A formula of Perrin-Riou and characteristic power series of signed Selmer groups
Francesc Castella
TL;DR
The paper proves a p-adic Birch–Swinnerton-Dyer type formula for elliptic curves at supersingular primes with a_p=0, relating the leading term of Kobayashi's signed Selmer groups to explicit arithmetic invariants. It builds on Perrin-Riou’s big exponential to produce an arithmetic p-adic L-function in the Dieudonné module, then extracts signed p-adic L-functions via Lei’s signed Coleman maps and computes their coordinates in a natural Dieudonné basis. Under finiteness assumptions on the p–part of the Tate–Shafarevich group and a nondegenerate strict regulator, the paper gives an explicit leading-term formula involving $( ext{log}_p κ(γ))^{-r}$, p-adic regulators, Tamagawa numbers, and torsion data, aligning with conjectures of Bernardi–Perrin-Riou and Kato’s Main Conjecture. As a consequence, the signed p-adic L-functions generate the characteristic ideals of the signed Selmer groups, supporting cases of Kundu–Ray’s conjecture and connecting Iwasawa-theoretic invariants to classical arithmetic data with potential broader implications for p-adic BSD-type results.
Abstract
We prove a conjecture of Kundu--Ray, following from the $p$-adic Birch--Swinnerton-Dyer conjecture for supersingular primes by Bernardi--Perrin-Riou and Kato's Main Conjecture, predicting an expression for the leading term (up to a $p$-adic unit) of a characteristic power series of Kobayashi's signed Selmer groups attached to elliptic curves $E/\mathbb{Q}$ with supersingular reduction at a prime $p>2$ with $a_p=0$. The proof is deduced from a similar formula due to Perrin-Riou for a generator of her module of arithmetic $p$-adic $L$-functions with values in the Dieudonné module of $E$.