We present a dynamical analog of the Mordell-Lang conjecture for integral points. We are able to prove this conjecture in the case of endomorphisms of semiabelian varieties.

The simplest non-trivial 5d superconformal field theories (SCFT) are the famous rank-one with $E_n$ flavour symmetry. We study their $U$-plane, which is one-dimensional Coulomb branch of theory on $\mathbb{R}^4 \times S^1$. total space Seiberg-Witten (SW) geometry -- SW curve fibered over $U$-plane described as a rational elliptic surface singular fiber type $I_{9-n}$ at infinity. A classificat...

A conjecture of Mordell, recently proven by Faltings [7], states that a curve of genus at least 2 has only finitely many rational points. Faltings' proof is not effective, although a careful reworking of his proof, combined with some further ideas of Faltings, Mumford, Parshin, and Raynaud, allows one to give an upper bound for the number of rational points. (See [19, XI, §2].) Unfortunately, t...

We explain how to use results from Iwasawa theory to obtain information about p-parts of Tate-Shafarevich groups of specific elliptic curves over Q. Our method provides a practical way to compute #X(E/Q)(p) in many cases when traditional p-descent methods are completely impractical and also in situations where results of Kolyvagin do not apply, e.g., when the rank of the Mordell-Weil group is g...

Let E be an elliptic curve having Complex Multiplication by the ring OK of integers of K = Q( √−D), let H = K(j(E)) be the Hilbert class field of K. Then the Mordell-Weil group E(H) is an OK-module. Its Steinitz class St(E) is studied here. In particular, when D is a prime number, St(E) is determined: If D ≡ 3 (mod 4) then St(E) = 1; if D ≡ 1 (mod 4) then St(E) = [P]t, where P is any prime-idea...

The generalized Mordell-Lang conjecture (GML) is the statement that the irreducible components of the Zariski closure of a subset of a group of finite rank inside a semi-abelian variety are translates of closed algebraic subgroups. In [6], M. McQuillan gave a proof of this statement. We revisit his proof, indicating some simplifications. This text contains a complete elementary proof of the fac...