List of Publications (with Abstracts)
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 Ed be the elliptic curve
y2 = x3 + 21dx2 + 112d2 x with complex multiplication
by the ring of integers in Q(sqrt(-7)). Let phi be the inverse Mellin transform of L(E1,s). We
construct explicit weight 3/2 cusp forms which are sent to phi by the Shimura correspondence.
We can then calculate L(Ed,1) in terms of the coefficients in the q-expansions of those forms.
As a consequence, the set of rational points on Ed is finite if the appropriate coefficient in
one of those forms is nonzero.
- "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 chiq. 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.
- "Rational eigenvectors in spaces of ternary forms.''
Mathematics of Computation.
Volume 66, Number 218, April, 1997,
- 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 phiE under the Shimura correspondence.
The tables which are linked below are intended to supplement the 1997
Mathematics of Computation article  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  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
Algorithms for Modular Elliptic Curves by
and his unpublished tables
obtained via ftp.
The 1992 Mathematics of Computation article  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
firstname.lastname@example.org if you have questions, comments, or corrections.
List of Tables
Last updated: May 27, 1997
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