We classify, up to automorphisms, the elliptic fibrations on the singular K3 surface X whose transcendental lattice is isometric to 6 ⊕ 2 .

In this paper, we characterize vertex algebras (in the appropriate sense) as algebras over a certain graded co-operad. We also discuss some applications and categorical ramifications of this characterization.

We give an asymptotic formula for the number of elliptic curves over Q with bounded Faltings height. Silverman [10] has shown that the Faltings height for elliptic curves over number fields can be expressed in terms of modular functions and the minimal discriminant of the elliptic curve. We use this to recast the problem as one of counting lattice points in a particular region in R 2 .

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