Arithmetic on elliptic threefolds

Abstract

In a recent paper, Rosen and Silverman showed that Tate's conjecture on algebraic cycles implies a formula of Nagao, which gives the rank of an elliptic surface in terms of a weighted average of fibral Frobenius trace values. In this article, we extend their result to the case of elliptic threefolds. The main ingredients of our argument are a Shioda-Tate-like formula for elliptic threefolds, and a relation between the 'average' number of rational points on singular fibers and the Galois action on those fibers.

F-theory and AdS3/CFT2

“…It will be very important in the following to determine the possible divisors (4d submanifolds) in such a geometry, which is the content of the Shioda-Tate-Wazir theorem [65], which implies that the divisors of an elliptic Calabi-Yau threefold Y 3 with a section, fall into the following three classes:…”

Fibrations et conjecture de Tate

“…Nous prouvons ici une formule (Proposition 2.6) analogue à celle de Shioda-Tate pour les surfaces elliptiques ayant une section généralisant [20,26,9]. Nous utilisons la théorie d'intersection et prenons comme référence Fulton [6] ou Hartshorne [8, Appendix A].…”

“…Dans le cas présent, au lieu d'analyser chaque fibre, la proposition donne un contrôle de la somme totale des contributions du nombre des points rationnels des fibres singulières. Mais en employant la même méthode de décompte, on peut aussi démontrer la généralisation suivante de [26,Theorem 4.1]. En effet on en tire que…”

Towards the Standard Model in F-theory

“…S 0 , can be interpreted as the zero-section. 3 The image of the remaining two independent sections under the Shioda map [81][82][83] then identifies the generators of two independent U (1) gauge groups as [57,58,60,61] 5) where K = π −1 K B is the pre-image of the anti-canonical bundle of the base B. Here and in the sequel our notation does not distinguish between a divisor (class) and its dual 2-form.…”