Let $p$ be a prime number, and let $k$ be an imaginary quadratic number field in which $p$ decomposes into two primes $𝔭$ and $\overline{𝔭}$. Let $k_{\infty }$ be the unique ${\mathbb{Z}}_p$-extension of $k$ which is unramified outside of $𝔭$, and let $K_{\infty }$ be a finite extension of $k_{\infty }$, abelian over $k$. Let ${𝒰}_{\infty }/{𝒞}_{\infty }$ be the projective limit of principal semi-local units modulo elliptic units. We prove that the various modules of invariants and coinvariants of ${𝒰}_{\infty }/{𝒞}_{\infty }$ are finite. Our approach uses distributions and the $p$-adic $\mathrm{L}$-function, as defined in [5].

For $n\in {𝔽}_q\left[T\right]$ let $G$ be a subgroup of the Atkin-Lehner involutions of the Drinfeld modular curve $X_0\left(𝔫\right)$. We determine all $𝔫$ and $G$ for which the quotient curve $G\setminus X_0\left(𝔫\right)$ is rational or elliptic.

Les valeurs aux entiers pairs (strictement positifs) de la fonction $\zeta$ de Riemann sont transcendantes, car ce sont des multiples rationnels de puissances de $\pi$. En revanche, on sait très peu de choses sur la nature arithmétique des $\zeta (2k+1)$, pour $k\ge 1$ entier. Apéry a démontré en 1978 que $\zeta \left(3\right)$ est irrationnel. Rivoal a prouvé en 2000 qu’une infinité de $\zeta (2k+1)$ sont irrationnels, mais sans pouvoir en exhiber aucun autre que $\zeta \left(3\right)$. Il existe plusieurs points de vue sur la preuve d’Apéry ; celui des séries hypergéométriques...

We prove that if a curve of a nonisotrivial family of abelian varieties over a curve contains infinitely many isogeny orbits of a finitely generated subgroup of a simple abelian variety, then it is either torsion or contained in a fiber. This result fits into the context of the Zilber-Pink conjecture. Moreover, by using the polyhedral reduction theory we give a new proof of a result of Bertrand.

Let $E$ be an elliptic curve over $\mathbb{Q}$, let $K$ be an imaginary quadratic field, and let $K_{\infty }$ be a ${\mathbb{Z}}_p$-extension of $K$. Given a set $\Sigma$ of primes of $K$, containing the primes above $p$, and the primes of bad reduction for $E$, write $K_{\Sigma }$ for the maximal algebraic extension of $K$ which is unramified outside $\Sigma$. This paper is devoted to the study of the structure of the cohomology groups $H^i(K_{\Sigma }/K_{\infty },E_{p^{\infty }})$ for $i=1,2,$ and of the $p$-primary Selmer group Sel${}_{p^{\infty }}(E/K_{\infty })$, viewed as discrete modules over the Iwasawa algebra of $K_{\infty }/K.$

We propose a solution to the hyperelliptic Schottky problem, based on the use of Jacobian Nullwerte and symmetric models for hyperelliptic curves. Both ingredients are interesting on its own, since the first provide period matrices which can be geometrically described, and the second have remarkable arithmetic properties.

We give a complete answer to the question of which polynomials occur as the characteristic polynomials of Frobenius for genus-$2$ curves over finite fields.

We determine the distribution over square-free integers $n$ of the pair $({dim}_{{𝔽}_2}{\mathrm{Sel}}^{\Phi }(E_n/\mathbb{Q}),{dim}_{{𝔽}_2}{\mathrm{Sel}}^{\widehat{\Phi }}(E_n^{\text{'}}/\mathbb{Q}))$, where $E_n$ is a curve in the congruent number curve family, $E_n^{\text{'}}:y^2=x^3+4n^{2}x$ is the image of isogeny $\Phi :E_n\to E_n^{\text{'}}$, $\Phi (x,y)=(y^2/x^2,y(n^2-x^2)/x^2)$, and $\widehat{\Phi }$ is the isogeny dual to $\Phi$.