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Unconditional class group tabulation of imaginary quadratic fields to $ \Vert\Delta\Vert < 2^{40}$

Authors: A. S. Mosunov and M. J. Jacobson, Jr.
Journal: Math. Comp. 85 (2016), 1983-2009
MSC (2010): Primary 11R29; Secondary 11R11, 11Y40
Published electronically: November 3, 2015
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Abstract: We present an improved algorithm for tabulating class groups of imaginary quadratic fields of bounded discriminant. Our method uses classical class number formulas involving theta-series to compute the group orders unconditionally for all $ \Delta \not \equiv 1 \pmod {8}.$ The group structure is resolved using the factorization of the group order. The $ 1 \bmod 8$ case was handled using the methods of Jacobson, Ramachandran, and Williams including the batch verification method based on the Eichler-Selberg trace formula to remove dependence on the Extended Riemann Hypothesis. Our new method enabled us to extend the previous bound of $ \vert\Delta \vert < 2 \cdot 10^{11}$ to $ 2^{40}$. Statistical data in support of a variety of conjectures is presented, along with new examples of class groups with exotic structures.

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Additional Information

A. S. Mosunov
Affiliation: University of Waterloo, 200 University Avenue W, Waterloo, Ontario, Canada N2L 3G1

M. J. Jacobson, Jr.
Affiliation: University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4

Keywords: Binary quadratic form, class group, tabulation, out-of-core multiplication, Cohen-Lenstra heuristics
Received by editor(s): October 21, 2014
Received by editor(s) in revised form: February 12, 2015
Published electronically: November 3, 2015
Additional Notes: The first author’s research was supported by “Alberta Innovates Technology Futures”, Canada.
The second author’s research was supported by NSERC of Canada.
Article copyright: © Copyright 2015 American Mathematical Society