# Seminars this semester

Series:

 Feb 8 Thu Christopher Williams (Imperial) 14:00 F24 p-adic Asai L-functions of Bianchi modular forms Abstract: The Asai (or twisted tensor) L-function attached to a Bianchi modular form is the 'restriction to the rationals' of the standard L-function. Introduced by Asai in 1977, subsequent study has linked its special values to the arithmetic of the corresponding form. In this talk, I will discuss joint work with David Loeffler in which we construct a p-adic Asai L-function -- that is, a measure on Z_p* that interpolates the critical values L^As(f,chi,1) -- for ordinary weight 2 Bianchi modular forms. The method makes use of techniques from the theory of Euler systems, namely Kato's system of Siegel units, building on the rationality results of Ghate. I will start by giving a brief introduction to p-adic L-functions and Bianchi modular forms. Feb 15 Thu Neil Dummigan (Sheffield) 14:00 J11 Automorphic forms on Feit's Hermitian lattices Abstract: This is joint work with Sebastian Schoennenbeck. Feit showed, in 1978, that the genus of unimodular hermitian lattices of rank 12 over the Eisenstein integers contains precisely 20 classes. Complex-valued functions on this finite set are automorphic forms for a unitary group. Using Kneser neighbours, we find a basis of Hecke eigenforms, for each of which we propose a global Arthur parameter. This is consistent with several kinds of congruences involving classical modular forms and critical L-values, and also produces some new examples of Eisenstein congruences for U(2,2). Feb 20 Tue Thanasis Bouganis (Durham) 14:00 F24 On the standard L function attached to Siegel-Jacobi modular forms of higher index Abstract: The standard L function attached to a Siegel modular form is one of the most well-studied L functions, both with respect to its analytic properties and to the algebraicity of its special L-values. Siegel-Jacobi modular forms are closely related to Siegel modular forms, and it was Shintani who first studied the standard L function attached to them. In this talk, I will start by introducing Siegel-Jacobi modular forms and then discuss joint work with Jolanta Marzec on the analytic properties of their standard L function, extending results of Murase and Sugano, and on the algebraicity of its special L values. I will also discuss some open questions. Feb 22 Thu Adel Betina (Sheffield) 14:00 F24 Classical and overconvergent modular forms - CANCELLED Abstract: I will explain the proof of Kassaei of Coleman’s theorem via analytic continuation on the modular curve. Mar 2 Fri Mladen Dimitrov (Lille) 14:00 LT-5 $p$-adic L-functions for nearly finite slope Hilbert modular forms and the Exceptional Zero Conjecture Abstract: We attach $p$-adic L-functions and improved variants theoreof to families of nearly finite slope cohomological Hilbert modular forms, and use them to prove the Greenberg-Benois exceptional zero conjecture at the central point for forms which are Iwahori spherical at $p$. This is a joint work with Daniel Barrera and Andrei Jorza. Mar 8 Thu David Spencer (Sheffield) 14:00 F35 Congruences of local origin for higher levels- CANCELLED Abstract: There are many kinds of congruences between different types of modular forms. The most well known of which is Ramanujan's mod 691 congruence. This is a congruence between the Hecke eigenvalues of the weight 12 Eisenstein series and the Hecke eigenvalues of the weight 12 cusp form (both at level 1). This type of congruence can be extended to give congruences of ''local origin''. In this talk I will explain what is meant by such a congruence while focusing on the case of weight 1. The method of proof in this case is very different to that of higher weights and involves working with Galois representations and ray class characters. Mar 15 Thu Matthew Bisatt (King's College) 14:00 LT A The generalised Birch--Swinnerton-Dyer conjecture and twisted L-functions- CANCELLED Abstract: The Birch and Swinnerton-Dyer conjecture famously connects the rank of an elliptic curve to the order of vanishing of its L-function. We combine this with a conjecture of Deligne to study twisted L-functions and derive several interesting properties of them using tools from representation theory. We show that, under certain conditions, these conjectures predict that the order of vanishing of the twisted L-function is always a multiple of a given prime and provide analogous statements for L-functions of modular forms. Mar 22 Thu Netan Dogra (Imperial) 14:00 J11 Unlikely intersections and the Chabauty-Kim method over number fields Abstract: Chabauty's method is a method for proving finiteness of rational points on curves under assumptions on the rank of the Jacobian. Recently, Kim has shown that one can extend this to prove finiteness of rational points on curves over Q, under slightly weaker assumptions on the dimension of certain Galois cohomology groups. A conjecture of Beilinson-Bloch-Kato implies these assumptions are always satisfied. In this talk I will explain Kim's construction, and how to extend his results to general number