Abstract
The modularity of elliptic curves always intrigues number theorists. Recently, Thorne had proved a marvelous result that for a prime p, every elliptic curve defined over a p-cyclotomic extension of \( {\mathbb {Q}} \) is modular. The method is to use some automorphy lifting theorems and study non-cusp points on some specific elliptic curves by Iwasawa theory for elliptic curves. Since the modularity of elliptic curves over real quadratic was proved, one may ask whether it is possible to replace \({\mathbb {Q}}\) with a real quadratic field K. Following Thorne’s idea, we give some assumptions first and prove the modularity of elliptic curves over the \({\mathbb {Z}}_p\)-extension of some real quadratic fields.
Similar content being viewed by others
References
Cremona, J.E.: Algorithms for Modular Elliptic Curves, 2nd edn. Cambridge University Press, Cambridge (1997)
Derickx, M., Najman, F., Siksek, S.: Elliptic curves over totally real cubic fields are modular. Algebra Number Theory 14(7), 1791–1800 (2020)
Freitas, N., Le Hung, B.V., Siksek, S.: Elliptic curves over real quadratic fields are modular. Invent. Math. 201(1), 159–206 (2015)
Greenberg, R.: Iwasawa theory for elliptic curves. In: Arithmetic Theory of Elliptic Curves, pp. 51–144. Springer, Berlin (1999)
Iovita, A., Pollack, R.: Iwasawa theory of elliptic curves at supersingular primes over \(\mathbb{Z} _p\)-extensions of number fields. J. Reine Angew. Math. 2006(598), 71–103 (2006)
Langlands, R.: Base change for GL(2). Number 96. Princeton University Press, Princeton (1980)
Le Hung, B.V.: Modularity of some elliptic curves over totally real fields. arXiv preprint (2013). arXiv:1309.4134
Silverman, J.H.: The Arithmetic of Elliptic Curves, vol. 106. Springer, Berlin (2009)
The LMFDB Collaboration: The L-functions and modular forms database (2022). http://www.lmfdb.org
Thorne, J.A.: Elliptic curves over \( \mathbb{Q} _\infty \) are modular. J. Eur. Math. Soc. 21(7), 1943–1948 (2019)
Yoshikawa, S.: On the modularity of elliptic curves over a composite field of some real quadratic fields. Res. Number Theory 2(1), 1–6 (2016)
Acknowledgements
The author would like to express appreciation to his supervisor Professor Takeshi Saito for his comments and suggestions. Also, the author would like to thank Professor Jack Thorne and Dr. Sho Yoshikawa for some helpful communications.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Zhang, X. On the modularity of elliptic curves over the cyclotomic \({\mathbb {Z}}_p\)-extension of some real quadratic fields. Ramanujan J 62, 545–550 (2023). https://doi.org/10.1007/s11139-022-00686-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-022-00686-x