Communications in Number Theory and Physics

Volume 16 (2022)

Number 4

Elliptic threefolds with high Mordell–Weil rank

Pages: 733 – 759



Antonella Grassi (Dipartimento di Matematica, Università di Bologna, Italy; and Department of Mathematics, University of Pennsylvania, Philadelphia, Penn., U.S.A.)

Timo Weigand (II. Institut für Theoretische Physik, Universität Hamburg, Germany; and Zentrum für Mathematische Physik, Universität Hamburg, Germany)


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 $\mathbb{P}^2$ of the Calabi–Yau threefolds as a log canonical model and compare it to a construction by Elkies and classical results of Burkhardt.

2010 Mathematics Subject Classification

14D99, 14G05, 14J27, 14N35, 81T30

The work of A.G. is partially supported by PRIN “Moduli and Lie Theory”. A.G. is a member of GNSAGA of INDAM.

The work of T.W. is supported in part by Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy EXC 2121 Quantum Universe 390833306.

Received 2 December 2021

Accepted 19 September 2022

Published 21 October 2022