An interesting elliptic surface over an elliptic curve

Abstract

Abstract. We study the elliptic modular surface attached to the commutator subgroup of the modular group. This has an elliptic curve as base and only one singular fibre. We employ an algebraic approach and then consider some arithmetic questions.

The Barth quintic surface has Picard number 41

“…Here we reverse this argument and verify the Tate conjecture for the quintics S through K3 quotients. A similar technique was applied in [12], but the situation here is more complicated since the surfaces in question have different geometric genus.…”

Explicit Algebraic Coverings of a Pointed Torus

“…Up to some scaling factors, this is the elliptic surface studied in [6]. By construction the discriminant of the polynomial x 3 +ax+b ∈ C(B)[x] is a nonzero constant.…”

“…In this note we use a particular elliptic surface described by Schütt and Shioda [6] in order to obtain some examples. Namely, they take the elliptic curve B/C with j-invariant 0 and an elliptic surface E → B which has precisely one singular fibre.…”

Étude des fibrations elliptiques d’une surface K3

“…On peut calculer la hauteur du point correspondant à (3A) ←→ −P k + −k 2 , 0 en utilisant le graphe et [10] Table 4. On compte quelle composante des fibres I 10 , I 3 et I * 2 est coupé par (3A).…”