# Research Projects

## Reductions of Algebraic Groups

Antonella Perucca, Pietro Sgobba

Consider an algebraic group defined over a number field K. For all but finitely many primes p of K we can reduce the algebraic group modulo p. We thus replace the given algebraic group by a collection of algebraic groups which are easier to study (both theoretically and computationally). Similarly we may reduce any given point modulo almost all primes of K and study instead an infinite family of torsion points. Several questions arise, for example one can study the reductions of a given point: which is the largest integer that divides the order of the reduction modulo p, for almost all primes p? Another direction of research is considering local-global principles i.e. investigating whether a property holds for the given algebraic group if and only if the analogous property holds for almost all its reductions. For example, if a point belongs to a group of points, then clearly the reductions of the point belong to the reductions of the group, however the converse does not hold (this problem is called the problem of detecting linear dependence).

## Arithmetic of Algebraic Groups

Antonella Perucca, Sebastiano Tronto, in collaboration with Peter Bruin (Leiden)

Important examples of algebraic groups are elliptic curves, abelian varieties, and tori. There are many interesting open questions in the arithmetic of algebraic groups. For example, we can fix a number field and study the group structure of the points defined on this field, e.g. the rank of this group. In particular, we consider the extensions that are generated by torsion points of the algebraic group. We also study Galois representations and the cohomology of algebraic groups, which give insight on its arithmetic. We use mathematical softwares such as Sagemath and PARI/GP to collect experimental data: this allows us to produce heuristics and formulate conjectures. We also contribute to the development of these softwares.

## Bianchi modular forms of varying discriminant, deeper level, and heavier weight

Rahm, Tsaknias, collaboration with Sengun (Warwick)

Bianchi modular forms are automorphic forms over an imaginary quadratic field, associated to a Bianchi group. Even though modern studies of Bianchi modular forms go back to the mid 1960's, most of the fundamental problems surrounding their theory are still wide open. Only for certain types of Bianchi modular forms, which we will call "generic", it is possible at present to develop dimension formulas: They are (twists of) those forms which arise from elliptic cuspidal modular forms via the Langlands base-change procedure, or arise from a quadratic extension of the imaginary quadratic field via automorphic induction (so-called CM-forms). The remaining, non-"generic" Bianchi modular forms are what we call "genuine", and they are of interest for an extension of the modularity theorem (formerly the Taniyama-Shimura conjecture, crucial in the proof of Femat's Last Theorem) to imaginary quadratic fields. The state of the art is a present paper by two of the members of this project, in which we report on our heavy machine computations that show the extreme paucity of "genuine" level one cuspidal Bianchi modular forms. But all of those and previous computations are limited to level One. As part of this project, P. Tsaknias is extending the formulas for the "generic" Bianchi modular forms to deeper levels, and we are able to spot the first, rare instances of "genuine" forms at deeper level and heavier weight.

## Higher companion form theorems and the local structure of Galois representations (FNR-AFR project)

Rajender Adibhatla

This project aims to generalize and extend previously established results in modular forms in order to study the local structure of Galois representations attached to automorphic forms. As in the classical case, a combination of well-entrenched techniques from the deformation theory of Galois representations, along with more recently developed automorphy lifting results, is used to prove higher companion form theorems. These companions are important objects because they potentially allow for a detailed, and computationally amenable, study of the fine structure of the local Galois representations attached to automorphic forms.

Cognate to this project is a recent result on the the realisability over Q of PSL_2(Z/p^nZ), p>3 and ongoing work, using potential modularity, on companion forms mod pq (and, more generally, for any odd composite modulus not divisible by 3.)

## Adelic openness without the Mumford-Tate conjecture

Chun Yin Hui, in collaboration with Michael Larsen (Indiana)

Let *X* be a smooth projective variety defined over a number field *K* and *i* an integer. The absolute Galois group of *K* acts on the *i*th etale cohomology with coefficients in the ring of finite adeles over the rationals. Assuming the MTC (Mumford-Tate conjecture), an adelic openness conjecture is formulated which describes the largeness of adelic Galois image inside the Mumford-Tate group corresponding to the *i*th cohomology. A weaker conjecture, which does not require the MTC but which, together with the MTC, implies the adelic openness conjecture, is stated and proved in some cases (the monodromy groups satisfy certain type A conditions or *X* is an abelian variety)

## The structure of the kernel on modular Jacobians

Yoo, collaboration with Ribet (Berkeley)

The Jacobian of modular curves has its own interest in many contexts. When Wiles proves the modularity conjecture to get Fermat's Last theorem, the structure of the kernel of a maximal ideal of the Hecke ring on modular Jacobians was crucial.

