# Trimester Seminar

**Venue:** HIM lecture hall, Poppelsdorfer Allee 45

## Tuesday, January 8

11:00-13:00 Jeanine Marie Van Order (EPFL, Lausanne): Iwasawa main conjectures for GL_{2} via Howard's criterion

Abstract: In this talk, I will present the Iwasawa main conjectures for Hilbert modular eigenforms of parallel weight two in dihedral or anticyclotomic extensions of CM fields. The first part will include an overview of known results, as well as some discussion of open problems and applications (e.g. to bounding Mordell-Weil ranks), and should be accessible to the non-specialist. The second part will describe the p-adic L-functions in more detail, as well as the nonvanishing criterion of Howard (and its implications for the main conjectures).

## Tuesday, January 15

11:00-13:00 Oliver Lorscheid (IMPA, Rio de Janeiro): A blueprinted view on F1-geometry

Abstract: A blueprint is an algebraic structure that "interpolates" between multiplicative monoids and semirings. The associated scheme theory applies to several problems in F1-geometry: Tits' idea of Chevalley groups and buildings over F1, Euler characteristics as the number of F1-rational points, total positivity, K-theory, Arakelov compactifications of arithmetic curves; and it has multiple connections to other branches of algebraic geometry: Lambda-schemes (after Borger), log schemes (after Kato), relative schemes (after Toen and Vaquie), congruence schemes (after Berkovich and Deitmar), idempotent analysis, analytic spaces and tropical geometry.

After a brief overview and an introduction to the basic definitions of this theory, we focus on the combinatorial aspects of blue schemes. In particular, we explain how to realize Jacques Tits' idea of Weyl groups as Chevalley groups over F1 and Coxeter complexes as buildings over F1. The central conepts are the rank space of a blue scheme and the Tits category, which make the idea of "F1-rational points" rigorous.

## Wednesday, January 16

11:00-13:00 Jean-Pierre Wintenberger (Strasbourg): Introduction to Serre's modularity conjecture

Abstract: This lecture is intended for non-specialists. We state Serre's modularity conjecture and give some consequences and hints on its proof.

## Thursday, January 17

10:00-12:00 Henri Carayol (Strasbourg): Realization of some automorphic forms and rationality questions (Part I)

Abstract: In this first (and mostly introductory) talk I shall recall some (well-known) facts on the realization of automorphic forms in the cohomology groups of some geometric objects, and the relation with the arithmetic properties of such forms. I shall introduce locally symmetric varieties, Shimura varieties and the more exotic Griffiths-Schmid varieties. I shall discuss the case of automorphic forms whose archimedean component is a limit of discrete series. In the case of degenerate limits, the only known realization uses the coherent cohomology of Griffiths-Schmid varieties.

## Monday, January 21

10:00-12:00 Günter Harder (Bonn): Modular construction of mixed motives and congruences (Part I)

Abstract: Starting from a Shimura variety S and its compactification S^{∨} we construct certain objects, which be thought of as being mixed motives. These mixed motives give rise to certain elements of Ext^{1} groups. We can use the theory of Eisenstein cohomology to compute the Hodge-de-Rham extension classes of these extension. We also have some conjectural formulas for these extensions as Galois-modules. Assuming the correctness of these formulas for the Galois-extension class we can derive congruences between eigenvalues of Hecke-operators acting on the cohomology of different arithmetic groups, these congruences are congruences modulo primes l dividing certain special values of L functions.

These congruences have been verified experimentally in many cases. They imply the reduciblity of certain Galois-representations mod l.

## Tuesday, January 22

11:00-12:00 Yuri I. Manin (Bonn): Non-commutative generalized Dedekind symbols

Abstract: Classical Dedekind symbol was introduced and studied in connection with functional equation of Dedekind eta-function. Later it was generalized and had multiple applications, in particular to topological invariants. I will define and study generalized Dedekind symbols with values in non-necessarily commutative groups, extending constructions of Sh. Fukuhara done in the commutative context. Basic examples of such symbols are obtained by replacing period integrals of modular forms by iterated period integrals. I will also explain the interpretation of such symbols in terms of non-commutative 1-cocycles.

## Wednesday, January 23

11:00-12:00 Henri Carayol (Strasbourg): Realization of some automorphic forms and rationality questions (Part II)

Abstract: This talk is a continuation of part I.

