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Transactions of the American Mathematical Society

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Rank-two Euler systems for symmetric squares


Authors: Kâzım Büyükboduk and Antonio Lei
Journal: Trans. Amer. Math. Soc. 372 (2019), 8605-8619
MSC (2010): Primary 11R23; Secondary 11F11, 11R20
DOI: https://doi.org/10.1090/tran/7860
Published electronically: June 25, 2019
MathSciNet review: 4029706
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Abstract: Let $ p\ge 7$ be a prime number, and let $ f$ be a normalized eigen-newform with good reduction at $ p$ such that its $ p$th Fourier coefficient vanishes. We construct a rank-two Euler system attached to the $ p$-adic realization of the symmetric square motive of $ f$. Furthermore, we show that the nontriviality is guaranteed by the nonvanishing of the leading term of the relevant $ L$-value and the nonvanishing of a certain $ p$-adic period modulo $ p$.


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Additional Information

Kâzım Büyükboduk
Affiliation: UCD School of Mathematics and Statistics, University College Dublin, Dublin, Ireland
Email: kazim.buyukboduk@ucd.ie

Antonio Lei
Affiliation: Département de Mathématiques et de Statistique, Université Laval, Pavillion Alexandre-Vachon, 1045 Avenue de la Médecine, Québec, Québec G1V 0A6, Canada
Email: antonio.lei@mat.ulaval.ca

DOI: https://doi.org/10.1090/tran/7860
Keywords: Iwasawa theory, elliptic modular forms, symmetric square representations, nonordinary primes
Received by editor(s): November 12, 2018
Received by editor(s) in revised form: February 1, 2019, and March 27, 2019
Published electronically: June 25, 2019
Additional Notes: The first author received funding from the European Union’s Horizon 2020 research and innovation program under Marie Skłodowska-Curie Grant Agreement No. 745691 (CriticalGZ)
The second author was supported by the NSERC Discovery Grants Program 05710.
Article copyright: © Copyright 2019 American Mathematical Society