Iwasawa invariants of Bertolini--Darmon Theta Elements

Technical Summary: Iwasawa Invariants of Bertolini–Darmon Theta Elements

Problem Statement
This article investigates the Iwasawa invariants ($\mu$ and $\lambda$) of Bertolini–Darmon theta elements attached to weight two modular forms $f \in S_2(\Gamma_0(N))$ within the anticyclotomic $\mathbb{Z}_p$-extension of an imaginary quadratic field $K$. The study addresses two distinct reduction scenarios at a prime $p \geq 5$: the ordinary case and the non-ordinary case.

The work aims to extend known results from the cyclotomic setting (specifically those of Pollack–Weston and Leonard–Lei) to the anticyclotomic setting. In the cyclotomic context, the Iwasawa invariants of Mazur-Tate elements are known to relate to the invariants of $p$-adic $L$-functions. This paper seeks to establish analogous relationships for Bertolini–Darmon theta elements ($L_{f,n}$) and the associated anticyclotomic $p$-adic $L$-functions ($L_p(f, K_\infty)$ or $L_p(f, K)^\ast$).

Methodology
The authors employ a combination of algebraic Iwasawa theory and the arithmetic of modular forms on definite quaternion algebras.

1. Setup and Definitions:

   • The authors fix an imaginary quadratic field $K$ and a prime $p$ unramified in $K$. The level $N$ is decomposed into $N = N^+ N^-$, where primes dividing $N^+$ split in $K$ and primes dividing $N^-$ are inert. $N^-$ is assumed to be square-free and a product of an odd number of primes.
   • Bertolini–Darmon theta elements $L_{f,n} \in \mathcal{O}[Gal(K_n/K)]$ are defined via the Jacquet–Langlands correspondence, identifying the modular form $f$ with a function on the Bruhat–Tits tree associated with a definite quaternion algebra of discriminant $N^-$.
   • These elements satisfy a three-term recurrence relation involving the Hecke eigenvalue $a_p(f)$ and norm maps.

2. Ordinary Case:

   • When $f$ is ordinary at $p$, the authors utilize $p$-stabilized theta elements ($L_{f\alpha, n}$) to define the anticyclotomic $p$-adic $L$-function $L_p(f/K_\infty)$ as a projective limit.
   • They analyze the relationship between the invariants of the finite-level elements $L_{f,n}$ and the infinite-level $L$-function using properties of the Iwasawa algebra $\Lambda \cong \mathcal{O}[[T]]$ and specific lemmas regarding the behavior of $\mu$ and $\lambda$ under projection and norm maps.

3. Non-Ordinary Case:

   • In this setting, the authors assume $p$ splits in $K$. They construct "Sprung-type" logarithm matrices ($M_{f,n}$) to relate the theta elements to the integral $\sharp/\flat$ $p$-adic $L$-functions ($L_p^\sharp(f, K_\infty)$ and $L_p^\flat(f, K_\infty)$).
   • A key technical step involves analyzing the congruence of the theta elements modulo $\omega_n$ (where $\omega_n = (1+T)^{p^n}-1$) in terms of the $\sharp/\flat$ $L$-functions. This allows the authors to express $L_{f,n}$ as a product of a cyclotomic polynomial factor ($\omega_n^\pm$) and the $L$-function elements.
   • The authors distinguish cases based on whether $a_p(f) = 0$ or $a_p(f) \not\equiv 0 \pmod{\varpi}$, utilizing the fact that $a_p(f) \equiv 0 \pmod{\varpi}$ in the non-ordinary setting to simplify matrix relations.

Key Contributions and Results

• Ordinary Case (Theorem A / Theorem 4.1):
  The authors prove an analogue of Pollack–Weston's result for the anticyclotomic setting. Assuming $f$ has good ordinary reduction and $\mu(L_p(f/K_\infty)) = 0$, they establish that for sufficiently large $n$:

  1. $\mu(L_{f,n}) = 0$.
  2. $2\lambda(L_{f,n}) = \lambda(L_p(f/K_\infty))$.
     This confirms that the $\lambda$-invariant of the theta element is exactly half that of the $p$-adic $L$-function, while the $\mu$-invariant vanishes if the $L$-function's $\mu$-invariant vanishes.

• Non-Ordinary Case (Theorem B / Theorem 5.10):
  The authors extend the results of Leonard–Lei to the anticyclotomic setting. Assuming $p$ splits in $K$ and $\mu(L_p^\sharp) = \mu(L_p^\flat) \neq \infty$, they relate the invariants of $L_{f,n}$ to those of the $\sharp/\flat$ $L$-functions ($L_p(f, K)^\ast$).
  For sufficiently large $n$ (where the parity of $n$ determines whether the $\sharp$ or $\flat$ branch is active):

  1. $\mu(L_p(f, K)^\ast) = 2\mu(L_{f,n})$.
  2. $\lambda(L_p(f, K)^\ast) = 2\lambda(L_{f,n}) = 2\lambda(L^\ast_f) + 2q_n$.
     Here, $q_n$ is a specific polynomial in $p$ depending on the parity of $n$ (e.g., $p^{n-1} - p^{n-2} + \dots - 1$ for even $n$).

Significance and Claims
The paper claims to successfully generalize the known cyclotomic results of Pollack–Weston and Leonard–Lei to the anticyclotomic extension of an imaginary quadratic field. By establishing precise formulas linking the Iwasawa invariants of finite-level Bertolini–Darmon theta elements to the invariants of the corresponding $p$-adic $L$-functions, the work provides a deeper understanding of the arithmetic properties of these elements in the non-ordinary case.

The authors note that the proof strategy relies on expressing the theta elements in terms of $\sharp/\flat$ $L$-functions using Sprung-type matrices, thereby allowing the transfer of invariant properties from the $L$-functions to the theta elements.

Future Directions
The authors explicitly state that the analogue of Theorem B for the case where $p$ is inert in $K$ is a work in progress and is not covered in this article.