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International Mathematics Research Notices (2007) Vol. 2007 : article ID rnm002, 18 pages, doi:10.1093/imrn/rnm002 published on May 24, 2007
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Copyright © The Author 2007. Published by Oxford University Press.

Reflection Principles and Bounds for Class Group Torsion

Jordan S. Ellenberg1 and Akshay Venkatesh2,

1 Department of Mathematics, University of Wisconsin, Madison
2 Courant Institute of Mathematical Sciences, New York University, New York

Correspondence: Correspondence to be sent to: Akshay Venkatesh, Courant Institute of Mathematical Sciences, New York University, New York. e-mail: akshay.venkatesh{at}gmail.com

We introduce a new method to bound {ell}-torsion in class groups, combining analytic ideas with reflection principles. This gives, in particular, new bounds for the 3-torsion part of class groups in quadratic, cubic and quartic number fields, as well as bounds for certain families of higher degree fields and for higher {ell}. Conditionally on GRH, we obtain a nontrivial bound for {ell}-torsion in the class group of a general number field.

Communicated by Bjorn Poonen


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