Mark Watkins

Class numbers of imaginary quadratic fields

The classical class number problem of Gauss asks for a classification of all imaginary quadratic fields with a given class number N. The first complete results were for N = 1 by Heegner, Baker, and Stark. After the work of Goldfeld and Gross-Zagier, the task was a finite decision problem for any N. Indeed, after Oesterle handled N = 3, in 1985 Serre wrote, No doubt the same method will work for other small class numbers, up to 100, say. However, more than ten years later, after doing N = 5, 6, 7, Wagner remarked that the N = 8 case seemed impregnable. We complete the classification for all N < 100, an improvement of four powers of 2 (arguably the most difficult case) over the previous best results. The main theoretical technique is a modification of the Goldfeld-Oesterle work, which used an elliptic curve L-function with an order 3 zero at the central critical point, to instead consider Dirichlet L-functions with low-height zeros near the real line (though the former is still required in our proof). This is numerically much superior to the previous method, which relied on work of Montgomery-Weinberger. Our method is still quite computer-intensive, but we are able to keep the time needed for the computation down to about seven months. In all cases, we find that there is no abnormally large exceptional modulus of small class number, which agrees with the prediction of the Generalised Riemann Hypothesis.

Average Ranks of Elliptic Curves: Tension between Data and Conjecture

Rational points on elliptic curves are the gems of arithmetic: they are, to diophantine geometry, what units in rings of integers are to algebraic number theory, what algebraic cycles are to algebraic geometry. A rational point in just the right context, at one place in the theory, can inhibit and control—thanks to ideas of Kolyvagin [Kol88]—the existence of rational points and other mathematical structures elsewhere. Despite all that we know about these objects, the initial mystery and excitement that drew mathematicians to this arena in the first place remains in full force today. We have a network of heuristics and conjectures regarding rational points, and we have massive data accumulated to exhibit instances of the phenomena. Generally, we would expect that our data support our conjectures; and if not, we lose faith in our conjectures. But here there is a somewhat more surprising interrelation between data and conjecture: they are not exactly in open conflict one with the other, but they are no great comfort to each other either. We discuss various aspects of this story, including recent heuristics and data that attempt to resolve this mystery. We shall try to convince the reader that, despite seeming discrepancy, data and conjecture are, in fact, in harmony.

A Database of Elliptic Curves - First Report

In the late 1980s, Brumer and McGuinness [2] undertook the construction of a database of elliptic curves whose absolute discriminant |Δ| was both prime and satisfied |Δ| ≤ 108. While the restriction to primality was nice for many reasons, there are still many curves of interest lacking this property. As ten years have passed since the original experiment, we decided to undertake an extension of it, simultaneously extending the range for the type of curves they considered, and also including curves with composite discriminant. Our database can be crudely described as being the curves with |Δ| ≤ 1012 which either have conductor smaller than 108 or have prime conductor less than 1010—but there are a few caveats concerning issues like quadratic twists and isogenous curves. For each curve in our database, we have undertaken to compute various invariants (as did Brumer and McGuinness), such as the Birch—Swinnerton-Dyer L-ratio, generators, and the modular degree. We did not compute the latter two of these for every curve. The database currently contains about 44 million curves; the end goal is find as many curves with conductor less than 108 as possible, and we comment below on this direction of growth of the database. Of these 44 million curves, we have started a first stage of processing (computation of analytic rank data), with point searching to be carried out in a later second stage of computation.

Computing the Modular Degree of an Elliptic Curve

We review previous methods of computing the modular degree of an elliptic curve, and present a new method (conditional in some cases), which is based upon the computation of a special value of the symmetric square L-function of the elliptic curve. Our method is sufficiently fast to allow large-scale experiments to be done. The data thus obtained on the arithmetic character of the modular degree show two interesting phenomena. First, in analogy with the class number in the number field case, there seems to be a Cohen-Lenstra heuristic for the probability that an odd prime divides the modular degree. Secondly, the experiments indicate that 2r should always divide the modular degree, where r is the Mordell–Weil rank of the elliptic curve. We also discuss the size distribution of the modular degree, or more exactly of the special L-value which we compute, again relating it to the number field case.

Number Theory and Algebraic Geometry: Constructing elements in Shafarevich–Tate groups of modular motives

We study Shafarevich-Tate groups of motives attached to modular forms on Γ0(N) of weight bigger than 2. We deduce a criterion for the existence of nontrivial elements of these Shafarevich-Tate groups, and give 16 examples in which a strong form of the Beilinson-Bloch conjecture implies the existence of such elements. We also use modular symbols and observations about Tamagawa numbers to compute nontrivial conjectural lower bounds on the orders of the Shafarevich-Tate groups of modular motives of low level and weight at most 12. Our methods build upon the idea of visibility due to Cremona and Mazur, but in the context of motives instead of abelian varieties.

Some Heuristics about Elliptic Curves

We give some heuristics for counting elliptic curves with certain properties. In particular, we rederive the Brumer–McGuinness heuristic for the number of curves with positive/negative discriminant up to X, which is an application of lattice-point counting. We then introduce heuristics that allow us to predict how often we expect an elliptic curve E with even parity to have L(E, 1) = 0. We find that we expect there to be about c 1 X 19/24(logX)3/8 curves with |Δ| < X with even parity and positive (analytic) rank; since Brumer and McGuinness predict cX 5/6 total curves, this implies that, asymptotically, almost all even-parity curves have rank 0. We then derive similar estimates for ordering by conductor, and conclude by giving various data regarding our heuristics and related questions.

Real zeros of real odd Dirichlet L -functions

Let χ be a real odd Dirichlet character of modulus d, and let L(s, χ) be the associated Dirichlet L-function. As a consequence of the work of Low and Purdy, it is known that if d ≤ 800000 and d ≠ 115147, 357819, 636184, then L(s, χ) has no positive real zeros. By a simple extension of their ideas and the advantage of thirty years of advances in computational power, we are able to prove that if d ≤ 300 000 000, then L(s, χ) has no positive real zeros.

Ranks of Elliptic Curves and Random Matrix Theory: Secondary terms in the number of vanishings of quadratic twists of elliptic curve L -functions

We examine the number of vanishings of quadratic twists of the L-function associated to an elliptic curve. Applying a conjecture for the full asymptotics of the moments of critical L-values we obtain a conjecture for the first two terms in the ratio of the number of vanishings of twists sorted according to arithmetic progressions.

Elliptic Curves of Large Rank and Small Conductor

For r=6,7,...,11 we find an elliptic curve E/Q of rank at least r and the smallest conductor known, improving on the previous records by factors ranging from 1.0136 (for r=6) to over 100 (for r=10 and r=11). We describe our search methods, and tabulate, for each r=5,6,...,11, the five curves of lowest conductor, and (except for r=11) also the five of lowest absolute discriminant, that we found.

On the Extremality of an 80-Dimensional Lattice

We show that a specific even unimodular lattice of dimension 80, first investigated by Schulze-Pillot and others, is extremal (i.e., the minimal nonzero norm is 8). This is the third known extremal lattice in this dimension. The known part of its automorphism group is isomorphic to SL 2(F 79), which is smaller (in cardinality) than the two previous examples. The technique to show extremality involves using the positivity of the Θ-series, along with fast vector enumeration techniques including pruning, while also using the automorphisms of the lattice.