Related papers: Congruences between modular forms and lowering the…

A new proof of Imprimitivity theorem for transitive systems of covariance is given and a definition of square-integrable representation modulo a subgroup is proposed. This clarifies the relation between coherent states, wavelet transforms…

In~\cite{CS04}, Calegari and Stein studied the congruences between classical cusp forms $S_k(\Gamma_0(p))$ of prime level and made several conjectures about them. In~\cite{AB07} (resp., ~\cite{BP11}) the authors proved one of those…

We prove in some cases a formula for the Greenberg-Benois $\mathcal{L}$-invariant of the spin, standard and adjoint Galois representations associated with Siegel-Hilbert modular forms. In order to simplify the calculation, we give a new…

This paper generalises previous work of the author to the setting of overconvergent $p$-adic automorphic forms for a definite quaternion algebra over a totally real field. We prove results which are analogues of classical `level raising'…

For the linearized setting of the dynamics of complex bodies we construct variational integrators and prove their convergence by making use of BV estimates on the rate fields. We allow for peculiar substructural inertia and internal…

We give explicit structure of the graded ring of modular forms with respect to Gamma(N) (N=1,2,3,4,5,6,7,8,9,10,12,16,18) and for some other congruence groups. We also study the modular forms of half-integer weight for certain groups.

We give a complete theta expression of a pair of Hermitian modular forms as an inverse period mapping of lattice polarized $K3$ surfaces. Our result gives a non-trivial relation among moduli of $K3$ surfaces, theta functions and the finite…

Let T be a free Z_p-module of finite rank equipped with a continuous Z_p-linear action of the absolute Galois group of a number field K satisfying certain conditions. In this article, by using a Selmer group corresponding to T, we give a…

In studying the structure of derived categories of module categories of group algebras or their blocks, it is fundamental to classify support $\tau$-tilting modules. Koshio and Kozakai showed that the structure of support $\tau$-tilting…

The set of modular invariants that can be obtained from Galois transformations is investigated systematically for WZW models. It is shown that a large subset of Galois modular invariants coincides with simple current invariants. For…

In this paper, we study the combinatorics of congruence subgroups of the modular group by generalizing results obtained in the non-modular case. For this, we define a notion of irreducible solutions from which we can build all the…

We prove a new Hilbertianity criterion for fields in towers whose steps are Galois with Galois group either abelian or a product of finite simple groups. We then apply this criterion to fields arising from Galois representations. In…

We prove some cases of a conjecture of Lewis, Reiner and Stanton regarding Hilbert series corresponding to the action of $Gl_n(\mathbb{F}_q)$ on a polynomial ring modulo Frobenius powers. We also give a few conjectures about the invariant…

We develop the representation theory for reductive linear differential algebraic groups (LDAGs). In particular, we exhibit an explicit sharp upper bound for orders of derivatives in differential representations of reductive LDAGs, extending…

We prove a criterion for the irreducibility of an integral group representation \rho over the fraction field of a noetherian domain R in terms of suitably defined reductions of \rho at prime ideals of R. As applications, we give…

The goal of the paper is to achieve - in the special case of the linear group SL_2 - some understanding of the relation between group homology and its A^1-invariant replacement. We discuss some of the general properties of A^1-invariant…

We determine the type of the zeta functions and the range of the dimensions of the moduli spaces of finite flat models of two-dimensional local Galois representations over finite fields. This gives a generalization of Raynaud's theorem on…

We construct moduli stacks of two-dimensional mod p representations of the absolute Galois group of a p-adic local field, and relate their geometry to the weight part of Serre's conjecture for GL(2).

We attempt to develop a general algebro-geometric study of the moduli stack of commutative, 1-parameter formal Lie groups. We emphasize the pro-algebraic structure of this stack: it is the inverse limit, over varying n, of moduli stacks of…

This is a book about computational aspects of modular forms and the Galois representations attached to them. The main result is the following: Galois representations over finite fields attached to modular forms of level one can, in almost…