Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Mazur--Tate refined conjecture for the anticyclotomic tower at inert primes

Ryota Shii

TL;DR

This work extends the weak Mazur–Tate refined conjecture to the inert case for the anticyclotomic $oldsymbol{ ext{Z}}_p$-tower over an imaginary quadratic field, for a supersingular elliptic curve $E$ at $p eq 2,3$. By leveraging Bertolini–Darmon elements and plus/minus Iwasawa theory, together with the Rubin conjecture resolution by BKO21, the authors prove that the interpolated $p$-adic $L$-values $L_p(E/K_n)$ lie in the Fitting ideals of the dual plus/minus Selmer groups over the towers, i.e. $L_p(E/K_n)\in ext{Fitt}_{oldsymbol{ extLambda}_n}( ext{Sel}_{p^ fty}(E/K_n)^ op)$. The proof mirrors strategies from Kim–Kurihara and Kim–Kimura in the cyclotomic and split cases, adapted to the inert anticyclotomic setting via the Rubin framework for local conditions. As an arithmetic consequence, the result yields a weak vanishing statement for twists and connects refined Iwasawa theory to the anticyclotomic tower in the supersingular regime, expanding the scope of Mazur–Tate refinements beyond the split or ordinary cases.

Abstract

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with supersingular reduction at $p \geq 5$, and $K$ be an imaginary quadratic field such that $p$ is inert in $K/\mathbb{Q}$. In this paper, we prove the analogous of the ``weak'' Mazur--Tate refined conjecture for an anticyclotomic tower over $K$ using the result by A. Burungale--K. Büyükboduk--A. Lei.

On the Mazur--Tate refined conjecture for the anticyclotomic tower at inert primes

TL;DR

This work extends the weak Mazur–Tate refined conjecture to the inert case for the anticyclotomic -tower over an imaginary quadratic field, for a supersingular elliptic curve at . By leveraging Bertolini–Darmon elements and plus/minus Iwasawa theory, together with the Rubin conjecture resolution by BKO21, the authors prove that the interpolated -adic -values lie in the Fitting ideals of the dual plus/minus Selmer groups over the towers, i.e. . The proof mirrors strategies from Kim–Kurihara and Kim–Kimura in the cyclotomic and split cases, adapted to the inert anticyclotomic setting via the Rubin framework for local conditions. As an arithmetic consequence, the result yields a weak vanishing statement for twists and connects refined Iwasawa theory to the anticyclotomic tower in the supersingular regime, expanding the scope of Mazur–Tate refinements beyond the split or ordinary cases.

Abstract

Let be an elliptic curve defined over with supersingular reduction at , and be an imaginary quadratic field such that is inert in . In this paper, we prove the analogous of the ``weak'' Mazur--Tate refined conjecture for an anticyclotomic tower over using the result by A. Burungale--K. Büyükboduk--A. Lei.

Paper Structure

This paper contains 4 sections, 13 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Bertolini--Darmon elements
  3. The plus/minus Selmer group
  4. The proof of Theorem \ref{['thm:mthm']}

Key Result

Theorem 1.2

Assume (cp), (def), and the following: Then, we obtain $L_{p}(E/K_{n}) \in \mathop{\mathrm{Fitt}}\nolimits_{\Lambda_{n}}\left( \mathop{\mathrm{Sel}}\nolimits_{p^{\infty}}(E/K_{n})^{\vee} \right)$.

Figures (1)

  • Figure 1: Previous research for the definite type

Theorems & Definitions (20)

  • Conjecture 1.1: "weak" main conjecture
  • Theorem 1.2
  • Corollary 1.3
  • Proposition 2.1: Jacquet--Langrands correspondence
  • proof
  • Proposition 2.2: Ber-Dar05
  • Proposition 2.3: Dar-Iov08
  • Definition 3.1
  • Theorem 3.2: Rubin conjecture
  • Theorem 3.3: BKOpre
  • ...and 10 more