Gauss' conjecture on equivalence classes of quadratic forms with positive discriminant (Led by Giuliana Davidoff).
Our summer topic was suggested by a mathematician at Princeton, who had directed a senior thesis aimed at putting specific numbers into a certain well-known conjecture by Gauss on the number of equivalence classes of quadratic forms with positive discriminant. This unsolved problem has a long history, but it has recently raised new interest because of its emergence in other contexts. Kwan, the Princeton student had obtained a good result, but her work, which ended upon her graduation, needed to be confirmed and augmented by further calculations. This was what my REU group set out to do, examining the class number both directly and through a related quantity called the fundamental unit, as well as through its deep connection to hyperbolic geometry.
This material is extremely beautiful, involving both classical number theory and early 20th century geometry, augmented by some relatively recent results. The REU students learned the extensive background material well enough to take on the problems involved in obtaining further numerical results. For discriminants associated to binary quadratic forms, one works not only with class groups of quadratic number fields, but with class groups of their associated orders, which are subrings of the ring of integers. The closely related theory of groups of genera, which was of primary interest to Gauss, sheds light on the class number problem and reduces it to a study of fewer cases, but still leaves open the classical question of small class number. My group of REU students studied the structure of both class groups and genera and then attacked the problem of writing efficient computer programs to produce and collate the large amount of data needed to see the asymptotic behavior of the class number and the fundamental unit.
In the end, after carrying the calculations to discriminants, d = b2 – 4ac, of size 20,000,000 and more, we were able to confirm Kwan’s coefficients for Gauss’ conjecture, though with corrections in the lower order terms she had obtained. We also looked at Hooley’s conjecture on similar asymptotics for the classical case, d = b2 – ac, requiring much larger discriminant sizes to see significant results. Ian Petrow continued our summer work when he returned to Princeton in the fall, and based on our initial computations, he was able to get very close to Hooley’s numbers for that case. Finally, the main difficulty in the positive discriminant case arises from the presence of fundamental units with negative norm unpredictably distributed as the discriminant of the quadratic number field grows. We also examined the the proportion of negative-to-positive norm fundamental units among allowable discriminants up to 350,000,000 and confirmed asymptotics previously conjectured by Stevenhagen, who had stated in his much earlier paper that the actual calculations would be extremely difficult to obtain. A full report with details will be posted on the Mount Holyoke College REU Web site.
The students from our number theory group gave two talks in August, 2006, at the Young Mathematicians Conference sponsored by the VIGRE grant at Ohio State University’s Department of Mathematics. One of our group presented the talk for which he was one of the two authors at both the MAA’s Mathfest and at the undergraduate poster session at the joint meetings in New Orleans in January. (The Mathfest presentation was awarded one of the undergraduate prizes.)
List of students:
- Lee Kennard, Kenyon College
- Jennifer Koonz, Wellesley College
- Ian Petrow, Princeton University
- Katharine Shultis, Scripps College
- Haokun (Sam) Xu, University of Arizona