We present the first examples of smooth elliptic Calabi-Yau threefolds with Mordell-Weil rank 10, the highest currently known value.They are given by the Schoen threefolds introduced by Namikawa; there are six isolated fibers of Kodaira Type IV.We explicitly compute the Shioda homomorphism and the induced height pairing.Compactification of F-theory on these threefolds gives an effective theory in six dimensions which contains ten abelian gauge group factors.We compute the massless matter spectrum.In particular, we show that the charged singlet matter need not reside at enhancement loci of Type I 2 , as previously believed.We relate the multiplicities of the massless spectrum to genus-zero Gopakumar-Vafa invariants and other geometric quantities of the Calabi-Yau.We show that the gravitational and abelian anomaly cancellation conditions are satisfied.We prove a Geometric Anomaly Cancellation equation and we deduce birational equivalence for the quantities in the spectrum.We explicitly describe a Weierstrass model over P 2 of the Calabi-Yau threefolds as a log canonical model and compare it to a construction by Elkies and classical results of Burkhardt.* (f )

Antonella Grassi

Timo Weigand

Communications in Number Theory and Physics

专注于最广泛意义上的数论在理论物理学中的应用。通过主要发表有关这两个领域之间关系和动态的研究、评论和说明性文章，为数论和理论物理学研究人员之间的交流提供一个论坛。

This journal focuses on the broad applications of number theory in theoretical physics. We primarily publish research, reviews, and expository articles concerning the relationships and dynamics between these two fields, providing a forum for communication among researchers in number theory and theoretical physics.

本期刊专注于数论在理论物理学中的广泛应用。我们主要发表关于这两个领域之间关系和动态的研究、评论和说明性文章，为数论和理论物理学研究人员之间的交流提供一个论坛。

Category	Quartile
数学	3
数学, 数学	3
数学, 应用数学	3
数学, 物理数学物理	3

Elliptic threefolds with high Mordell–Weil rank

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Category	Quartile
PHYSICS, MATHEMATICAL	3