We consider a smooth system of two homogeneous quadratic equations over Q in n > 13 variables. In this case, the Hasse principle is known to hold, thanks to the work of Mordell in 1959. The only local obstruction is over R. In this paper, we give an explicit algorithm to decide whether a nonzero rational solution exists and, if so, compute one.

We describe the possible Mordell-Weil groups for degree 1 elliptic threefold with rational base and constant j-invariant. Moreover, we classify all such elliptic threefolds if the j-invariant is nonzero. We can use this classification to describe a class of singular hypersurfaces in P(2, 3, 1, 1, 1) that admit no variation of Hodge structure. (Remark 9.3.)

Computing a lower bound for the canonical height is a crucial step in determining a Mordell–Weil basis of an elliptic curve. This paper presents a new algorithm for computing such lower bound, which can be applied to any elliptic curves over totally real number fields. The algorithm is illustrated via some examples.

In previous work the present authors and others have studied Mordell-Tornheim-Witten sums and their connections with multiple-zeta values. In this note we describe the numerical computation of derivatives at zero of a specialization originating in a preprint by Romik, and the experimental evaluation of these numerical values in terms of well-known constants.

This paper assumes no background on elliptic curves and culminates with a proof of the Mordell-Weil theorem. The Riemann-Roch and Dirichlet unit theorem are recalled but used without proof, but everything else is self-contained. After some elementary properties of elliptic curves are given, the group structure is explored in detail.