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Let p be an odd prime number, E an elliptic curve over a number field k, and F/k a Galois extension of degree twice a power of p. We study the Z_p-corank rk_p(E/F) of the p-power Selmer group of E over F. We obtain lower bounds for…

We give several new moduli interpretations of the fibers of certain Shimura varieties over several prime numbers. As a consequence (of our theorem 9.1) one obtains that for every prescribed odd prime characteristic $p$ every bounded…

Let $k=k_0(\sqrt[3]{d})$ be a cubic Kummer extension of $k_0=\mathbb{Q}(\zeta_3)$ with $d>1$ a cube-free integer and $\zeta_3$ a primitive third root of unity. Denote by $C_{k,3}^{(\sigma)}$ the $3$-group of ambiguous classes of the…

The classical theory of elliptic curves with complex multiplication is a fundamental tool for studying the arithmetic of abelian extensions of imaginary quadratic fields. While no direct analogue is available for real quadratic fields, a…

If $D$ is the definite quaternion algebra over $\qu$ of discriminant $p$, we compute, for any prime $p>3$, the number of infinite dimensional cusp forms on $D^*$ which are trivial at infinity, tamely ramified at $p$, and have given…

We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose $K/k$ is a quadratic extension of number fields, $E$ is an elliptic curve defined over $k$, and $p$ is an odd prime. Let $F$…

Let $p$ be a prime. The $2$-primary part of the class group of the pure quartic field $\mathbb{Q}(\sqrt[4]{p})$ has been determined by Parry and Lemmermeyer when $p \not\equiv \pm 1\bmod 16$. In this paper, we improve the known results in…

Let $p$ be a prime number. If a number field $k$ has at least one complex place, there are infinitely many $\mathbb{Z}_p$-extensions over $k$, and some authors studied the behavior of Iwasawa invariants of these $\mathbb{Z}_p$-extensions.…

Let $p$ be an irregular prime and $K=\Q(\zeta)$ the $p$-cyclotomic field. Let $\sigma$ be a $\Q$-isomorphism of $K$ generating $Gal(K/\Q)$. Let $S/K$ be a cyclic unramified extension of degree $p$, defined by $S= K(A^{1/p})$ where $A\in…

Nondegenerate quadratic forms over $p$-adic fields are classified by their dimension, discriminant, and Hasse invariant. This paper uses these three invariants, elementary facts about $p$-adic fields and the theory of quadratic forms to…

Let $p$ be an odd prime number. For a degree $p$ extension of $p$-adic fields $L/K$, we give a complete characterization of the condition for the ring of integers $\mathcal{O}_L$ to be free as a module over its associated order in the…

In this paper it is proved, that for every prime number p, the set of cyclic p-roots in C^p is finite. Moreover the number of cyclic p-roots counted with multiplicity is equal to (2p-2)!/(p-1)!^2. In particular, the number of complex…

Given a $p$-adic field $K$ and a prime number $\ell$, we count the total number of the isomorphism classes of $p^\ell$-extensions of $K$ having no intermediate fields. Moreover for each group that can appear as Galois group of the normal…

Let p be an odd prime. Let K = Q(zeta) be the p-cyclotomic field. Let v be any primitive root mod p. Let sigma be a Q-isomorphism of K. Let P(sigma) = sigma^{p-2}v^{-(p-2)}+ ... + sigma v^{-1} +1 \in Z[G] where 1 \leq v^n \leq p-1 is a…

In this article, we prove a modulo $p$ congruence which connects the class number of the quadratic field $\mathbb{Q}(\sqrt{(-1)^{(p-1)/2}p})$ and the trace of a certain monomial in a root $\theta$ of the Artin-Schreier polynomial…

Let p be a prime number and M a quadratic number field, M not equal to Q(\sqrt{p}) if p is congruent to 1 modulo 4. We will prove that for any positive integer d there exists a Galois extension F/Q with Galois group D_{2p} and an elliptic…

Let $E$ be an elliptic curve over $\Q$. It is well known that the ring of endomorphisms of $E_p$, the reduction of $E$ modulo a prime $p$ of ordinary reduction, is an order of the quadratic imaginary field $Q(\pi_p)$ generated by the…

We compute the $p$-adic regulator of cyclic cubic extensions of $\mathbb Q$ with discriminant up to $10^{16}$ for $3<p<100$, and observe the distribution of the $p$-adic valuation of the regulators. We find that for almost all primes, the…

Let k be a perfect field of characteristic p>0. When p>2, Fontaine and Laffaille have classified p-divisibles groups and finite flat p-groups over the Witt vectors W(k) in terms of filtered modules. Still assuming p>2, we extend these…

Let p be an odd prime. Let F_p^* be the no-null part of the finite field of p elements. Let K=\Q(zeta) be a p-cyclotomic field and O_K be its ring of integers. Let pi be the prime ideal of K lying over p. Let sigma : zeta --> zeta^v be the…