Mod $\ell$ non-vanishing of self-dual Hecke $L$-values over CM fields and applications

1. Problem Statement and Context

The Core Problem:
The paper addresses the non-vanishing of central values of Hecke $L$-functions associated with self-dual Hecke characters $\lambda$ over a CM field $K$. Specifically, it investigates the behavior of these values within a $p$-adic family of twists by finite-order characters $\nu$ of the Galois group $\Gamma_p$ of the anticyclotomic $\mathbb{Z}_p$-extension of $K$.

There are two main questions:

1. Complex Non-Vanishing (Q1): Are the central values $L(1, \lambda\nu)$ non-zero for all but finitely many characters $\nu$ in the family where the global epsilon factor $\varepsilon(\lambda\nu) = +1$?
2. Mod $\ell$ Non-Vanishing (Q2): For a prime $\ell \neq p$ (ordinary for the extension $K/F$), is the $\ell$-adic valuation of the algebraic part of the $L$-value, $L_{alg}(1, \lambda\nu)$, bounded (specifically, equal to a specific invariant $\mu_\ell(\lambda)$) for all but finitely many $\nu$?

The Gap in Previous Work:
Prior results by Hida (and subsequent corrections by Hsieh) established non-vanishing for a Zariski dense subset of characters. However, a gap was discovered in Hida's proof regarding the Zariski density of CM points on the self-product of a Hilbert modular Shimura variety. Consequently, Hida's results were weakened to guarantee non-vanishing for infinitely many characters, but not necessarily all but finitely many. This gap rendered incomplete Hsieh's proof of the Eisenstein congruence divisibility towards the CM Iwasawa Main Conjecture.

2. Methodology and Strategy

The authors employ a novel approach that diverges from Hida's geometry of Hilbert modular varieties. Instead, they utilize the arithmetic of Shimura sets associated with totally definite quaternion algebras and ergodic theory.

Key Methodological Steps:

1. Quaternionic Transfer (Jacquet-Langlands):
   The problem is recast from the unitary group $U(1)$ (associated with the CM field) to a totally definite quaternion algebra $B$ over the maximal totally real subfield $F$. The self-dual Hecke character $\lambda$ corresponds to a cuspidal automorphic representation $\pi_\lambda$ of $B^\times(\mathbb{A})$.

2. Explicit Waldspurger Formula:
   The authors use an explicit Waldspurger formula (based on the work of Cai–Shu–Tian) to link the central Rankin–Selberg $L$-values $L(1/2, \pi_\lambda \otimes \chi)$ to toric periods of a specific test vector $f_\lambda$ on the Shimura set $X_U = B^\times \backslash \mathbb{A}_B^\times / U$.
   $$L(1/2, \pi_\lambda \otimes \chi) \propto |P_\chi(f_\lambda)|^2$$
   where $P_\chi(f_\lambda)$ is a sum of the form $\sum \chi(t) f_\lambda(x(t))$.

3. Equidistribution and Ergodicity:
   To prove non-vanishing of the toric periods modulo $\ell$, the authors rely on Ratner's ergodicity of unipotent flows. They analyze the distribution of special points (Gross points) on the Shimura set.

   • Challenge: In the CM case, the test vector $f_\lambda$ is "exceptional" (it vanishes on certain components of the Shimura set depending on the epsilon factor).
   • Solution: The authors refine the strategy by partitioning the Shimura set into components $X_U^+$ and $X_U^-$ based on the reduced norm. They prove that $f_\lambda \pmod \ell$ is non-Eisenstein on exactly one of these components (determined by $\varepsilon(\lambda)$). This allows them to apply equidistribution arguments to show that the periods do not vanish for almost all twists.

4. Period Comparison (The Bridge):
   A critical technical contribution is the comparison between the quaternionic period $\Omega_\lambda$ (arising from the Shimura set) and the CM period $\Omega_\infty$ (arising from the abelian variety).

   • They establish an $\ell$-integral relation between these periods involving Gauss sums and Gamma factors.
   • This comparison allows them to transfer the non-vanishing result from the Rankin–Selberg $L$-values (proven via ergodicity) to the original Hecke $L$-values.

5. Auxiliary Characters:
   To handle the period comparison and establish the exact $\ell$-adic valuation, the authors introduce auxiliary Hecke characters and primes to construct specific families where the lower bounds for valuations become equalities.

3. Key Contributions and Results

Theorem 1.1 (Complex Non-Vanishing):
Let $K$ be a CM field, $F$ its maximal totally real subfield, and $\lambda$ a self-dual Hecke character. Let $p$ be a degree one prime of $F$ unramified in $K$. Then:
$$L(1, \lambda\nu) \neq 0$$
for all but finitely many finite order characters $\nu$ of $\Gamma_p$ such that $\varepsilon(\lambda\nu) = +1$.
Significance: This strengthens previous results from "infinitely many" to "all but finitely many," filling the gap in Hida's theory.

Theorem 1.4 (Mod $\ell$ Non-Vanishing):
Let $\ell \neq p$ be an ordinary prime for $K/F$. Then for all but finitely many $\nu \in \Xi^+_{\lambda,p}$:
$$v_\ell(L_{alg}(1, \lambda\nu)) = \mu_\ell(\lambda)$$
where $\mu_\ell(\lambda)$ is a specific invariant related to the local behavior of $\lambda$ at inert primes.
Significance: This determines the exact $\ell$-adic valuation, implying the boundedness of the $\ell$-part of the Tate–Shafarevich groups in the anticyclotomic family.

Theorem 1.6 (CM Iwasawa Main Conjecture):
The authors complete Hsieh's proof of the divisibility towards the CM Iwasawa Main Conjecture. Specifically, they prove that the Katz $p$-adic $L$-function $L_\Sigma(\psi)$ divides the characteristic power series $F_\Sigma(\psi)$ of the Selmer group:
$$L_\Sigma(\psi) \mid F_\Sigma(\psi)$$
Significance: This resolves the incompleteness in Hsieh's 2014 proof caused by the gap in Hida's non-vanishing results.

Corollary 1.2 (Mordell-Weil Ranks):
Under the conditions of Theorem 1.1, the Mordell-Weil group $A_\lambda(K_{p^\infty})$ of the associated CM abelian variety is finitely generated.

4. Significance and Impact

1. Resolution of a Major Gap: The paper successfully patches a critical flaw in the arithmetic of Hilbert modular forms that had stalled progress on the CM Iwasawa Main Conjecture for over a decade.
2. New Methodological Paradigm: By shifting from the geometry of Hilbert modular varieties (Hida's approach) to the arithmetic of Shimura sets and Ratner's ergodicity, the authors provide a robust framework that handles the "exceptional" CM case more effectively. This approach is less dependent on the specific geometry of the moduli space and more on the dynamical properties of unipotent flows.
3. Uniformity: The results hold for "all but finitely many" characters, which is a stronger statement than the "Zariski dense" or "infinitely many" results previously available. This uniformity is essential for applications in Iwasawa theory where one needs to control the behavior of the entire family.
4. Applications to BSD and Iwasawa Theory: The results provide concrete evidence for the Birch and Swinnerton-Dyer (BSD) conjecture in the context of CM fields and establish the necessary divisibility for the Iwasawa Main Conjecture, a central pillar of modern number theory.

In summary, this paper represents a significant breakthrough in the arithmetic of $L$-functions over CM fields, combining deep automorphic representation theory with ergodic theory to solve long-standing problems in Iwasawa theory.