# Seminars this semester

Series:

 Oct 9 Tue Angelo Rendina (Sheffield) 14:00 J11 Nearly holomorphic Siegel modular forms and applications Abstract: Nearly holomorphic modular forms were introduced by Shimura as a generalization of modular forms to study a more general class of Eisenstein series. I will introduce some of the tools that we use to work with them, such as the Shimura-Maass differential operator and holomorphic projection, and present some applications: some formulae for the sum of divisor $s_r$ and Ramanujan $\tau$ functions and then congruences of critical $L$-values attached to Siegel modular forms, the latter being part of my research project. Oct 23 Tue Kim Klinger-Logan (Minnesota) 14:00 J11 The Riemann Hypothesis and periods of Eisenstein series Abstract: This summer at Perspectives on the Riemann Hypothesis, Bombieri and Garrett discussed modifications to the invariant Laplacian $\Delta=y^2(\partial_x^2+\partial_y^2)$ on $SL_2(\mathbb{Z})\backslash\mathfrak{H}$ possibly relevant to RH. We will present a $GL(2)$ $L$-function as a period of Eisenstein series which can, in turn, be thought of as a linear functional on an Eisenstein series and we will discuss how such functionals may be use to analyze the zeros of the $L$-function. This idea is an extension of recent work of Bombieri and Garrett and uses techniques from functional analysis and spectral theory of automorphic forms. Oct 30 Tue Adel Betina (Sheffield) 14:00 J11 On the p-adic periods of semi-stable modular curves Abstract: I will present a joint work with E.Lecouturier in which we prove a variant of Oesterlé's conjecture about $p$-adic periods of the modular curve $X_0(p)$, with an additional $Γ(2)$-structure. We use de Shalit's techniques and $p$-adic uniformization of Mumford curves whose reduction is semi-stable. Nov 6 Tue Vlad Serban (Vienna) 14:00 J11 A finiteness result for families of Bianchi modular forms Abstract: We develop a p-adic "unlikely intersection” result and show how it can be used to examine which Hida families over imaginary quadratic fields interpolate a dense set of modular forms for GL2 over an imaginary quadratic field. In this way we arrive at the first proven examples where only finitely many classical automorphic forms are on a p-adic family. Nov 27 Tue Jack Shotton (Durham) 14:00 J11 Shimura curves and Ihara's lemma Abstract: Ihara's lemma is a statement about the structure of the mod l cohomology of modular curves that was the key ingredient in Ribet's results on level raising. I will motivate and explain its statement, and then describe joint work with Jeffrey Manning on its extension to Shimura curves. Dec 4 Tue Ciaran Schembri (Sheffield) 14:00 J11 Modularity of abelian surfaces over imaginary quadratic fields Abstract: In this talk I will discuss the modularity of abelian surfaces with quaternionic endomorphisms. This includes a discussion of how they correspond to Bianchi newforms and how to prove this for individual cases using the Faltings-Serre method. Furthermore, we give explicit examples which do not arise as the base-change of a GL(2)-type surface, which settles a question posed by J. Cremona in 1992. Dec 13 Thu Steffen Kionke (Karlsruhe Institute of Technology) 11:00 J11 The first Betti number of arithmetic hyperbolic 3-manifolds Abstract: An arithmetic hyperbolic 3-manifold is the quotient of the 3-dimensional hyperbolic space by an action of a discrete arithmetically defined subgroup of SL(2,C). The cohomology of these manifolds contains number theoretic information and it is of interest to understand the dimension of the cohomology. We discuss some known results about the first Betti numbers of arithmetic hyperbolic 3-manifolds. In particular, we review a method to obtain lower bounds which was developed by Harder, Rohlfs and others. Finally, we explain how the representation theory of compact p-adic Lie groups can be used to obtain significantly stronger results. Dec 17 Mon Alice Pozzi (UCL) 14:00 J11 The eigencurve at Eisenstein weight one points Abstract: In 1973, Serre observed that the Hecke eigenvalues of Eisenstein series can be $p$-adically interpolated. In other words, Eisenstein series can be viewed as specializations of a $p$-adic family parametrized by the weight. The notion of $p$-adic variations of modular forms was later generalized by Hida to include families of ordinary cuspforms. In 1998, Coleman and Mazur defined the eigencurve, a rigid analytic space classifying much more general $p$-adic families of Hecke eigenforms parametrized by the weight. The local nature of the eigencurve is well-understood at points corresponding to cuspforms of weight $k \geq 2$, while the weight one case is far more intricate. In this talk, we discuss the geometry of the eigencurve at weight one Eisenstein points. In particular, we focus on the unusual phenomenon in which cuspidal Hida families specialize to Eisenstein series at weight one. Our approach consists in studying the deformation rings of certain (deceptively simple!) Artin representations. We discuss how this Galois-theoretic method yields some new insight on Gross’s formula relating the leading term of the $p$-adic L-function to $p$-adic logarithms of units of certain number fields.