Related papers: Infinite Hilbert Class Field Towers from Galois Re…

We study the definability of maximal towers and of inextendible linearly ordered towers (ilt's), a notion that is more general than that of a maximal tower. We show that there is, in the constructible universe, a $\Pi^1_1$ definable maximal…

We relate two different proposals to extend the \'etale topology into homotopy theory, namely via the notion of finite cover introduced by Mathew and via the notion of separable commutative algebra introduced by Balmer. We show that finite…

Let $\ell$ be an odd prime and $d$ a positive integer. We determine when there exists a degree-$d$ number field $K$ and an elliptic curve $E/K$ with $j(E)\in\mathbb{Q}\setminus\{0,1728\}$ for which $E(K)_{\mathrm{tors}}$ contains a point of…

We give effective bounds for the class number of any algebraic function field of genus $g$ defined over a finite field. These bounds depend on the possibly partial information on the number of places on each degree $\leq g$. Such bounds are…

Let (k1,k2,k3,k4) be a quartet of cyclic cubic number fields sharing a common conductor c=pqr divisible by exactly three prime(power)s p,q,r. For those components k of the quartet whose 3-class group Cl(3,k) = Z/3Z x Z/3Z is elementary…

Let $k$ be a number field and $G$ be a finite group. Let $\mathfrak{F}_{k}^{G}(Q)$ be the family of number fields $K$ with absolute discriminant $D_K$ at most $Q$ such that $K/k$ is normal with Galois group isomorphic to $G$. If $G$ is the…

For each prime $\ell$, let $|\cdot|_\ell$ be an extension to $\bar \Q$ of the usual $\ell$-adic absolute value on $\Q$. Suppose $g(z) = \sum_{n=0}^\infty c(n)q^n \in M_{k+\half}(N)$ is an eigenform whose Fourier coefficients are algebraic…

Let $K$ be a totally real field. In this article we present an asymptotic formula for the number of Hilbert modular cusp forms $f$ with given ramification at every place $v$ of $K$. When $v$ is an infinite place, this means specifying the…

Let F be a global function field and let F^ab be its maximal abelian extension. Following an approach of D.Hayes, we shall construct a continuous homomorphism \rho: Gal(F^ab/F) \to C_F, where C_F is the idele class group of F. Using class…

The deformation theory of ordinary representations of the absolute Galois groups of totally real number fields (over a finite field $k$) has been studied for a long time, starting with the work of Hida, Mazur and Tilouine, and continued by…

This article deals with the Galois representation attached to elliptic curves with an isogeny of prime degree over a number field. We first determine uniform criteria for the irreducibility of Galois representations attached to elliptic…

We give formulas for 3-fold Massey products in the \'etale cohomology of the ring of integers of a number field and use these to find the first known examples of imaginary quadratic fields with class group of $p$-rank two possessing an…

This article starts a computational study of congruences of modular forms and modular Galois representations modulo prime powers. Algorithms are described that compute the maximum integer modulo which two monic coprime integral polynomials…

The fundamental theorem of arithmetic factorizes any integer into a product of prime numbers. The Jordan-Holder theorem dissolves many groups by their normal series which can be refined into composition series. The main topic of this thesis…

Let E/F be a CM field split above a finite place v of F, let l be a rational prime number which is prime to v, and let S be the set of places of E dividing lv. If E_S denotes a maximal algebraic extension of E unramified outside S, and if u…

Using the link between mod $p$ Galois representations of $\qu$ and mod $p$ modular forms established by Serre's Conjecture, we compute, for every prime $p\leq 1999$, a lower bound for the number of isomorphism classes of continuous Galois…

Weinberger in 1972, proved that the ring of integers of a number field with unit rank at least $1$ is a principal ideal domain if and only if it is a Euclidean domain, provided the generalised Riemann hypothesis holds. Lenstra extended the…

We prove that every odd semisimple reducible (2-dimensional) mod l Galois representation arises from a cuspidal eigenform. In addition, we investigate the possible different types (level, weight, character) of such a modular form. When the…

This note presents Galois theory for finite fields. It was written as a handout for the MAT401 course ``Polynomial equations and fields'' taught at the University of Toronto in Spring 2026. We use without proofs some basic properties of…

Let $K$ be an imaginary quadratic field. For an order $\mathcal{O}$ in $K$ and a positive integer $N$, let $K_{\mathcal{O},\,N}$ be the ray class field of $\mathcal{O}$ modulo $N\mathcal{O}$. We deal with various subjects related to…