothetical local correspondence.

This talk will be a survey describing what's known for GL2 over Q, and where obstacles lie ahead even for GL2 over number fields. In particular, I'll discuss work of Emerton making the local-global compatility statement precise for GL2 over Q, and work of Breuil and Paskunas illustrating the difficulty of constructing a mod p local Langlands correspondence for GL2 of a p-adic field.

## Ivan Fesenko

### Geometry and analysis on regular models of elliptic curves and their arithmetic

The current state of two dimensional adelic analysis programme on arithmetic schemes, some of the main issues and open problems will be discussed, for a review please see ade.pdf.

## Gerhard Frey

### Curves of genus 2 with elliptic differentials and associated Hurwitz spaces (joint work with E. Kani)

In the lecture we discuss the Hurwitz space related to covers φ : {\PP}1K → {\PP}1K of degree n prime to char(K) ramified in 5 points P1,…, P5 with ramification order at most 2 such that the ramification cycle corresponding to P5 in the Galois closure of the cover is a transposition and explain that it is isomorphic to the moduli space related to covers f : C → E where C is a curve of genus 2, E is an elliptic curve and f is a normalized morphism of degree n. This result enables us to combine methods from group theory and from algebraic and arithmetic geometry to get results about diophantine properties of the moduli spaces like rigidity numbers, special curves and rational points closely related to conjectures (and results) about elliptic curves with Galois isomorphic torsion structures.

## Hidekazu Furusho

### Double shuffle relations for p-adic multiple zeta values

This is on my joint works with Amnon Besser and with Amir Jafari. In my talk I will explain a construction of p-adic (analogue of) multiple zeta values and show that they satisfy double shuffle relations and reguralization relations.

## Krzysztof Górnisiewicz

### Linear dependence in Mordell-Weil groups

We consider a local to global principle for detecting linear dependence of nontorsion points, by reduction maps rv, in the Mordell-Weil group of an abelian variety A defined over a number field F. Especially, we show that in every nonempty isogeny class of abelian varieties over a number field F, there exists an abelian variety A, which nontorsion points P0, P1, …, Pr satisfy the following condition: P0 = f1P1 + … + f1Pr, if $rv(P0) ∈ rv(L)$, where Pi are linearly independent over EndA and generate L \subset A(F) and fi ∈ EndA for i ∈ {1, …, r }.

## Zbigniew Jelonek

### On the cancellation problem

I show that for every n \geq 8 there exists a Zariski open subvariety V of kn and an affine variety F such that V × k = F × k, but V \not= F (k is an algebraically closed field).

## Krzysztof Klosin

### Congruences among automorphic forms on unitary groups and the Bloch-Kato conjecture

The Bloch-Kato conjecture predicts a precise relationship between an L-value attached to a motif and the order of its Selmer group. The aim of this talk is to prove statements towards this conjecture for adjoint motives of modular forms. More precisely, for a prime p, we relate the p-adic valuation of Lalg(Sym2 f, k) to the p-adic valuation of the order of Sel(ad0 \rhof(-1))\vee, where f is a modular form of weight k-1, and \rhof is the p-adic Galois representation attached to f. Our approach is a variation of a method introduced by Ribet (1976) and later developed by Wiles (1990) and Skinner and Urban (2002) in the sense that to achieve our goal we introduce an intermediate step and construct congruences between two different kinds (CAP and non-CAP) of automorphic forms on a certain unitary group.

## Ignazio Longhi

### Coleman's power series for Drinfeld modules

By exploiting the analogy between Lubin-Tate formal groups and rank 1 Drinfeld modules, one can associate power series to inverse systems of units in certain towers of characteristic p function fields, extending an idea of Coleman's. This allows to study analogues of the classical theory over number fields, such as Wiles' explicit reciprocity law.

## Jan Nekovář

### Some questions on Hilbert modular varieties

We formulate several questions on Hilbert modular varieties and answer a different, but related, question.