## Thursday, January 24

11:00-12:30 Michael Harris (Paris 7): Eisenstein cohomology and construction of Galois representations (Part I)

Abstract: I will report on some aspects of the joint work with Lan, Taylor, and Thorne, which attaches compatible families of l-adic Galois representations to a cuspidal cohomological automorphic representation of GL(n) of a CM field. Earlier work by many authors had treated the case where the automorphic representation is dual to its image under complex conjugation; under this hypothesis, the Galois representations in question, or closely related representations, can be obtained directly in the cohomology with twisted coefficients of Shimura varieties attached to unitary groups. Without the duality hypothesis, this is no longer possible; instead, the representations are constructed by p-adic approximation of Eisenstein cohomology classes by cuspidal classes in an appropriate (infinite-dimensional) space of p-adic modular forms. The lectures will concentrate on the construction of Eisenstein classes, the relation to p-adic modular forms, and the definition of Galois representations by p-adic approximation.

## Friday, January 25

11:00-12:30 Michael Harris (Paris 7): Eisenstein cohomology and construction of Galois representations (Part II)

Abstract: This talk is a continuation of part I.

## Monday, January 28

10:00-12:00 Günter Harder (Bonn): Modular construction of mixed motives and congruences (Part II)

Abstract: This talk is a continuation of part I.

## Tuesday, January 29

11:00-13:00 Fred Diamond (King's College, London): The weight part of Serre's conjecture for GL_{2} over totally real fields

Abstract: I will review the statement of the weight part of Serre's conjecture for GL(2) over totally real fields. I will describe what has been proved by Gee and his coauthors, and give a brief overview of the methods.

## Wednesday, January 30

11:00-13:00 Luis Dieulefait (Universitat de Barcelona): Non-solvable base change for GL(2)

Abstract: We will show that any classical cuspidal modular form can be lifted to any totally real number field F. The proof uses a recent Modularity Lifting Theorem proved by Barnet-Lamb, Gee, Geraghty and Taylor (plus a variant of it proved by Gee and the speaker) and another one by Kisin that is used in the "killing ramification" step. The core of the proof is the construction of a "safe" chain of congruences linking to each other any given pair of cuspforms. The safe chain that we will construct is also a key input in the proof of other cases of Langlands functoriality, but this will be explained in another talk (see the abstracts for the conference week).

## Thursday, January 31

11:00-12:30 Kai-Wen Lan (University of Minnesota): Galois representations for regular algebraic cuspidal automorphic representations over CM fields (part I)

Abstract: I will report on my joint work with Michael Harris, Richard Taylor, and Jack Thorne on the construction of p-adic Galois reprensentations for regular algebraic cuspidal automorphic representations of GL(n) over CM (or totally real) fields, without hypotheses on self-duality or ramification. (This should be considered as part III of a series of four talks, the first two being given by Michael Harris in the previous week.)

## Friday, February 1

11:00-12:30 Kai-Wen Lan (University of Minnesota): Galois representations for regular algebraic cuspidal automorphic representations over CM fields (part II)

Abstract: This is a continuation of part I.

## Tuesday, February 12

11:00-12:30 David Geraghty (Princeton): The Breuil-Mezard Conjecture for Quaternion Algebras

Abstract: The Breuil-Mezard conjecture relates the complexity of certain deformation rings for mod p representations of the Galois group of Q_{p} with the representation theory of GL_{2}(F_{p}). Most cases of the conjecture were proved by Kisin who established a link between the conjecture and modularity lifting theorems. In this talk I will discuss a generalization of the conjecture to quaternion algebras (over arbitrary finite extensions of Q_{p}) and show how it follows from the original conjecture for GL_{2}. This is a joint work with Toby Gee.

## Monday, February 18

10:00-12:00 B.E.Kunyavskii (Universität Bar-Ilan): Geometry and arithmetic of word maps in simple matrix groups

Abstract: We wil discuss various geometric and arithmetic properties of matrix equations of the form where the left-hand side is an associative noncommutative monomial in 's and their inverses, and the right-hand side is a fixed matrix. Solutions are sought in some group . We will focus on the case where the group is simple, or close to such.

