Congruence Property In Conformal Field Theory

The congruence subgroup property is established for the modular representations associated to any modular tensor category. This result is used to prove that the kernel of the representation of the modular group on the conformal blocks of any rational, C_2-cofinite vertex operator algebra is a congruence subgroup. In particular, the q-character of each irreducible module is a modular function on the same congruence subgroup. The Galois symmetry of the modular representations is obtained and the order of the anomaly for those modular categories satisfying some integrality conditions is determined.Comment: References are updated. Some typos and grammatical errors are correcte

Higher congruence companion forms

For a rational prime $p \geq 3$ we consider $p$-ordinary, Hilbert modular newforms $f$ of weight $k\geq 2$ with associated $p$-adic Galois representations $\rho_f$ and $\mod{p^n}$ reductions $\rho_{f,n}$. Under suitable hypotheses on the size of the image, we use deformation theory and modularity lifting to show that if the restrictions of $\rho_{f,n}$ to decomposition groups above $p$ split then $f$ has a companion form $g$ modulo $p^n$ (in the sense that $\rho_{f,n}\sim \rho_{g,n}\otimes\chi^{k-1}$).Comment: 13 page

Distributive semilattices as retracts of ultraboolean ones; functorial inverses without adjunction

A (v,0)-semilattice is ultraboolean, if it is a directed union of finite Boolean (v,0)-semilattices. We prove that every distributive (v,0)-semilattice is a retract of some ultraboolean (v,0)-semilattices. This is established by proving that every finite distributive (v,0)-semilattice is a retract of some finite Boolean (v,0)-semilattice, and this in a functorial way. This result is, in turn, obtained as a particular case of a category-theoretical result that gives sufficient conditions, for a functor $Pi$, to admit a right inverse. The particular functor $Pi$ used for the abovementioned result about ultraboolean semilattices has neither a right nor a left adjoint

Computations of Galois Representations Associated to Modular Forms

We propose an improved algorithm for computing mod $\ell$ Galois representations associated to a cusp form $f$ of level one. The proposed method allows us to explicitly compute the case with $\ell=29$ and $f$ of weight $k=16$, and the cases with $\ell=31$ and $f$ of weight $k=12,20, 22$. All the results are rigorously proved to be correct. As an example, we will compute the values modulo $31$ of Ramanujan's tau function at some huge primes up to a sign. Also we will give an improved higher bound on Lehmer's conjecture for Ramanujan's tau function.Comment: This paper has been published in Acta Arithmetic