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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Congruences modulo $ p$ between $ \rho$-twisted Hasse-Weil $ L$-values


Authors: Daniel Delbourgo and Antonio Lei
Journal: Trans. Amer. Math. Soc. 370 (2018), 8047-8080
MSC (2010): Primary 11R23; Secondary 11G40, 19B28
DOI: https://doi.org/10.1090/tran/7240
Published electronically: July 12, 2018
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Abstract: Suppose $ E_1$ and $ E_2$ are semistable elliptic curves over $ \mathbb{Q}$ with good reduction at $ p$, whose associated weight two newforms $ f_1$ and $ f_2$ have congruent Fourier coefficients modulo $ p$. Let $ R_S(E_{\star \;},\rho )$ denote the algebraic $ p$-adic $ L$-value attached to each elliptic curve $ E_{\star }$, twisted by an irreducible Artin representation, $ \rho $, factoring through the Kummer extension $ \mathbb{Q}\big (\mu _{p^\infty },\Delta ^{1/p^{\infty }}\big )$.

If $ E_1$ and $ E_2$ have good ordinary reduction at $ p$, we prove that

$\displaystyle R_S(E_1,\rho )\equiv R_S(E_2,\rho ) \mod p ,$    

under an integrality hypothesis for the modular symbols defined over the field cut out by $ \text {\rm Ker}(\rho )$. Under this hypothesis, we establish that $ E_1$ and $ E_2$ have the same analytic $ \lambda $-invariant at $ \rho $.

Alternatively, if $ E_1$ and $ E_2$ have good supersingular reduction at $ p$, we show that

$\displaystyle \big \vert R_S(E_1,\rho )- R_S(E_2,\rho ) \big \vert _p \;<\; p^{\text {\rm ord}_p(\text {\rm cond}(\rho ))/2} .$    

These congruences generalise some earlier work of Vatsal [Duke Math. J. 98 (1999), pp. 399-419], Shekhar-Sujatha [Trans. Amer. Math. Soc. 367 (2015), pp. 3579-3598], and Choi-Kim [Ramanujan J. 43 (2017), p. 163-195], to the false Tate curve setting.

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Additional Information

Daniel Delbourgo
Affiliation: Department of Mathematics, University of Waikato, Private Bag 3105, Hillcrest, Hamilton 3240, New Zealand
Email: daniel.delbourgo@waikato.ac.nz

Antonio Lei
Affiliation: Département de mathématiques et de statistique, Université Laval, Pavillon Alex- andre-Vachon, 1045 avenue de la Médecine, Québec, G1V 0A6 Canada
Email: antonio.lei@mat.ulaval.ca

DOI: https://doi.org/10.1090/tran/7240
Received by editor(s): November 2, 2016
Received by editor(s) in revised form: March 2, 2017
Published electronically: July 12, 2018
Additional Notes: The second author’s research was supported by the NSERC Discovery Grants Program 05710
Article copyright: © Copyright 2018 American Mathematical Society

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