We will give a survey of classical and recent results and open problems concerning this equation, concentrating around the following questions (posed for geometrically and/or arithmetically interesting rings and fields ):

- is it solvable for any A?
- is it solvable for a "typical" A?
- does it have "many'' solutions?
- does the set of solutions possess "good'' local-global properties?
- to what extent does the set of solutions depend on A?

The last question will be discussed in some detail for the case and , where criteria for equidistribution were obtained in our recent joint work with T. Bandman.

## Tuesday, February 19

10:00-12:00 S.Haran (Technion, Haifa): Non-additive geometry

Abstract: We give a language for algebraic geometry based on non-additive generalized rings. In this language, number fields look more like curves over a finite field. The initial object of generalized rings is the "field with one element". This language "sees" the real and complex primes of a number field, and there is a compactificaton

of

being the ring of integers of a number field . The arithmetic surface

exists and is not reduced to its diagonal. And yet most of the Grothendieck algebraic geometry works with generalized rings replacing commutative rings.

## Wednesday, February 20

10:00-12:00 A.L.Smirnov (Steklov Mathematical Institute, St. Petersburg): The Internal and External Problems of Algebraic Geometry over F_{1}

Abstract: An introduction to Durov's approach will be given. The theory will be illustrated with several explicit examples. Besides we plan to discuss some problems caused by both the development of the theory and demands of its applications.

## Thursday, February 21

10:00-12:00 M.S.Viazovska (Universität Köln): CM values of higher Green's functions and regularized Petersson products

Abstract: Higher Green functions are real-valued functions of two variables on the upper half-plane, which are bi-invariant under the action of a congruence subgroup, have a logarithmic singularity along the diagonal, and satisfy the equation ; here is a hyperbolic Laplace operator and is a positive integer. The significant arithmetic properties of these functions were disclosed in the paper of B. Gross and D. Zagier "Heegner points and derivatives of -series" (1986). In the particular case when and one of the CM points is equal to , the conjecture has been proved by A. Mellit in his Ph.D. thesis. In this lecture we prove that conjecture for arbitrary , assuming that all the pairs of CM points lie in the same quadratic field. The two main parts of the proof are as follows. We first show that the regularized Petersson scalar product of a binary theta-series and a weight one weakly holomorphic cusp form is equal to the logarithm of the absolute value of an algebraic integer and then prove that the special values of weight Green's function, occurring in the conjecture of Gross and Zagier, can be written as the Petersson product of that type, where the form of weight one is the -st Rankin-Cohen bracket of an explicitly given holomorphic modular form of weight and a binary theta-series. Algebraicity of regularized Petersson products was also proved at about the same time by W. Duke and Y. Li by a different method; however, our result is stronger since we also give a formula for the factorization of the algebraic number in question.

## Tuesday, February 26

10:00 - 12:00 D. Mendes da Costa (University of Bristol) : Integral Points on Elliptic Curves and the Bombieri-Pila Bounds

Abstract: In 1989, Bombieri and Pila found upper bounds for the number of integer points of (naive exponential) height at most lying on a degree affine plane curve . In particular, these bounds are both uniform with respect to the curve and the best possible with this constraint. It is conjectured though that if we restrict to curves with positive genus then the bounds can be broken. In this talk we shall discuss progress towards this conjecture in the case of elliptic curves and an application to counting rational points on degree 1 del Pezzo surfaces.

## Wednesday, February 27

10:00 - 12:00 L. Kühne (SNS Pisa): Effective and uniform results of André-Oort type

Abstract: The André-Oort Conjecture (AOC) states that the irreducible components of the Zariski closure of a set of special points in a Shimura variety are special subvarieties. Here, a special variety means an irreducible component of the image of a sub-Shimura variety by a Hecke correspondence. The AOC is an analogue of the classical Manin-Mumford conjecture on the distribution of torsion points in abelian varieties. In fact, both conjectures are considered as special instances of the far-reaching Zilber-Pink conjecture(s).

I will present a rarely known approach to the AOC that goes back to Yves André himself: Before the model-theoretic proofs of the AOC in certain cases by the Pila-Wilkie-Zannier approach, André presented in 1998 the first non-trivial proof of the AOC in a non-trivial case, namely, a product of two modular curves. In my talk, I discuss several results in the style of André's method, allowing to compute all special points in a non-special curve of a product of two modular curves.