## Antonella Perucca

### On the order of the reductions of points on an abelian variety

Consider an abelian variety A defined over a number field K and non-torsion points R1,…,Rn in A(K). Fix a non-zero integer m. We find infinitely many primes p of K such that the orders of R1,…,Rn modulo p are all divisible by m (respectively coprime to m under some assumptions on the points).

In the second part of the talk we use similar techniques to weaken the hypotheses in Larsen's theorem on the support problem (2003). Our main tool is the following result by Banaszak, Gajda and Krasoń (2005): if l is a prime number and if a point R in A(K) is killed only by the zero endomorphism, then there exist infinitely many primes p of K such that the order of R modulo p is coprime to l.

## Piotr Pragacz

### "Point" and "diagonal" properties of algebraic schemes

This is a report on a joint work with V. Srinivas and V. Pati. We shall discuss the following 2 properties of an algebraic scheme X:

Point property: for a point x ∈ X there exist a vector bundle E of rank dim X, and a section s of E such that {x}=Z(s).

Diagonal property: there exist a vector bundle E on X × X of rank dim X and a section s of E such that the diagonal of X is Z(s).

Recent results and a conjecture of O. Debarre on abelian varieties will be also mentioned.

## Karl Rubin

### Growth of ranks of elliptic curves in Galois extensions of number fields

Suppose E is an elliptic curve defined over a number field k, K/k is a quadratic extension, and F is a Galois extension of k containing K. The Parity Conjecture can be used to predict a lower bound for the rank of E(F) that sometimes can be quite large. For example, if F/k is dihedral and the rank of E(K) is odd, then under mild assumptions the rank of E(F) should be at least [F:K].

In this talk I will discuss recent joint work with Barry Mazur, where we prove some lower bounds of this type when p is an odd prime, F/K is a p-extension, and with "rank" replaced by "p-Selmer rank".

## René Schoof

### Semistable abelian varieties and modular curves

For every squarefree natural numner n, the Jacobian J0(n) of the modular curve X0(n) is a semistable abelian variety with good reduction outside n that is defined over Q. We show, conversely, that for every odd squarefree n < 30, any semistable abelian variety over Q with good reduction outside n is isogenous to a power of J0(n)

## V. Srinivas

### Algebraic cycles on products of elliptic curves over p-adic fields

In this talk, based on joint work with A. Rosenschon, I will first discuss examples of Schoen of complex varieties with infinite Chow groups mod l, and infinite l-torsion subgroups, for certain primes l. Then I will explain how his examples can be modified to yield similar examples over certain p-adic fields, for any odd prime p. This modification relies on the cuspidal geometry of the Deligne-Rapoport models for modular curves.

## Yoichi Uetake

### Spectral scattering theory for automorphic forms

We construct a scattering process for L2-modular forms on the arithmetic quotient of the upper half-plane by the full modular group. The construction is algebraic in the sense that it depends on some relations satisfied by the Eisenstein series. We discuss some spectral properties of an operator obtained from this process.

## Jean-Pierre Wintenberger

### p-adic lifts of Galois representations.

For a representation of the Galois group of Q in a finite dimensional vector space over a finite field of characteristic p, we shall discuss the existence of a p-adic lift with conditions on ramification.

## Andrei Yafaev

### Manin-Mumford and Andre-Oort conjectures

In this lecture we explain how the combination of Galois-theoretic and ergodic-theoretic techniques imply the Manin-Mumford (work of Ullmo and Ratazzi) and the Andre-Oort (work of Klingler, Ullmo and Yafaev) conjectures.

## Hong You

### Prestability for Quadratic K1 of Λ-1-Fold

The general quadratic group U2n and its elementary subgroup EU2n are analogs in the theory of the general linear group GLn and its elementary subgroup En. It is known, for GLn, K1=⟨(a+c+abc)(a+c+cba)-1 | a+c+abc∈ GL1 under the first Bass stable range condition on the ring of entries. In this article we give a description for KU1 under the Λ-1-fold stable range condition on the ring of entries.