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The p-adic Gross-Zagier formula for elliptic curves at supersingular primes

Abstract

Let p be a prime number and let E be an elliptic curve defined over ℚ of conductor N. Let K be an imaginary quadratic field with discriminant prime to pN such that all prime factors of N split in K. B. Perrin-Riou established the p-adic Gross-Zagier formula that relates the first derivative of the p-adic L-function of E over K to the p-adic height of the Heegner point for K when E has good ordinary reduction at p. In this article, we prove the p-adic Gross-Zagier formula of E for the cyclotomic ℤ p -extension at good supersingular prime p. Our result has an application for the full Birch and Swinnerton-Dyer conjecture. Suppose that the analytic rank of E over ℚ is 1 and assume that the Iwasawa main conjecture is true for all good primes and the p-adic height pairing is not identically equal to zero for all good ordinary primes, then our result implies the full Birch and Swinnerton-Dyer conjecture up to bad primes. In particular, if E has complex multiplication and of analytic rank 1, the full Birch and Swinnerton-Dyer conjecture is true up to a power of bad primes and 2.

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Correspondence to Shinichi Kobayashi.

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Dedicated to the memory of Toshiyuki Ikegaya

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Kobayashi, S. The p-adic Gross-Zagier formula for elliptic curves at supersingular primes. Invent. math. 191, 527–629 (2013). https://doi.org/10.1007/s00222-012-0400-9

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Mathematics Subject Classification

  • 11F85
  • 11G05
  • 11G40
  • 11G50
  • 14G10
  • 14L05