These results are effective - in stark contrast to those obtainable by the Pila-Wilkie-Zannier approach - and have also the further advantage of being uniform in the degrees of the curve and its definition field. For example, this allows to show that there are actually no two singular moduli x and y satisfying x+y=1.

## Thursday, February 28

10:00 - 12:00 N. Freitas (Universitat de Barcelona): Fermat-type equations of signature

Abstract: In this talk I plan to discuss how a modular approach via the Hilbert cusp-forms can be used to attack equations of the form , where is a fixed prime and varies. We first relate a possible solution of that equation to solutions of several related Diophantine equations over certain totally real fields F. Then we attach Frey curves over to the solutions of the latter equations. After proving modularity of and irreducibility of certain Galois representations attached to we can use the modular approach. We apply the method to solve equations in the particular case of signature .

## Monday, March 4

10:00 - 12:00 L.V. Kuzmin (Kurchatov Institute, Moscow): -adic regulator of an algebraic number field and Iwasawa theory.

Abstract: We give a new definition of the -adic regulator, which makes sense for any (not necessarily totally real) algebraic number field, present a few results and conjectures, relating to that notion, and discuss the behaviour of the -adic regulator in a -cyclotomic extension of the field.

14:00 - 16:00 A. Ivanov (Universität Heidelberg): Arithmetic and anabelian geometry of stable sets of primes in number fields.

Abstract: We define a new class of sets -- stable sets -- of primes in number fields. For example, Chebotarev sets are very often stable. These sets have positive (but arbitrary small) Dirichlet density and generalize sets with density 1 in the sense that arithmetic theorems like certain Hasse principles, the Grunwald-Wang theorem, the Riemann's existence theorem, etc. hold for them. Geometrically this allows to give examples of infinite sets with arbitrary small positive density, such that corresponding arithmetic curves are algebraic and, using some further ideas, to generalize (a part of the) birational anabelian theorem of Neukirch-Uchida to stable sets.

## Tuesday, March 5

10:00 - 12:00 Tuan Ngo Dac (Universite de Paris 13): On the problem of counting shtukas.

Abstract: I will introduce the stacks of shtukas, explain its role in the Langlands program and then report on my work on the problem of counting shtukas.

## Thursday, March 7

10:00 - 12:00 R. Adibhatla (Universität Regensburg): Modularity of certain 2-dimensional mod representations of .

Abstract: For an odd rational prime and integer , we consider certain continuous representations of into with fixed determinant, whose local restrictions "look" like arising from modular Galois representations, and whose mod $p$ reductions are odd and irreducible. Under suitable hypotheses on the size of their images, we use deformation theory to lift to in characteristic 0. We then invoke a modularity lifting theorem of Skinner-Wiles to show that is modular.

## Monday, March 11

10:00 - 12:00 F. Gounelas (Universität Heidelberg): Rationally connected varieties and free curves

Abstract: The first part of this talk will be a general introduction to rationally connected varieties. I will then discuss various ways in which a variety can be "connected by curves of a fixed genus", mimicking the notion of rational connectedness. At least in characteristic zero, in the specific case of the existence of a single curve with a large deformation space of morphisms to a variety implies that the variety is in fact rationally connected. Time permitting I will discuss attempts to show this result in positive characteristic.

## Tuesday, March 12

10:00 - 12:00 T. Centeleghe (Universität Heidelberg): On the decomposition of primes in torsion fields of an elliptic curve.

Abstract: Let be an elliptic curve over a number field and be a positive integer. In this talk we consider the problem of describing how primes of of good reduction for and away from decompose in the extension . As it turns out, the class in can be completely described, apart of finitely many primes , in terms of the error term and the -invariant of . The Hilbert class polynomials, associated to imaginary quadratic orders, play a role in the description. The main result relies on a theorem on elliptic curves over finite fields.

## Wednesday, March 13

10:00 - 12:00 S. Gupta (University of Iowa): Noether's Problem and Rationality of Invariant Spaces.

Abstract: In the early 1900's Emmy Noether asked the following question: If a group acts faithfully on a vector space (over a field ), is the field of invariants rational, i.e. purely transcendental over ? The answer (for or a number field) in general is no, and we will discuss some consequences and variants of Noether's Problem. We will also discuss the question when the field is algebraically closed, and techniques of testing rationality of invariant fields using unramified cohomology groups.

