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p-converse to a theorem of Gross–Zagier, Kolyvagin and Rubin

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Abstract

Let E be a CM elliptic curve over the rationals and \(p>3\) a good ordinary prime for E. We show that

$$\begin{aligned} {\mathrm {corank}}_{{\mathbb {Z}}_{p}} {\mathrm {Sel}}_{p^{\infty }}(E_{/{\mathbb {Q}}})=1 \implies {\mathrm {ord}}_{s=1}L(s,E_{/{\mathbb {Q}}})=1 \end{aligned}$$

for the \(p^{\infty }\)-Selmer group \({\mathrm {Sel}}_{p^{\infty }}(E_{/{\mathbb {Q}}})\) and the complex L-function \(L(s,E_{/{\mathbb {Q}}})\). In particular, the Tate–Shafarevich group \(\hbox {X}(E_{/{\mathbb {Q}}})\) is finite whenever \({\mathrm {corank}}_{{\mathbb {Z}}_{p}} {\mathrm {Sel}}_{p^{\infty }}(E_{/{\mathbb {Q}}})=1\). We also prove an analogous p-converse for CM abelian varieties arising from weight two elliptic CM modular forms with trivial central character. For non-CM elliptic curves over the rationals, first general results towards such a p-converse theorem are independently due to Skinner (A converse to a theorem of Gross, Zagier and Kolyvagin, arXiv:1405.7294, 2014) and Zhang (Camb J Math 2(2):191–253, 2014).

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Notes

  1. Strictly speaking, \(\lambda \) is not self-dual as an automorphic representation of \({\mathrm {GL}}_{1/K}\). It is so after an automorphic induction to \({\mathbb {Q}}\). We follow this unconventional terminology throughout.

  2. We refer to [49] for details.

  3. We refer to [13, §3.3] for details.

  4. The existence follows from g being p-ordinary.

  5. We refer to [3, §1.2].

  6. We also refer to [33, §11.1].

  7. We also refer to [3, §4.2] and references therein.

  8. Note that we are ignoring interpolation factors at places away from p appearing elsewhere in the literature, since those, while non-integral, can be interpolated by polynomial functions on \(W[[\Gamma _{K}^{\sharp }]]\).

  9. As the reader may note, hypothesis (ord) is inessential in this approach as well.

  10. We also refer to [27, Intro].

  11. We also refer to [29, Thm. 1.1].

  12. As \(L(s, \lambda ^{*}\cdot \frac{\chi }{\chi ^{*}})=L(s,\theta (\lambda ^{*}\cdot \frac{\chi }{\chi ^{*}}))\), the finiteness (4.4) also follows from [28].

  13. Usually referred as a ‘control theorem’.

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Acknowledgements

We are grateful to Karl Rubin, Chris Skinner and Wei Zhang for inspiring conversations and encouragement. We are also grateful to Francesc Castella, Laurent Clozel, Haruzo Hida, Chandrashekhar Khare and Peter Sarnak for insightful conversations. We thank Adebisi Agboola, Ben Howard and Xin Wan for helpful correspondence. We also thank Li Cai, John Coates, Henri Darmon, Daniel Disegni, Ralph Greenberg, Yukako Kezuka, Shinichi Kobayashi, Chao Li, Richard Taylor and Shou-Wu Zhang for instructive conversations about the topic. We are grateful to organisers of the program ‘Euler Systems and Special Values of L-functions’ held at CIB Lausanne during July–December 2017 for stimulating atmosphere. Part of this work was done while the authors were visiting CIB during an early part of the program. The first named author is also grateful to MCM Beijing for persistent warm hospitality. The article was conceived in Beijing during the summer of 2017. Finally, we are indebted to the referee. The current form of the article owes much to the perceptive comments and incisive suggestions.

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Authors and Affiliations

  1. California Institute of Technology, 1200 E California Blvd, Pasadena, CA, 91125, USA

    Ashay A. Burungale

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Ye Tian

  3. MCM, HLM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Ye Tian

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  1. Ashay A. Burungale
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Correspondence to Ashay A. Burungale.

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Burungale, A.A., Tian, Y. p-converse to a theorem of Gross–Zagier, Kolyvagin and Rubin. Invent. math. 220, 211–253 (2020). https://doi.org/10.1007/s00222-019-00929-7

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