Congruences modulo $p$ between $\rho$-twisted Hasse-Weil $L$-values
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- by Daniel Delbourgo and Antonio Lei PDF
- Trans. Amer. Math. Soc. 370 (2018), 8047-8080 Request permission
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 \begin{equation*} R_S(E_1,\rho )\equiv R_S(E_2,\rho ) \mod p , \end{equation*} under an integrality hypothesis for the modular symbols defined over the field cut out by $\operatorname {Ker}(\rho )$. 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 \begin{equation*} \big | R_S(E_1, \rho ) - R_S(E_2, \rho ) \big |_p < p^{ \operatorname {ord}_p(\operatorname {cond}(\rho )) /2 } . \end{equation*} 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
- MR Author ID: 355277
- 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
- MR Author ID: 902727
- ORCID: 0000-0001-9453-3112
- Email: antonio.lei@mat.ulaval.ca
- 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
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 8047-8080
- MSC (2010): Primary 11R23; Secondary 11G40, 19B28
- DOI: https://doi.org/10.1090/tran/7240
- MathSciNet review: 3852457