fields by proving an 'unlikely intersection' result for the zeroes of p-adic iterated integrals. Apr 19 Thu Gary McConnell (Imperial) 14:00 J11 Empirical connections between "crystals" of complex equiangular lines and Hilbert's twelfth problem for real quadratic fields Abstract: Let K be a real quadratic field of discriminant D or 4D, and set d to be one of the infinitely many integers for which the square-free part of $(d-1)^2 - 4$ is D. Over the past ten years it has become evident from many calculations that there is a profound connection between certain maximal sets of equiangular lines in complex d-dimensional Hilbert space on the one hand, and the ray class field of K of conductor d on the other. I will give a short outline of what has been discovered and where we believe it may be heading. Apr 26 Thu Emmanuel Lecouturier (Paris) 14:00 LT 6 Higher Eisenstein elements in weight 2 and prime level Abstract: In his classical work, Mazur considers the Eisenstein ideal $I$ of the Hecke algebra $T$ acting on cusp forms of weight $2$ and level $\Gamma_0(N)$ where $N$ is prime. When $p$ is an Eisenstein prime, i.e. $p$ divides the numerator of $\frac{N−1}{12}$, denote by $\mathbf{T}$ the completion of $T$ at the maximal ideal generated by $I$ and $p$. This is a $\mathbb{Z}_p$-algebra of finite rank $g_p ≥ 1$ as a $\mathbb{Z}_p$-module. Mazur asked what can be said about $g_p$. Merel proved a criterion for when $g_p \geq 2$. We will give criteria for $g_p \geq 3, 4$ and prove higher Eichler formulas. May 3 Thu Abhishek Saha (Queen Mary) 14:00 J11 On the critical values for the standard L-function of a Siegel modular form Abstract: I will talk about some joint work with Pitale and Schmidt where we prove an explicit pullback formula that gives an integral representation for the twisted standard L-function for a holomorphic vector-valued Siegel cusp form of degree n and arbitrary level. In contrast to all previously proved pullback formulas in this situation, our formula involves only scalar-valued functions despite being applicable to L-functions of vector-valued Siegel cusp forms. Further, by specializing our integral representation to the case n=2, we prove an algebraicity result for the critical L-values in that case (generalizing previously proved critical-value results for the full level case). Time permitting, I will also talk of some further applications and works in progress. May 9 Wed Matthew Bisatt (King's College) 15:00 LT 9 The generalised Birch--Swinnerton-Dyer conjecture and twisted L-functions Abstract: The Birch and Swinnerton-Dyer conjecture famously connects the rank of an elliptic curve to the order of vanishing of its L-function. We combine this with a conjecture of Deligne to study twisted L-functions and derive several interesting properties of them using tools from representation theory. We show that, under certain conditions, these conjectures predict that the order of vanishing of the twisted L-function is always a multiple of a given prime and provide analogous statements for L-functions of modular forms. May 10 Thu Fredrik Stromberg (Nottingham) 14:00 F24 Spectral theory and Maass forms for noncongruence subgroups Abstract: The spectral theory for congruence subgroups of the modular group is fairly well understood since Selberg and the development of the Selberg trace formula. In particular it is known that congruence subgroups has an infinite number of discrete eigenvalues (corresponding to Maass cusp forms) and there is extensive support towards Selberg’s conjecture that there are no small eigenvalues for congruence subgroups. In contrast to this setting, much less is known for noncongruence subgroups of the modular group even though these groups are clearly arithmetic. In fact, it can be shown that under certain circumstances small eigenvalues must exist. And even the existence of infinitely many “new” discrete eigenvalues is not known for these groups. The main obstacle for developing the spectral theory here is that there is in general no explicit formula for the scattering determinant. In this talk I will present sufficient conditions for an “odd” discrete spectrum to exist and I will also give experimental support for the conjecture that these conditions are also necessary. I will also present an experimental version of Turing’s method for certifying correctness of the spectral counting. May 17 Thu Frazer Jarvis (Sheffield) 14:00 F24 $p$-adic periods of genus 2 curves via the AGM Abstract: The arithmetic-geometric mean provides the fastest way to compute periods of elliptic curves, both over the complex and $p$-adic numbers. There is an isogeny of genus 2 curves which looks like it might play the same role to compute periods for curves of genus 2. In this talk, we will discuss progress in developing an algorithm for the $p$-adic case, where $p$-adic periods were defined and first investigated in Teitelbaum's thesis. It is as yet incomplete, but the only missing step is an explicit Tate uniformisation for genus 2 curves. This is joint work with Rudolf Chow, and relates to the final chapter of his PhD thesis.