Mathematics Commons^™

Automorphism-Preserving Color Substitutions On Profinite Graphs, Michal Cizek

Electronic Thesis and Dissertation Repository

Profinite groups are topological groups which are known to be Galois groups. Their free product was extensively studied by Luis Ribes and Pavel Zaleskii using the notion of a profinite graph and having profinite groups act freely on such graphs. This thesis explores a different approach to study profinite groups using profinite graphs and that is with the notion of automorphisms and colors. It contains a generalization to profinite graphs of the theorem of Frucht (1939) that shows that every finite group is a group of automorphisms of a finite connected graph, and establishes a profinite analog of the theorem …

Reduction Of L-Functions Of Elliptic Curves Modulo Integers, Félix Baril Boudreau

Electronic Thesis and Dissertation Repository

Let $\mathbb{F}_q$ be a finite field of size $q$, where $q$ is a power of a prime $p \geq 5$. Let $C$ be a smooth, proper, and geometrically connected curve over $\mathbb{F}_q$. Consider an elliptic curve $E$ over the function field $K$ of $C$ with nonconstant $j$-invariant. One can attach to $E$ its $L$-function $L(T,E/K)$, which is a generating function that contains information about the reduction types of $E$ at the different places of $K$. The $L$-function of $E/K$ was proven to be a polynomial in $\mathbb{Z}[T]$.

In 1985, Schoof devised an algorithm to compute the zeta function of an …

Polynomial Identities Of Algebras With Actions: A Unified Combinatorial Approach, Mayecxiliana Cardenas Montoya

Electronic Thesis and Dissertation Repository

Fix a field k. In this thesis, we establish different criteria for when a given (associative or Lie) k-algebra is a PI-algebra.

Fix a unitary associative k-algebra R. Our focus is on associative k-algebras A endowed with an R-module action ρ:R\to End_k(A) with the property that ρ(R) is finite-dimensional. Our main result asserts that if such an algebra is endowed with an R-module action that is `compatible’ with the multiplicative structure of A, then A is a PI-algebra if (and only if), for some positive integer d and all a_1,...,a_d in A, the product a_1···a_d is a linear combination of …