2003/10/02 by Mark E. Watkins · 1 citation
Mathematics · #Analytic Number Theory Research #Algebraic Geometry and Number Theory #Advanced Mathematical Identities #Mathematics #Elliptic curve #Quadratic field #Class (philosophy) #Quadratic equation #Riemann hypothesis #Gauss #Order (exchange) #Algebraic number field #Dirichlet distribution #Character (mathematics) #Discrete mathematics #Combinatorics #Arithmetic #Pure mathematics #Quadratic function #Mathematical analysis #Computer science
paper · pdf · doi:10.1090/s0025-5718-03-01517-5
openalex publication_date 2003/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/05/29
The classical class number problem of Gauss asks for a classification of all imaginary quadratic fields with a given class number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The first complete results were for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N equals 1"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">N=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by Heegner, Baker, and Stark. After the work of Goldfeld and Gross-Zagier, the task was a finite decision problem for any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Indeed, after Oesterlé handled <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N equals 3"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">N=3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N equals 5 comma 6 comma 7"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>5</mml:mn> <mml:mo>,</mml:mo> <mml:mn>6</mml:mn> <mml:mo>,</mml:mo> <mml:mn>7</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">N=5,6,7</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , Wagner remarked that the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N equals 8"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>8</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">N=8</mml:annotation> </mml:semantics> </mml:math> </inline-formula> case seemed impregnable. We complete the classification for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N less-than-or-equal-to 100"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mn>100</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">N≤ 100</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , 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-Oesterlé work, which used an elliptic curve <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -function with an order 3 zero at the central critical point, to instead consider Dirichlet <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -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.