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

研究成果: Contribution to journalArticle

12 引用 (Scopus)

抜粋

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 ℤp-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.

元の言語英語
ページ(範囲)527-629
ページ数103
ジャーナルInventiones Mathematicae
191
発行部数3
DOI
出版物ステータス出版済み - 1 1 2013
外部発表Yes

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

フィンガープリント The p-adic Gross-Zagier formula for elliptic curves at supersingular primes' の研究トピックを掘り下げます。これらはともに一意のフィンガープリントを構成します。

  • これを引用