## Thursday, March 14

10:00 - 12:00 M. Saidi (Exeter University): Some problems/results related to the Grothendieck anabelian section conjecture.

Abstract: I will discuss some (major) problems related to the Grothendieck anabelian section conjecture. I will discuss two new results related to these problems and the section conjecture.

First result: there exists a local-global principle for torsors under the geometric prosolvable fundamental group of a proper hyperbolic curve over a number field. Second result: the passage in the section conjecture from number fields to finitely generated fields is possible under the assumption of finiteness of suitable Shafarevich-Tate groups.

## Monday, March 18

10:00 - 12:00 S.O. Gorchinskiy (Steklov Math. Institute, Moscow): Parameterized differential Galois theory.

Abstract: Classical Galois theory studies symmetry groups of solutions of algebraic equations. Differential Galois theory studies symmetry groups of solutions of linear differential equations. We discuss the so-called parameterized differential Galois theory which studies symmetry groups of solutions of linear differential equations with parameters. The groups that arise are linear differential groups given by differential equations (not necessarily linear) on functions in parameters. We also discuss, in this connection, derivations on Abelian categories and differential Tannakian categories.

## Tuesday, March 19

10:00 - 12:00 A. Buium (University of New Mexico) : The concept of linearity for arithmetic differential equation.

Abstract: The concept of ordinary differential equation has an arithmetic analogue in which the derivation operator is replaced by a Fermat quotient operator. We would like to understand which arithmetic differential equations should be considered as being "linear". Classical linear differential equations arise from differential cocycles of linear algebraic groups into their Lie algebras and their differential Galois groups are algebraic groups with coefficients in the field of constants. On the other hand one can prove that there are no such cocycles in the arithmetic context. This leads one to introduce, in the arithmetic context, a new concept of "Lie algebra", "cocycles", "linear" equations, and "differential Galois groups"; the latter can be viewed as subgroups of the general linear group with coefficients in the algebraic closure of the "field with one element".

## Wednesday, March 20

10:00 - 12:00 R.Ya. Budylin (Steklov Math. Institute, Moscow): Adelic Bloch formula.

**Abstract:** Chern class is involved in the functional equation for 2-dimensional schemes. To get functional equation by Tate method we need local decomposition of Chern class satisfying some properties. Bloch proves that the second Chern class of a vector bundle with trivial determinant can be obtained by boundary homomorphism for the universal central extension of the sheaf . In the talk, this construction will be used to get an adelic formula for the second Chern class in terms of trivializations in scheme points. We will also discuss a generalization of this formula for of vector bundles with for .

## Thursday, March 21

10:00 - 12:00 A.N. Parshin (Steklov Math. Institute, Moscow): A generalization of the Langlands correspondence and zeta-functions of two-dimensional scheme.

Abstract: We introduce Abelian Langlands correspondence for algebraic surfaces defined over a finite field. When the surface is a semi-stable fibration over an algebraic curve, we define two operations, automorphic induction and base change, which connect this correspondence with the classical Langlands correspondence on the curve. Some conjectural properties of these operations imply the standard theorems for zeta- and L-functions on the surface (analytical continuation and functional equation). In this approach we do not need to use the etale cohomology theory.

## Friday, March 22

10:00 - 12:00 D.V. Osipov (Steklov Math. Institute, Moscow): Unramified two-dimensional Langlands correspondence.

Abstract: We will describe the local unramified Langlands correspondence for two-dimensional local fields (following an approach of M. Kapranov). For this goal, we will construct a categorical analogue of principal series representations of general linear groups of even degrees over two-dimensional local fields and describe their properties. The main ingredient of this construction is some central extension of a general linear group defined over a two-dimensional local field or over an adelic ring of a two-dimensional arithmetic scheme. We will prove reciprocity laws for such central extensions, i.e. splittings of the central extensions over some subgroups defined over rings constructed by means of points or by integral one-dimensional subschemes of a two-dimensional arithmetic scheme.

