Vol. 7, No. 7, 2013

Download this article
Download this article For screen
For printing
Recent Issues

Volume 10
Issue 9, 1845–2052
Issue 8, 1601–1843
Issue 7, 1373–1600
Issue 6, 1147–1371
Issue 5, 939–1146
Issue 4, 695–938
Issue 3, 451–694
Issue 2, 215–450
Issue 1, 1–214

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
Cover
Editorial Board
Editors' Addresses
Editors' Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Subscriptions
Editorial Login
Contacts
Author Index
To Appear
 
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Multiplicative excellent families of elliptic surfaces of type $E_7$ or $E_8$

Abhinav Kumar and Tetsuji Shioda

Vol. 7 (2013), No. 7, 1613–1641
Abstract

We describe explicit multiplicative excellent families of rational elliptic surfaces with Galois group isomorphic to the Weyl group of the root lattices E7 or E8. The Weierstrass coefficients of each family are related by an invertible polynomial transformation to the generators of the multiplicative invariant ring of the associated Weyl group, given by the fundamental characters of the corresponding Lie group. As an application, we give examples of elliptic surfaces with multiplicative reduction and all sections defined over for most of the entries of fiber configurations and Mordell–Weil lattices described by Oguiso and Shioda, as well as examples of explicit polynomials with Galois group W(E7) or W(E8).

Keywords
rational elliptic surfaces, multiplicative invariants, inverse Galois problem, Weyl group, Mordell–Weil group
Mathematical Subject Classification 2010
Primary: 14J27
Secondary: 11G05, 12F10, 13A50
Milestones
Received: 8 April 2012
Revised: 13 December 2012
Accepted: 10 January 2013
Published: 12 October 2013
Authors
Abhinav Kumar
Department of Mathematics
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, MA 02139
United States
Tetsuji Shioda
Department of Mathematics
Rikkyo University
3-34-1 Nishi-Ikebukuro Toshima-ku
Tokyo 171-8501
Japan
http://www.rkmath.rikkyo.ac.jp/math/shioda/