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November 12, 2024: $p$-converse theorems for odd Eisenstein primes of potentially good ordinary reduction

This is a follow-up to my blog entry from February 26, 2024. Mulun Yin (University of California Santa Barbara) managed to generalize our $p$-converse theorem for odd Eisenstein primes from the good ordinary to the potentially good ordinary case, i.e., allowing additive reduction at $p$. Note that the construction of $p$-adic $L$-functions for additive $p$ is an open problem in general.

Why this matters. The Birch–Swinnerton-Dyer conjecture predicts that two invariants of an elliptic curve agree: its algebraic rank $r$ (the number of independent rational points) and its analytic rank $r_\mathrm{an}$ (the order of vanishing of the $L$-function). It is a theorem of Gross–Zagier–Kolyvagin that $r_\mathrm{an}\in \{0,1\}$ implies $r = r_\mathrm{an}$; the converse is still open in general. A $p$-converse theorem supplies it via the $p$-Selmer rank $r_p$: it shows that $r_p \in \{0,1\}$ implies $r_\mathrm{an}= r = r_p$. The generalization described here reaches elliptic curves with additive reduction at $p$, where even the construction of the $p$-adic $L$-function is an open problem in general.

Statement of the result. The $p$-converse theorem (to the theorems of Gross–Zagier–Kolyvagin) states that in the situation above, if the $p$-Selmer rank $r_p$ equals $0$ or $1$, then the analytic and algebraic ranks equal $r_p$.

A trick. One trick we are using is a proposition due to Nekovář that our modular form in question has an ‘untwist’ that has good reduction. Then we are using ‘Heegner pairs’ due to Jetchev–Loeffler–Zerbes (2021).

Consequences. Together with results of Ari Shnidman et al. on the average $3$-Selmer rank in the presence of a $3$-isogeny, the result implies that the elliptic curve 13a3 (with a $\mathbf{Q}$-rational $3$-torsion point) has $\geq \frac{2}{3}$ of its quadratic twists satisfying the BSD rank conjecture.