Research Information

Larry Lehman
Mary Washington College

List of Publications (with Abstracts)

  1. "Rational points on elliptic curves with complex multiplication by the ring of integers in Q(sqrt(-7))." Journal of Number Theory. Volume 27, Number 3, November, 1987, pages 253-272.


  2. "Levels of positive definite ternary quadratic forms." Mathematics of Computation. Volume 58, Number 197, January, 1992, pages 399-417, with supplementary tables, pages S17-S22.


  3. "Rational eigenvectors in spaces of ternary forms.'' Mathematics of Computation. Volume 66, Number 218, April, 1997, pages 833-839.


Tables

The tables which are linked below are intended to supplement the 1997 Mathematics of Computation article [3] listed above. Let E be a modular elliptic curve over the rationals, with rank zero and conductor N such that either N or N/4 is squarefree, and let phiE be its associated weight two newform. For each isogeny class of such curves with N less than 2000, these tables present a linear combination of ternary quadratic forms in a genus depending on invariants of E. In each case, the theta series (weight 3/2 cusp form) obtained from this linear combination provides a potential pre-image of phiE under the Shimura correspondence. (The remaining details can, theoretically, be verified by direct computation. Please see [3] for a more precise statement of these results.)

The algorithm which produces these linear combinations can also be successfully applied to elliptic curves of rank two (and presumably to any elliptic curve of even rank) and conductor N as above. The last table listed below presents the linear combinations of ternary quadratic forms which correspond to each elliptic curve of rank two and conductor N less than 2000 (N or N/4 squarefree). In each case however, the theta series obtained from this linear combination of ternary forms is identically zero.

For the representatives of the isogeny classes of elliptic curves, I used the tables in Algorithms for Modular Elliptic Curves by John Cremona and his unpublished tables obtained via ftp. The 1992 Mathematics of Computation article [2] above provides a method for finding representatives of all classes of ternary quadratic forms in a particular genus. For these particular tables, I used a more direct algorithm, which seems to be just as fast on a typical personal computer. Aside from Cremona's data, these tables were compiled using my own computer programs and I am solely responsible for their accuracy. Please contact me at llehman@mwc.edu if you have questions, comments, or corrections.


List of Tables

Last updated: May 27, 1997

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