Deformations of Galois representations

In this thesis we study a paper by Barry Mazur ([11]) about deforming Galois representations. In particular we will prove that, if $\bar{\rho}: \Pi \rightarrow \mathrm{GL}_N(k)$ is an absolutely irreducible residual representation, a universal deformation ring $R=R(\Pi,k,\bar{\rho})$ and a universal deformation $\boldsymbol{\rho}$ of $\bar{\rho}$ to $R$ exist. This result is part of the proof of the modularity conjecture. The modularity conjecture is of great importance since it states a connection between modular forms and elliptic curves over \Q, providing a great tool to study the arithmetic properties of those elliptic curves. Andrew Wiles studied the conjecture as a part of the more general problem of relating two-dimensional Galois representations and modular forms and used [11] to complete his construction. To better understand the proof of Mazur, we will analyze in detail the paper of Michael Schlessinger ([13]). This article, which is focused on functors over Artin rings, provides a criterion for a functor to be pro-representable. Moreover, it gives the definition of a "hull", which is a weaker property than pro-representability

Optimal Approximate Minimization of One-Letter Weighted Finite Automata

In this paper, we study the approximate minimization problem of weighted finite automata (WFAs): to compute the best possible approximation of a WFA given a bound on the number of states. By reformulating the problem in terms of Hankel matrices, we leverage classical results on the approximation of Hankel operators, namely the celebrated Adamyan-Arov-Krein (AAK) theory. We solve the optimal spectral-norm approximate minimization problem for irredundant WFAs with real weights, defined over a one-letter alphabet. We present a theoretical analysis based on AAK theory, and bounds on the quality of the approximation in the spectral norm and $\ell^2$ norm. Moreover, we provide a closed-form solution, and an algorithm, to compute the optimal approximation of a given size in polynomial time.Comment: 32 pages. arXiv admin note: substantial text overlap with arXiv:2102.0686

Optimal Spectral-Norm Approximate Minimization of Weighted Finite Automata

We address the approximate minimization problem for weighted finite automata (WFAs) over a one-letter alphabet: to compute the best possible approximation of a WFA given a bound on the number of states. This work is grounded in Adamyan-Arov-Krein Approximation theory, a remarkable collection of results on the approximation of Hankel operators. In addition to its intrinsic mathematical relevance, this theory has proven to be very effective for model reduction. We adapt these results to the framework of weighted automata over a one-letter alphabet. We provide theoretical guarantees and bounds on the quality of the approximation in the spectral and $\ell^2$ norm. We develop an algorithm that, based on the properties of Hankel operators, returns the optimal approximation in the spectral norm.Comment: 24 pages, authors appear in alphabetical order; minor correction in Theorem 3.2 and consequently updated notation in Section 3, the validity of the result is not affecte