Let E be an elliptic curve over a number field K. Descent calculations on E can be used to find upper bounds for the rank of the Mordell-Weil group, and to compute covering curves that assist in the search for generators of this group. The general method of 4-descent, developed in the PhD theses of Siksek, Womack and Stamminger, has been implemented in Magma (when K = Q) and works well for elli...

Let $E$ be an elliptic curve with positive rank over a number field $K$ and let $p$ odd prime number. $K_{cyc}$ the cyclotomic $\mathbb{Z}_p$-extension of $K_n$ denote its $n$-th layer. The Mordell--Weil is said to constant in tower if for all $n$, $E(K_n)$ equal $E(K)$. We apply techniques Iwasawa theory obtain explicit conditions above sense. then indicate potential applications Hilbert's ten...

To get a feeling for our level of ignorance in the face of such questions, consider that, before Faltings, there was not a single curve X (of genus > 1) for which we knew this statement to be true for all number fields K over which X is defined! Already in the twenties, Weil and Siegel made serious attempts to attack the problem. Siegel, influenced by Weil's thesis, used methods of diophantine ...

Let Y0(p) be the Drinfeld modular curve parameterizing Drinfeld modules of rank two over Fq[T ] of general characteristic with Hecke level p-structure, where p ⊳ Fq[T ] is a prime ideal of degree d. Let J0(p) denote the Jacobian of the unique smooth irreducible projective curve containing Y0(p). Define N(p) = q−1 q−1 , if d is odd, and define N(p) = q −1 q2−1 , otherwise. We prove that the tors...

We develop a general classification theory for Brumer’s dihedral quintic polynomials by means of Kummer theory arising from certain elliptic curves. We also give a similar theory for cubic polynomials.