## Monday, March 25

10:00 - 12:00 Rainer Dietmann (Department of Mathematics Royal Holloway): On quantitative versions of Hilbert's irreducibility Theorem

Abstract: If f(X,Y) is an irreducible rational polynomial, then by Hilbert's irreducibility Theorem for infinitely many rational specialisations of X the resulting polynomial in Y still is irreducible over the rationals. In this talk we want to discuss quantitative versions of this result, using recent advances from the determinant method on bounding the number of points on curves.

## Wednesday, March 27

10:00 - 12:00 Roger Heath-Brown (University of Oxford): Pairs of quadratic forms in 8 variables

Abstract: We show that a smooth intersection of two quadrics in P^{^7}, defined over a number field, satisfies the Hasse principle and weak approximation. The proof is based on the work of Colliot-Thelene, Sansuc and Swinnerton-Dyer on Chatelet surfaces, which enables one to reduce the problem to a purely local problem. The first part of the talk will discuss the background and the overall strategy of the proof, and the second part will look in a little more detail at some of the methods involved.

## Thursday, March 28

10:00 - 12:00 Daniel Loughran (University of Bristol): On the number of conics in a family which contain a rational point.

Abstract: Given a family of conics, it is natural to ask about the distribution of the conics in this family which contain a rational point. Serre considered the case of families of conics over projective spaces and proved upper bounds for the number of conics of bounded height in these families which contain rational points, and asked whether or not these upper bounds were sharp. In this talk we answer this question for some special families. We also consider certain families of conics over toric varieties and obtain results on the distribution of conics in these families which contain rational points. These results suggest a possible generalisation of Manin's conjecture to our setting.

## Tuesday, April 2

10:00- 12:00 Pankaj Vishe (University of York ): Cubic hypersurfaces and a version of the circle method over number fields

Abstract: A version of the Hardy-Littlewood circle method is developed for number fields and is used to show that any non-singular projective cubic hypersurface over , with dimension at least 8, always has a - rational point. This is joint work with T. Browning.

## Wednesday, April 3

10.00- 12:00 Efthymios Sofos (University of Bristol ): Counting rational points on the Fermat surface

Abstract: In this talk we shall discuss progress towards finding lower bound for the number of rational points of bounded height on the Fermat cubic surface. The argument is based on a uniform asymptotic estimate for the associated counting function on conics.

## Friday, April 5

10:00- 12:00 Jörg Brüdern (Universität Göttingen ) & Trevor Wooley (University of Bristol ): Systems of cubic forms at the convexity barrier

Abstract: We describe recent joint work concerning the validity of the Hasse principle for systems of diagonal cubic forms. The number of variables required meets the convexity barrier. Certain features of our methods are motivated by work of Gowers on Szemeredi's theorem.

## Monday, April 8

10:00- 12:00 Mike Swarbrick Jones (University of Bristol): Weak approximation on cubic hypersurfaces of large dimension

Abstract: A natural question in arithmetic geometry is to investigate weak approximation on varieties. If the dimension of the variety is large compared to the degree, our most successful tool is the circle method, however there are cases where using this is not feasible given our current state of knowledge. In this talk I will sketch a proof that weak approximation holds for generic cubic hypersurfaces of dimension at least 17, in particular discussing a fibration method argument that applies to the cases where the usual application of the circle method is not possible.

## Wednesday, April 10

10:00- 12:00 Arne Smeets (Université Paris-Sud 11): Local-global principles for fibrations in torsors under tori

Abstract: This talk is a report on work in progress about local-global principles for varieties fibred over the projective line. In particular, we will study the Brauer-Manin obstruction to the Hasse principle and weak approximation for certain fibrations in torsors under tori, e.g. (multi-)norm form equations. Our results are conditional on Schinzel's hypothesis.

## Friday, April 12

10:00- 12:00 Dave Mendes da Costa (University of Bristol): On Uniform Bounds for Integral Points on Elliptic Curves

Abstract: In 1989, Bombieri and Pila proved that given a plane algebraic affine curve of degree d there are no more than integral points on the curve within a box of size N by N. Moreover, the implied constant in their bound depended only on the degree of the curve and not on the equation. Such bounds are, in general, the best possible however it is believed that by restricting to curves which have positive genus that one can do much better. In this talk we consider the problem of improving these uniform bounds for integral points on elliptic curves. An application of this work to degree one del Pezzo surfaces will be presented.