Related papers: The distribution of class groups of function field…

We study hyperbolic curves and their Jacobians over finite fields in the context of anabelian geometry.

In the framework of infinite ergodic theory, we derive equidistribution results for suitable weighted sequences of cusp points of Hecke triangle groups encoded by group elements of constant word length with respect to a set of natural…

Let $E$ be an elliptic curve defined over a number field $K$ without complex multiplication. If $\Gamma \subset E(\overline{K})$ is a subgroup of finite rank, a very special case of a conjecture of R\'emond predicts that points of small…

Let $K$ be an imaginary quadratic field and $E/\mathbb{Q}$ an elliptic curves with complex multiplication by $\mathcal{O}_K$. Let $K_\infty/K$ be the anticyclotomic $\mathbb{Z}_p$-extension of $K$ and $K_n$ the intermediate layers. Under…

We propose (and prove under some restrictions) that the square class of the central value of the $L$-function of an everywhere unramified symplectic Galois representation is given by a universal cohomological formula. This phenomenon is…

Given a genus two curve $X: y^2 = x^5 + a x^3 + b x^2 + c x + d$, we give an explicit parametrization of all other such curves $Y$ with a specified symplectic isomorphism on three-torsion of Jacobians $\mbox{Jac}(X)[3] \cong…

Let $K$ be a number field, $A/K$ be an absolutely simple abelian variety of CM type, and $\ell$ be a prime number. We give explicit bounds on the degree over $K$ of the division fields $K(A[\ell^n])$, and when $A$ is an elliptic curve we…

We introduce an analogue of the Mertens conjecture for elliptic curves over finite fields. Using a result of Waterhouse, we classify the isogeny classes of elliptic curves for which this conjecture holds in terms the size of the finite…

Among abelian extensions of a congruence function field, an asymptotic relation of class number and genus is established. The proof is classical, employing well-known results from congruence function field theory.

Let $\Gamma$ be the fundamental group of a finite connected graph $\mathcal G$. Let $\mathfrak M$ be an abelian group. A {\it distribution} on the boundary $\partial\Delta$ of the universal covering tree $\Delta$ is an $\mathfrak M$-valued…

In this paper, we generalize the results presented in [5] for the case of real algebraic space curves. More precisely, given an algebraic space curve C (parametrically or implicitly defined), we show how to compute the generalized…

We study the distribution of algebraic points on curves in abelian varieties over finite fields.

Given two semistable, non potentially isotrivial elliptic surfaces over a curve $C$ defined over a field of characteristic zero or finitely generated over its prime field, we show that any compatible family of effective isometries of the…

This note uses some recent calculations of Conder and Bujalance (on classifying finite index group extensions of Fuchsian groups with abelian quotient and torsion free kernel) in order to determine the full automorphism groups of some cylic…

We prove Abelian and Tauberian theorems for regularised Cauchy transforms of positive Borel measures on the real line whose distribution functions grow at most polynomially at infinity. In particular, we relate the asymptotics of the…

We compute the automorphism groups of the Torelli complex and the complex of separating curves for all but finitely many compact orientable surfaces. As an application, we show that the abstract commensurators of the Torelli group and the…

Let k be a number field, let E/k be an elliptic curve, and let S be a finite set of places of k contianing the archimedean places. Let F be an algebraic closure of k. We prove that if a point P in E(F) is nontorsion, then there are only…

Let C be an algebraic curve in a power of an elliptic curve, both defined over the algebraic numbers. We show that the set of algebraic points of C which satisfy certain conditions is a finite set. This result has implications with the…

Motivated by the analogy between number fields and function fields, this paper extends the main result of \cite{janbazi2025unified} to the function field setting. Let $C$ be a smooth affine curve over a finite field, and let $\pi: S…

We strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on complex hyperbolic spaces, especially in dimension $2$.…