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arXiv · 2606.02254

On periods of Elliptic curves

Daniel Barrera Salazar·Juan-Pablo Llerena-Córdova
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Abstract

Let $E$ be an elliptic curve over $\mathbb{Q}$ having split multiplicative reduction at a prime number $p$. We describe the tame part of the $\mathcal{L}$-invariant of $E$ at $p$ in terms of automorphic $p$-adic periods introduced in the work of Darmon. More precisely, we prove an equality of refined $\mathcal{L}$-invariants using twisted versions of refined exceptional zero conjectures. When the conductor of the elliptic curve is exactly $p$ and the automorphic period is attached to an optimal embedding of conductor $1$ then we prove this equality unconditionally by using the work of de-Shalit.

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Daniel Barrera Salazar, Juan-Pablo Llerena-Córdova. 2026-06-01. On periods of Elliptic curves. https://arxiv.org/abs/2606.02254

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