Mod $\ell$ non-vanishing of self-dual Hecke $L$-values over CM fields and applications (2508.19706v1)

Abstract: Let $\lambda$ be a self-dual Hecke character over a CM field $K$. Let $\mathfrak{p}$ be a prime of the maximal totally real subfield $F$ of $K$ and $\Gamma_{\mathfrak{p}}$ the Galois group of the maximal anticyclotomic ${\mathbb{Z}}{p}^{deg \mathfrak{p}}$-extension of $K$ unramified outside $\mathfrak{p}$. We prove that $$L(1,\lambda\nu)\neq 0$$ for all but finitely many finite order characters $\nu$ of $\Gamma{\mathfrak{p}}$ such that $\epsilon(\lambda\nu)=+1$. For an ordinary prime $\ell$ with respect to the CM quadratic extension $K/F$, we also determine the $\ell$-adic valuation of the normalised Hecke $L$-values $L^{{\mathrm{alg}}}(1,\lambda\nu)$. As an application, we complete Hsieh's proof of Eisenstein congruence divisibility towards the CM Iwasawa main conjecture over $K$. Our approach and results complement the prior work initiated by Hida's ideas on the arithmetic of Hilbert modular Eisenstein series, studied via mod $\ell$ analogue of the Andr\'e--Oort conjecture. The previous results established the non-vanishing only for a Zariski dense subset of characters $\nu$. Our approach is based on the arithmetic of a CM modular form on a Shimura set, studied via arithmetic of the CM field and Ratner's ergodicity of unipotent flows.

• The paper establishes uniform mod ℓ non-vanishing for central Hecke L-values in p-adic self-dual families over CM fields.
• It employs refined Fourier analysis and Ratner's ergodicity to deduce ℓ-adic valuations aligning with a local invariant μℓ(λ).
• The results complete key steps toward the CM Iwasawa main conjecture and impact the study of Katz p-adic L-functions and Selmer groups.

Mod $\ell$ Non-Vanishing of Self-Dual Hecke $L$-Values over CM Fields and Applications

Introduction and Context

This paper establishes strong non-vanishing results for central Hecke $L$-values in $p$-adic self-dual families over CM fields, with a focus on their behavior modulo a prime $\ell \neq p$. The main objects of paper are self-dual Hecke characters $\lambda$ over a CM field $K$, and the central values $L(1, \lambda\nu)$ as $\nu$ varies over finite order characters of the Galois group $\Gamma_\fp$ of the maximal anticyclotomic $\mathbb{Z}_p^{\deg \fp}$-extension of $K$ unramified outside a prime $\fp$ of the maximal totally real subfield $F$ of $K$.

The results generalize and strengthen previous work by Hida, Hsieh, and Finis, which established non-vanishing for Zariski dense subsets of characters, but left open the question of uniform non-vanishing for all but finitely many characters, especially in the case $\deg \fp > 1$. The paper also determines the $\ell$-adic valuation of normalized Hecke $L$-values and applies these results to complete Hsieh's proof of Eisenstein congruence divisibility towards the CM Iwasawa main conjecture.

Main Results

Uniform Mod $\ell$ Non-Vanishing

The central theorem asserts that for a self-dual Hecke character $\lambda$ over $K$ and a prime $\fp$ of $F$ unramified in $K$, the central $L$-values $L(1, \lambda\nu)$ are nonzero for all but finitely many finite order characters $\nu$ of $\Gamma_\fp$ with global epsilon factor $+1$. This holds independently of the degree of $\fp$, and is the first such result for $\deg \fp > 1$.

For an ordinary prime $\ell$ (with respect to $K/F$), the paper determines the $\ell$-adic valuation of the normalized algebraic part of $L$-values, showing that for all but finitely many $\nu$, the valuation equals a local invariant $\mu_\ell(\lambda)$, which is closely related to the $\ell$-part of the Tamagawa number associated to $\lambda$.

