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

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.