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

  • Ashay A. BurungaleEmail author
  • Ye Tian
Article
  • 186 Downloads

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).

Notes

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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.California Institute of TechnologyPasadenaUSA
  2. 2.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina
  3. 3.MCM, HLM, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina

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