Abstracts
Michael Bennett (UBC)
Klein forms, Thue-Mahler equations and elliptic curves with bad reduction at a prescribed set of primes
Nicolas Billerey (Clermont-Ferrand)
Explicit large image results for modular Galois representations
Starting from the classical congruences of Ramanujan's tau function and after briefly recalling the history of the problem, I will give an explicit version of a large image theorem of Ribet for residual Galois representations attached to classical modular forms. In relation to this result I will also discuss the modularity of some specific reducible representations (joint works for Luis Dieulefait and Ricardo Menares).
Gebhard Boeckle (Heidelberg)
Computing multiplicative integrals over function fields and applications
Let $E$ be an elliptic curve over $\mathbb{Q}$. In the theoretical and experimental study of Heegner points on $E$, multiplicative integrals as introduced by Darmon play a central role. In the most basic case when $E$ has a uniformization by a Shimura curve, Heegner points as well as the Tate-period of $E$ can be expressed via multiplicative integrals. Work of Darmon-Greenberg-Pollack-Stevens, based on the theory of rigid analytic automorphic forms, show that such integrals can be computed in polynomial time and are thus well-suited for explicit computations.
In our talk we shall describe joint work in progress with the PhD student Yamidt Bermudez on multiplicative integrals for elliptic curves $E$ over a global function field. By a result of Drinfeld E corresponds to a function field automorphic form, and one disposes of multiplicative integrals as in the case over $\mathbb{Q}$. A main obstacle for algorithms in positive characteristic is the absence of a logarithm function, a minor obstacle that of a theory of overconvergent automorphic forms. We shall explain a polynomial time algorithm for the computation of such integrals in the absence of a logarithm and indicate possible applications.
Peter Bruin (Zurich)
Ranks of elliptic curves with prescribed level structure
I will explain joint work with J. Bosman, A. Dujella and F. Najman, and work in progress with F. Najman. We proved that every elliptic curve over a quadratic field with a point of order 13 or 18 has even rank, the same holds for every elliptic curve over a quartic field with a point of order 22, and similar (more complicated) results hold for elliptic curves over quadratic fields admitting a cyclic $n$-isogeny for certain values of $n$. The proofs rely on geometric and arithmetic properties of the modular curves $X_1(n)$ and $X_0(n)$.
Sander Dahmen (Utrecht)
Some generalized Fermat equations of the form $x^p+y^p=z^q$
We discuss how the method of Chabauty-Coleman and the modular method can be combined to attack some new cases of the generalized Fermat equation $x^p+y^p=z^q$. In particular, we shall show how to solve this equation (in coprime integers $x$, $y$, $z$) for $(p,q) \in \{(5,7), (5,19), (7,5)\}$. We will also indicate how to obtain more results assuming GRH and some facts about Hilbert modular forms, and discuss the possibility of making these results unconditional. This is joint work with Samir Siksek.
Lassina Dembele (Warwick)
Examples of abelian surfaces with everywhere good reduction
In this talk, we will present explicit examples of abelian surfaces with everywhere good reduction. One class of examples has a particular connection with the Paramodularity Conjecture of Brumer-Kramer, which will be discussed in the process. This is joint work with Abhinav Kumar.
Luis Dieulefait (Barcelona)
Langlands functoriality for $\mathrm{GL}(2) \otimes \mathrm{GL}(n)$ and other related cases
We will explain how to combine recent Automorphy Lifting Theorems with the method of "Propagation of Automorphy" to deduce new cases of Langlands functoriality.
Neil Dummigan (Sheffield)
Testing congruences using a trace formula for algebraic modular forms
I will exhibit a simple formula for the trace of a Hecke operator on a space of algebraic modular forms, which can be used to find eigenvalues of Hecke operators in cases where the space has small dimension but the forms take values in a representation of dimension prohibitively high for making explicit the operators themselves. Such situations arise naturally when testing various analogues of Harder's conjectural congruences between Hecke eigenvalues of genus 2 and genus 1 cusp forms. We will look at two cases, involving compact forms of $U(2,2)$ and $\mathrm{GSp}_4$.
Ian Kiming (Copenhagen)
Higher congruences between modular forms
I will review recent work on modular forms and Galois representations modulo prime powers, focusing primarily on work by myself, I. Chen, G. Wiese, as well as P. Tsaknias, R. Adibhatla. I will also go into various open problems and report on some experimental/computational work related to those, primarily by my student N. Rustom.
