Mary Washington College

- "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.- Let E
^{d}be the elliptic curve y^{2}= x^{3}+ 21dx^{2}+ 112d^{2}x with complex multiplication by the ring of integers in Q(sqrt(-7)). Let phi be the inverse Mellin transform of L(E^{1},s). We construct explicit weight 3/2 cusp forms which are sent to phi by the Shimura correspondence. We can then calculate L(E^{d},1) in terms of the coefficients in the q-expansions of those forms. As a consequence, the set of rational points on E^{d}is finite if the appropriate coefficient in one of those forms is nonzero.

- Let E
- "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.- The level N and squarefree character q of a positive definite ternary quadratic form
are defined so that its associated modular form has level N and character chi
_{q}. We define a collection of correspondences between classes of quadratic forms having the same level and different discriminants. This makes practical a method for finding representatives of all classes of ternary forms having a given level. We also give a formula for the number of genera of ternary forms with a given level and character.

- The level N and squarefree character q of a positive definite ternary quadratic form
are defined so that its associated modular form has level N and character chi
- "Rational eigenvectors in spaces of ternary forms.''
**Mathematics of Computation**. Volume 66, Number 218, April, 1997, pages 833-839.- We describe the explicit computation of linear combinations of
ternary quadratic forms which are eigenvectors, with rational eigenvalues,
under all Hecke operators. We use this process to construct, for each
elliptic curve E of rank zero and conductor N less than 2000 for which N
or N/4 is squarefree, a weight 3/2 cusp form which is (potentially) a
preimage of the weight two newform phi
_{E}under the Shimura correspondence.

- We describe the explicit computation of linear combinations of
ternary quadratic forms which are eigenvectors, with rational eigenvalues,
under all Hecke operators. We use this process to construct, for each
elliptic curve E of rank zero and conductor N less than 2000 for which N
or N/4 is squarefree, a weight 3/2 cusp form which is (potentially) a
preimage of the weight two newform phi

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 phi_{E} 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 phi_{E}
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.

- Sample Table with Annotations

- Elliptic Curves of Rank Zero, Conductor N (N or N/4 Squarefree)

- Elliptic Curves of Rank Two,
Conductor N < 2000 (N or N/4 Squarefree)