This article deals with the coherence of the model given by the Cohen-Lenstra heuristic philosophy for class groups and also for their generalizations to Tate-Shafarevich groups. More precisely, our first goal is to extend a previous result due to É. Fouvry and J. Klüners which proves that a conjecture provided by the Cohen-Lenstra philosophy implies another such conjecture. As a consequence of our work, we can deduce, for example, a conjecture for the probability laws of $p^j$-ranks of Selmer groups...

Let k be a real quadratic field and let ${}_k$ and ${}_k^{\times }$ be the ring of integers and the group of units, respectively. A method of solving the Diophantine equation X³ = u+v ($X{\in }_k$, $u,v{\in }_k^{\times }$) is developed.

Let $n$ be a positive odd integer. In this paper, combining some properties of quadratic and quartic diophantine equations with elementary analysis, we prove that if $n>1$ and both $6n^2-1$ and $12n^2+1$ are odd primes, then the general elliptic curve $y^2=x^3+(36n^2-9)x-2(36n^2-5)$ has only the integral point $(x,y)=(2,0)$. By this result we can get that the above elliptic curve has only the trivial integral point for $n=3,13,17$ etc. Thus it can be seen that the elliptic curve $y^2=x^3+27x-62$ really is an unusual elliptic curve which has large integral points.

We study the Ljunggren equation Y² + 1 = 2X⁴ using the "multiplication by 2" method of Chabauty.

Let $D_m$ be an elliptic curve over $\mathbb{Q}$ of the form $y^2=x^3-m^{2}x+m^2$, where $m$ is an integer. In this paper we prove that the two points $P_{-1}=(-m,m)$ and $P_0=(0,m)$ on $D_m$ can be extended to a basis for $D_m\left(\mathbb{Q}\right)$ under certain conditions described explicitly.

In this paper we show that for every prime $p\ge 5$ the dimension of the $p$-torsion in the Tate-Shafarevich group of $E/K$ can be arbitrarily large, where $E$ is an elliptic curve defined over a number field $K$, with $[K:\mathbb{Q}]$ bounded by a constant depending only on $p$. From this we deduce that the dimension of the $p$-torsion in the Tate-Shafarevich group of $A/\mathbb{Q}$ can be arbitrarily large, where $A$ is an abelian variety, with $dimA$ bounded by a constant depending only on $p$.

Let $K$ be a complete discretely valued field with perfect residue field $k$. Assuming upper bounds on the relation between period and index for WC-groups over $k$, we deduce corresponding upper bounds on the relation between period and index for WC-groups over $K$. Up to a constant depending only on the dimension of the torsor, we recover theorems of Lichtenbaum and Milne in a “duality free” context. Our techniques include the use of LLR models of torsors under abelian varieties with good reduction and...

We consider the expansion of the real field by the group of rational points of an elliptic curve over the rational numbers. We prove a completeness result, followed by a quantifier elimination result. Moreover we show that open sets definable in that structure are semialgebraic.