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The Gauss class number problem for imaginary quadratic fields
 AMER. MATH. SOC
, 1985
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CLASS NUMBERS OF IMAGINARY QUADRATIC FIELDS
, 2003
"... 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 GrossZagier, the task was a finite decision problem for any N. ..."
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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 GrossZagier, the task was a finite decision problem for any N. Indeed, after Oesterlé 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 GoldfeldOesterlé work, which used an elliptic curve Lfunction with an order 3 zero at the central critical point, to instead consider Dirichlet Lfunctions with lowheight 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 MontgomeryWeinberger. Our method is still quite computerintensive, 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.
Order computations in generic groups
 PHD THESIS MIT, SUBMITTED JUNE 2007. RESOURCES
, 2007
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A modular curve of level 9 and the Class Number One Problem
, 2008
"... In this note we give an explicit parametrization of the modular curve associated to the normalizer of a nonsplit Cartan subgroup of level 9. We determine all integral points of this modular curve. As an application, we give an alternative solution to the class number one problem. ..."
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In this note we give an explicit parametrization of the modular curve associated to the normalizer of a nonsplit Cartan subgroup of level 9. We determine all integral points of this modular curve. As an application, we give an alternative solution to the class number one problem.
Ranks of elliptic curves in the families of quadratic twists
 Experiment Math
"... Abstract. We show that the unboundedness of the ranks of the quadratic twists of an elliptic curve is equivalent to the divergence of certain infinite series. 1. ..."
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Abstract. We show that the unboundedness of the ranks of the quadratic twists of an elliptic curve is equivalent to the divergence of certain infinite series. 1.
Heegner points: the beginnings
, 2004
"... Dick Gross and I were invited to talk about Heegner points from a historical point of view, and we agreed that I should talk first, dealing with the period before they became well known. I felt encouraged to indulge in some personal reminiscence of that period, particularly where I can support it by ..."
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Dick Gross and I were invited to talk about Heegner points from a historical point of view, and we agreed that I should talk first, dealing with the period before they became well known. I felt encouraged to indulge in some personal reminiscence of that period, particularly where I can support it by documentary evidence. I was fortunate enough to be working on the arithmetic of elliptic curves when comparatively little was known, but when new tools were just becoming available, and when forgotten theories such as the theory of automorphic function were being rediscovered. At that time, one could still obtain exciting new results without too much sophisticated apparatus: one was learning exciting new mathematics all the time, but it seemed to be less difficult! To set the stage for Heegner points, one may compare the state of the theory of elliptic curves over the rationals, E/Q for short, in the 1960’s and in the 1970’s; Serre [15] has already done this, but never mind! Lest I forget, I should stress that when I say “elliptic curve ” I will always mean “elliptic curve defined over the rationals”.