Related papers: On some $p$-adic Galois representations and form c…

Let $N$ be a positive integer and $\Gamma$ be a subgroup of $\mathrm{SL}_2(\mathbb{Z})$ containing $\Gamma_1(N)$. Let $K$ be an imaginary quadratic field and $\mathcal{O}$ be an order of discriminant $D_\mathcal{O}$ in $K$. Under some…

We give a complete answer to the analogue of Grothendieck conjecture on p-curvatures for q-difference equations defined over K(x), where K is any finitely generated extension of Q and q\in K can be either a transcendental or an algebraic…

For a number field $K$, we consider $K^{\rm ta}$ the maximal tamely ramified algebraic extension of~$K$, and its Galois group $G^{\rm ta}_K= Gal(K^{ta}/K)$. Choose a prime $p$ such that $\mu_p \not \subset K$. Our guiding aim is to…

For an elliptic curve defined over a number field, the absolute Galois group acts on the group of torsion points of the elliptic curve, giving rise to a Galois representation in $\mathrm{GL}_2(\hat{\mathbb{Z}})$. The obstructions to the…

Fix distinct primes $p$ and $q$ and let $E$ be an elliptic curve defined over a number field $K$. The $(p,q)$-entanglement type of $E$ over $K$ is the isomorphism class of the group $\operatorname{Gal}(K(E[p])\cap K(E[q])/K)$. The size of…

In 1976, Onabe discovered that, in contrast to the Neukirch-Uchida results that were proved around the same time, a number field $K$ is not completely characterized by its absolute abelian Galois group $A_K$. The first examples of…

Given a finite abelian group $\Gamma$, we study the distribution of the $p$-part of the class group $\operatorname{Cl}(K)$ as $K$ varies over Galois extensions of $\mathbb{Q}$ or $\mathbb{F}_q(t)$ with Galois group isomorphic to $\Gamma$.…

Let n be an odd number and F an imaginary quadratic field with odd discriminant. We show that there exists infinitely many cubic fields K such that the class number of K is divisible by n and the Galois closure of K contains F.

Let $K$ be a non-cylotomic imaginary quadratic field of class number 1 and $E/K$ is an elliptic curve with $E(K)[2]\simeq \mathbb{Z}/2\mathbb{Z} \oplus\mathbb{Z}/2\mathbb{Z}.$ In this article, we determine the torsion groups that can arise…

Let k be an algebraically closed field of arbitrary characteristic,let K/k be a finitely generated field extension and let X be a separated scheme of finite type over K. For each prime ell, the absolute Galois group of K acts on the…

Let $K$ be a field of characteristic different from $2$ and let $E$ be an elliptic curve over $K$, defined either by an equation of the form $y^{2} = f(x)$ with degree $3$ or as the Jacobian of a curve defined by an equation of the form…

We study the Galois symbol map associated to the multiplicative group and an abelian variety which has good ordinary reduction over a $p$-adic field. As a byproduct, one can calculate the "class group" in the view of the class field theory…

Let F:K be a Galois extension of number fields and Q a prime ideal of O_F lying over the prime P of O_K. By analyzing the Q-adic closure of O_K in O_F we characterize those rings of integers O_K for which every residue class ring of…

We complete the classification of torsion subgroups $E(K)_{\text{tors}}$ that can occur for an elliptic curve $E/\mathbb{Q}$ over a sextic number field $K$. Previous work determined the complete set of these groups, leaving the existence of…

Let $E$ be an elliptic curve over $\mathbb{Q}$, $p$ an odd prime number and $n$ a positive integer. In this article, we investigate the ideal class group $\mathrm{Cl}(\mathbb{Q}(E[p^n]))$ of the $p^n$-division field $\mathbb{Q}(E[p^n])$ of…

This paper concerns our earlier conjecture about the equivalence of a derived completion construction applied to the representation spectrum of the absolute Galois group of a geometric field is equivalent to the algebraic K-theory of the…

Let $K$ be an imaginary quadratic field and $p$ a prime split in $K$. In this paper we construct an anticyclotomic Euler system for the adjoint representation attached to elliptic modular forms base changed to $K$. We also relate our Euler…

We study the growth of the rank of elliptic curves and, more generally, Abelian varieties upon extensions of number fields. First, we show that if $L/K$ is a finite Galois extension of number fields such that $\Gal(L/K)$ does not have an…

Let $k$ be an imaginary quadratic field and $p$ an odd prime number such that the $p$-rank of the class group of $k$ is one. Let $S$ be a finite set of places of $k$ distinct from $p$-adic places. We give sufficient conditions for the…

Let $K$ be an extension of $\mathbb{Q}$ and $A/K$ an elliptic curve. If $\mathrm{Gal}(\bar K/K)$ is finitely generated, then $A$ is of infinite rank over $K$. In particular, this implies the $g=1$ case of the Junker-Koenigsmann conjecture.…