In this project, the kernels of Eisenstein maximal ideals on modular Jacobians for square-free level are studied. In particular, one is interested in its dimension. This study will also help to understand the rational torsion of modular Jacobians. Moreover, this result can be useful to determine "non-optimal levels" of reducible Galois representation in the sense of Serre's conjecture.

## Equidistribution of signs of coefficients of half-integral modular forms

Wiese, Arias-de-Reyna, collaboration with Inam (Bilecik)

A very special case of the Sato-Tate theorem states that signs of coefficients of integral weight Hecke eigenforms are equidistributed. That such should also be the case for half-integral weight forms was conjectured by Kohnen and Bruinier.

In this project, the Shimura lift and the Sato-Tate theorem are exploited to obtain sign equidistribution for certain subsets of the coefficients of half-integral weight eigenforms. Some subtleties concerning different notions of density had to be overcome and an explicit error bound for Sato-Tate in the CM-case was established.

## Inverse Galois Problem through Galois Representations

Wiese, Arias-de-Reyna, collaboration with Dieulefait (Barcelona), Shin (MIT)

The Inverse Galois problem, first studied by David Hilbert, asks whether any finite group appears as the Galois group of a Galois extension of the rationals. Formulated in more modern language, it asserts that all finite groups appear as quotients of the absolute Galois group of the rationals, which is arguably (one of) the central object in Algebraic Number Theory. Despite many efforts, the Inverse Galois Problem is essentially still open.

In part of this project, the Inverse Galois Problem for projective symplectic groups is approached through compatible systems of Galois representations attached to automorphic forms. There are essentially three steps: (1) control the field of definition of the projectivised residual representations, (2) establish local conditions to ensure residually `large' projective image, (3) construct automorphic forms whose attached compatible system of Galois representations meets the requirements of (1) and (2). This strategy allowed to realise many infinite families of projective symplective groups as Galois groups (joint work of Arias-de-Reyna, Dieulefait, Shin, Wiese).

Due to poor control of coefficient fields of compatible systems attached to modular and automorphic forms, the previously mentioned results `only' give infinite families. However, exploiting the control on coefficient fields provided by a famous conjecture of Maeda, leads to density 1 results for PSL_{2}
(F_{q}
), when q runs through p^{d}
for any fixed even d (work of Wiese).

## Galois Representations of Weight One (GRWTONE, PUL)

Wiese, Yoo, collaboration with Dimitrov (Lille) and Dembélé (Warwick)

Modular forms and Galois representations play a central role in modern number theory and arithmetic geometry: They are at the heart of Wiles' proof of the world-famous Fermat's last theorem and since then spectacular results such as the proof of conjectures of Serre and Sato-Tate have been obtained. The subject is in full bloom, of which, for instance, the great recent advances towards mod p and p-adic Langlands correspondences testify. Modular forms of weight one occupy a somewhat special position in the theory: over a finite field of characteristic p, they correspond to mod-p Galois representations which are unramified at p; moreover, they are non-cohomological. Thus, though they are arithmetically highly interesting, geometrically they are quite elusive.

This project targets parallel and partial weight one Hilbert modular forms and their roles in Serre type conjectures, both theoretically and computationally.

## Computational aspects of modular forms and p-adic Galois representations (COMFGREP, INTER/DFG/12/10)

Wiese, Tsaknias, Adibhatla, Arias-de-Reyna, collaboration with Böckle (Heidelberg), Centeleghe (Heidelberg), Kiming (Copenhagen), Rustom (Copenhagen)

Recent breakthroughs in Arithmetic Geometry and various topical conjectures in the spirit of the Langlands programme establish and postulate deep correspondences between certain geometric objects: *modular and automorphic forms* and certain number theoretic objects: *Galois representations*. The geometric side is often amenable to calculations and by the explicit nature of the correspondences also number theoretic objects become computationally accessible.

The objectives of this project concern the investigation of these geometric and arithmetic objects either directly or through the correspondence. One main research line will lead to the computation of integral properties of p-adic modular Galois representations. Another research line will study the finiteness and describe the growth of sets of p-adic modular Galois representations modulo prime powers. Both lines are naturally related to level and weight optimisation modulo prime powers. They are also intimately linked to the p-adic coefficient fields of newforms about which numerous conjectures exist. These coefficient fields are strongly affected by the local coefficient field at p and a so called L-invariant. The latter will be explored by generating new data. Finally in weight two the integral local representation at good primes away from p will be studied by methods involving abelian varieties and their endomorphism rings.

The methods to be employed are experimental, algorithmic, and theoretical and progress is expected from the interplay of these. For the experimental study, algorithms will be developed and implemented in computer algebra systems and a publicly available database will be computed.

These new computer tools will be of service to other researchers as well.