Hartmut Monien (Bonn)
Modular forms for noncongruence subgroups of $\mathrm{PSL}_{2}\left(\mathbb{Z}\right)$
A powerful tool for investigating these non-congruence subgroups was introduced by Kulkarni in 1991 and is now known as Farey-Symbols. A complete and efficient implementation of it became available only recently. With the help of the Farey-Symbols it is possible to substantially extend methods from analytic number theory. We will discuss two methods. The first is a generalization of a numerical algorithm originally due to Hejhal. The second approach is based on series of papers by Rademacher and Zuckerman. We will present some noncongruence subgroups and and their modular forms.
Bartosz Naskręcki (Poznan)
On higher congruences between cusp forms and Eisenstein series
We will discuss certain new results of numerical investigation of congruences modulo prime powers between newforms and Eisenstein series at prime levels and with equal weights. In particular I would like to show the upper bound on the exponent of the congruence and formulate several observations based on the results of our computations, including a conjectural behavior of the congruence under the condition of ramification of prime ideals in coefficient fields of newforms.
Ariel Pacetti (Buenos Aires)
Half integral weight modular forms
During this talk we will recall the definition of half integral weight modular forms and some applications like the Shimura correspondence and its relation with special values of L-series. Then we will show how this theory generalizes to Hilbert modular forms over totally real fields.
Cris Poor (Fordham)
Paramodular cusp forms via Borcherds products
Evidence is given for the Paramodular Conjecture of Brumer and Kramer by using Borcherds products to construct examples of weight two paramodular cusp eigenforms that not are in the image of the Gritsenko lift.
Haluk Sengun (Warwick)
Cohomology of Bianchi Groups and Arithmetic
Given an imaginary quadratic field $K$ with ring of integers $R$, consider the Bianchi group $\mathrm{GL}(2,R)$. It is suspected since the late 1970's that there is a connection between the Hecke eigenclasses in the mod $p$ cohomology of (congruence subgroups of) Bianchi groups and the $2$-dimensional continuous mod $p$ representations of the absolute Galois group of $K$.
Most of the basic tools used for establishing this connection (and its surrounding problems) in the classical setting fail to work in the setting of Bianchi groups. The situation has an extra layer of complication by the fact that there are "genuinely mod $p$" Hecke eigenvalue systems, resulting from the existence of torsion in the integral cohomology. In this talk I will elaborate on the above paragraph, presenting numerical examples for illustration. Towards the end, I will also talk about how the "even" $2$-dimensional continuous mod $p$ representations of the absolute Galois group of $\mathbb{Q}$ come into the picture.
Nils Skoruppa (Siegen)
How to construct explicitly vector valued modular forms and Jacobi forms
In various theories one needs to construct explicitly spaces of vector valued elliptic modular forms. Examples for such theories are algebraic quantum field theory (where vector valued modular forms occur as traces of representations of infinite dimensional Lie algebras) or the geometry of moduli spaces in algebraic geometry (where vector valued modular forms occur in the construction of functions with distinguished divisors using Borcherds products or Gritsenko lifts). A recent theorem shows that vector valued modular forms can always be realized as Jacobi forms. For the latter there are various efficient constructions available. In this talk we explain the mentioned theorem, various constructions for Jacobi forms, and we show how to assemble everything to obtain useful explicit formulas for the objects mentioned in the title.
Fredrik Strömberg (Durham)
Dimension formulas for Hilbert modular forms
Dimension formulas for (scalar-valued) Hilbert modular forms have existed for a long time. However, due to the complex nature of the invariants which appear in the formulas, only a few cases have been worked out explicitly, in the sense of producing actual numbers. In a joint project together with N.-P. Skoruppa we have generalized the previous (theoretical) dimension formulas to the setting of vector-valued Hilbert modular forms of both integral and half-integral weights. Our goal is to also provide implementations of algorithms to compute the relevant invariants. The development of these algorithms is still in progress and I will give an overview of the necessary algorithms and present the current status of the project.
John Voight (Vermont)
Power series expansions for modular forms
We exhibit a method to numerically compute power series expansions of modular forms on a cocompact Fuchsian group, using the explicit computation a fundamental domain and linear algebra. As applications, we compute Shimura curve parametrizations of elliptic curves over a totally real field, including the image of CM points, and equations for Shimura curves. This is joint work with Michael Klug and John Willis.