## Congruences modulo ρ between ρ-wisted Hasse-Weil L-values

##### Citation

Export citationDelbourgo, D., & Lei, A. (2018). Congruences modulo ρ between ρ-wisted Hasse-Weil L-values. Transactions of The American Mathematical Society, 370(11), 8047–8080. https://doi.org/10.1090/tran/7240

Permanent Research Commons link: https://hdl.handle.net/10289/12155

##### Abstract

Suppose E₁ and E₂ are semistable elliptic curves over Q with good reduction at p, whose associated weight two newforms f₁ and f₂ have congruent Fourier coefficients modulo p. Let RS(E*, ρ) denote the algebraic padic L-value attached to each elliptic curve E, twisted by an irreducible Artin representation, ρ, factoring through the Kummer extension Q(μp∞, Δ1/p∞).
If E₁ and E₂ have good ordinary reduction at p, we prove that RS(E₁, ρ) ≡ RS(E₂, ρ) mod p, under an integrality hypothesis for the modular symbols defined over the field cut out by Ker(ρ). Under this hypothesis, we establish that E₁ and E₂ have the same analytic λ-invariant at ρ.
Alternatively, if E₁ and E₂ have good supersingular reduction at p, we show that |RS(E₁, ρ) − RS(E₂, ρ)|ₚ < p ᵒʳᵈᵖ⁽ᶜᵒⁿᵈ⁽ρ⁾⁾/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.

##### Date

2018##### Type

##### Publisher

American Mathematical Society

##### Rights

This is an author’s accepted version of an article published in the journal: Transactions of The American Mathematical Society. © 2018 Transactions of The American Mathematical Society.