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The number of elliptic curves over ℚ with conductorNwith conductorN

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Abstract

We prove that the number of elliptic curves E/ℚ with conductorN isO(N 1/2+ε). More generally, we prove that the number of elliptic curves E/ℚ with good reduction outsideS isO(M 1/2+ε), whereM is the product of the primes inS. Assuming various standard conjectures, we show that this bound can be improved toO(M c/loglogM).

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Authors and Affiliations

  1. Mathematics Department, Fordham University, 10458, Bronx, NY, USA

    Armand Brumer & Joseph H. Silverman

  2. Mathematics Department, Brown University, Box 1917, 02912, Providence, RI, USA

    Armand Brumer & Joseph H. Silverman

Authors
  1. Armand Brumer
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  2. Joseph H. Silverman
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Research partially supported by NSF DMS-9424642.

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Brumer, A., Silverman, J.H. The number of elliptic curves over ℚ with conductorNwith conductorN . Manuscripta Math 91, 95–102 (1996). https://doi.org/10.1007/BF02567942

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