## Seminar on Euler Systems/Arithmetic Geometry (ss2014)

It is devoted to the theory of Euler systems, and its relations to modular forms and p-adic L-functions. On occasions, it will host talks on various topics in Arithmetic Geometry.

**April 10**

*Speaker:*Marco Seveso (Milano)

*Title:*Derivatives of measures and Coleman integration

*Abstract:*Coleman's theory of primitives allows to attach a cocycle to a modular form f on a Mumford curve. This cocycle enters in the definition of the L-invariant of f and may be described in terms of p-adic integration theory, thanks to Teitelbaum. On the other hand, it is possible to move the modular form in a family of measures. We introduce the concept of "derivatives of a family of measures" and then we use this concept to give a second formula for the Coleman's cocycle in terms of this family.

**April 17**

*Speaker:*Shanwen Wang (Essen)

*Title:*Two variable p-adic L functions

*Abstract:*In 1996, Coleman and Mazur constructed a geometric object C, called the eigencurve, parametrising overconvergent modular forms of finite slope. We denote by C^0 the closed sub-curve of C, parametrising the cuspidal overconvergent modular forms of finite slope, and by C^1 the normalisation of C^0. Using the deformation of Kato's Euler system, we prove that the p-adic L function of a modular form f varies analytically along C^1.

**April 24**

*Speaker:*Xevi Guitart (Essen)

*Title:*Effective computation and some new instances of Darmon points

*Abstract:*Darmon points (also known as Stark-Heegner points) are a collection of conjectural generalizations of Heegner points on modular elliptic curves. Algorithms for their explicit calculation are useful in providing numerical evidence supporting their conjectured rationality, and can be used in practice as an efficient method for computing algebraic points. In this talk I will recall Greenberg's construction of Darmon points for curves over totally real number fields, and present some (co)homological methods that allow for their effective computation. If time permits, I will also report on some new constructions for curves defined over number fields of arbitrary signature. This is joint work with Marc Masdeu and Haluk Sengun.

**May 8**

No meeting

**May 15**

*Speaker:*Francesc Fite' (Bielefeld)

*Title:*Sato-Tate groups and Galois endomorphism modules in dimension 2

*Abstract:*The (general) Sato-Tate Conjecture for an abelian variety A of dimension g defined over a number field k predicts the existence of a compact real Lie subgroup ST(A) of the unitary symplectic group USp(2g) that is supposed to govern the limiting distribution of the normalized Euler factors of A at the primes of k at which A has good reduction. For the case g=1, there are 3 possibilities for ST(A) (only 2 of which occur for k=Q). In this talk, I will present a joint work with K.S. Kedlaya, V. Rotger, and A.V. Sutherland, in which we provide a precise statement of the Sato-Tate Conjecture for the case of abelian surfaces, by showing that if g=2, then ST(A) is limitted to a list of 52 possibilites, exactly 34 of which can occur if k=Q. Moreover, I will give a characterization of ST(A) in terms of the Galois-module structure of the R-algebra of endomorphisms of A defined over a Galois closure of k.

**May 22**

*Speaker:*Shanwen Wang (Essen)

*Title:*Compatibility of the étale realization and the de Rham realization for the torsion sections of the elliptic polylogarithm.

*Abstract:*G. Kings describes the étale realization of the torsion sections of the elliptic polylogarithm. Furthermore, he describes the de Rham realization of these torsion sections in a joint work with K. Bannai. In this talk, we will give an elementary construction for the étale realization of these torsion sections and we will explain an explicit reciprocity law which shows the compatibility of these two realizations. If time permits, we will explain how to recover Kato's Euler system from these torsion sections. This is a joint work with Francesco Lemma.

**June 5**

*Speaker:*Filippo Nuccio (Saint Etienne)

*Title:*Coleman map in Coleman families

*Abstract:*In the early 90’s Perrin-RIou, generalising a construction due to Coleman, constructed an exponential map interpolating Bloch-Kato exponentials of a crystalline representation along the cyclotomic Z_p-extension of a p-adic field. Such an interpolation property plays a crucial role in Iwasawa theory: in the same way Bloch-Kato exponential links the Dieudonné crystal of the representation to its Galois cohomology and allows one to read in this last group some special value of L functions, Perrin-Riou exponential can be fruitfully used to construct a p-adic L-function starting from (local) Euler systems of cohomology classes. In this joint work with T. Ochiai we extend Coleman -- Perrin-Riou exponentials to Coleman families of p-adic modular forms of finite slope.

**June 12**

*Speaker:*Luis Garcia (Imperial College, London)

*Title:*Regularized theta lifts and (1,1)-currents on orthogonal Shimura varieties.

*Abstract:*We will introduce a regularized theta lift for reductive dual pairs of the form Sp_4,O(V) with V a quadratic vector space over a totally real field F. The lift takes values in the space of (1,1)-currents on the Shimura variety attached to GSpin(V), and we will explain that its values are cohomologous to currents given by integration on special divisors against automorphic Green functions. We will give explicit examples of these currents when the Shimura variety is a product of Shimura curves over F. Finally, we will discuss the evaluation of these currents on certain closed forms; the resulting formula is very similar to the classical Rallis inner product formula for theta lifts.

**June 26**

*Speaker:*Sarah Zerbes (University College, London)

*Title:*Euler systems and the Birch-Swinnerton-Dyer conjecture

*Abstract:*I show how Beilinson's Eisenstein symbol can be used to construct motivic cohomology classes attached to pairs of modular forms of weight >= 2. These motivic cohomology classes can be used to construct an Euler system -- a compatible family of global cohomology classes -- attached to pairs of modular forms, related to the critical values of the corresponding Rankin-Selberg L-function. This is joint work with Kings and Loeffler, extending my previous work with Lei and Loeffler for weight 2 forms. This Euler system has several arithmetic applications, including one divisibility in the Iwasawa main conjecture for modular forms over imaginary quadratic fields, and cases of the finiteness of Tate--Shafarevich groups for elliptic curves twisted by dihedral Artin representations.

**July 3**

*Speaker:*Michael Spiess (Bielefeld)

*Title:*Eisenstein cocycle and a conjecture of Gross

*Abstract:*I will discuss two applications of the Eisenstein cocycle, Firstly, I will explain a proof of a conjecture of Gross regarding the vanishing order of the Stickelberger element in an abelian tower of fields. Secondly, we propose a conjectural cohomological construction of Gross-Stark units (this is joint work with Samit Dasgupta).

**July 10**

*Speaker:*Alexander D. Rahm (National University of Ireland at Galway)

*Title:*The image of the Borel-Serre bordification in algebraic K-theory

*Abstract:*We give a method for constructing explicit non-trivial elements in the third K-group (modulo torsion) of an imaginary quadratic number field. These arise from the relative homology of the map attaching the Borel-Serre boundary to the orbit space of the SL_2 group over the ring of imaginary quadratic integers on its symmetric space - hyperbolic three-space. We provide an algorithm which produces a chain of matrix quadruples specifying our element of K_3 of the field, modulo torsion. We carry out the algorithm for the Eisenteinian integers as well as for the imaginary quadratic integers of discriminant -7. This is joint work with Rob de Jeu (VU Amsterdam).

**July 17**

*Speaker:*Henri Darmon (McGill University, Montreal)

*Title:*Euler systems and the Birch Swinnerton Dyer conjecture

*Abstract:*I will discuss recent (and not-so-recent) progress on the Birch and Swinnerton-Dyer conjecture based on the theory of Euler systems.