Applications to Iwasawa Theory

The non-vanishing results have direct implications for the arithmetic of CM fields, including:

• Rigidity of Katz $p$-adic $L$-functions: The zero divisor of the anticyclotomic Katz $p$-adic $L$-function does not vanish on any $\mathbb{Z}_p$-line in the anticyclotomic tower, implying strong rigidity properties.
• Completion of Eisenstein congruence divisibility: The results fill a gap in Hsieh's proof, establishing the divisibility of the Katz $p$-adic $L$-function by the characteristic power series of the relevant Selmer group, as predicted by the CM Iwasawa main conjecture.
• Implications for Mordell–Weil ranks and Tate–Shafarevich groups: The non-vanishing implies boundedness of the $\ell$-part of Tate–Shafarevich groups in anticyclotomic families, and finite generation of Mordell–Weil groups over anticyclotomic extensions.

Technical Approach

Arithmetic of Shimura Sets and Ratner's Theorem

The proof strategy departs from previous approaches based on Zariski density and instead leverages the arithmetic of CM modular forms on Shimura sets associated to totally definite quaternion algebras over $F$. The key technical tools include:

• Explicit Waldspurger formula: Relates central Rankin–Selberg $L$-values to toric periods of test vectors on Shimura sets.
• Ratner's ergodicity of unipotent flows: Used to establish equidistribution and linear independence of CM values, which is crucial for uniform non-vanishing.
• Refined Fourier analysis on ring class groups: Systematic use of the Tunnell–Saito theorem allows partitioning the character space into subfamilies indexed by roots of unity and $\mathbb{Z}_p$-quotients, and uniform non-vanishing over each subfamily.

Period Comparison and Mod $\ell$ Non-Vanishing

A central technical achievement is the $\ell$-integral comparison of quaternionic and CM periods, which enables the translation of non-vanishing results for Rankin–Selberg $L$-values to Hecke $L$-values. The period comparison is established via auxiliary Hecke characters and primes, and is shown to be independent of the choice of $\fp$.

The paper also provides sharp lower bounds for the $\ell$-adic valuation of Hecke $L$-values, and proves that these bounds are achieved for all but finitely many characters, under mild hypotheses.

Numerical and Structural Implications

• Uniformity in $\deg \fp$: The non-vanishing results hold for all but finitely many characters, regardless of the rank of the anticyclotomic extension, in contrast to previous results which only held for Zariski dense subsets.
• Explicit determination of $\ell$-adic valuations: The valuation is shown to be exactly $\mu_\ell(\lambda)$ for almost all characters, confirming predictions from the Bloch–Kato conjecture.
• Rigidity phenomena: The zero divisor of Katz $p$-adic $L$-functions is shown to be rigid, not vanishing on any $\mathbb{Z}_p$-line, which is a new phenomenon in the context of Iwasawa theory.

Theoretical and Practical Implications

The results have several important implications:

• Iwasawa Main Conjecture: The completion of the divisibility part of the CM main conjecture provides new evidence for the conjecture in the general CM field setting, beyond the case of imaginary quadratic fields.
• Selmer groups and BSD conjecture: The boundedness of the $\ell$-part of Tate–Shafarevich groups and finite generation of Mordell–Weil groups over anticyclotomic extensions have consequences for the arithmetic of CM abelian varieties and the Birch–Swinnerton-Dyer conjecture.
• Potential for generalization: The methods developed, particularly the use of Ratner's theorem and refined Fourier analysis, suggest avenues for extending non-vanishing results to non-ordinary primes and more general settings.

Future Directions

The paper outlines several directions for further research:

• Non-ordinary primes: Extending mod $\ell$ non-vanishing results to non-ordinary primes, potentially using ideas from the arithmetic of theta functions with complex multiplication.
• Primitivity of Eisenstein series: Applications to the primitivity of Hida families of theta lifts and Eisenstein series on unitary groups, with potential removal of technical hypotheses in the main conjecture.
• Rigidity for other $p$-adic $L$-functions: Generalization of rigidity phenomena to anticyclotomic Rankin–Selberg $p$-adic $L$-functions over CM fields.

Conclusion

This work establishes uniform mod $\ell$ non-vanishing of self-dual Hecke $L$-values over CM fields, determines their $\ell$-adic valuations, and applies these results to complete key steps in the proof of the CM Iwasawa main conjecture. The approach combines explicit arithmetic of Shimura sets, ergodic theory, and refined Fourier analysis, and opens new directions for the paper of $L$-values and Iwasawa theory in the context of CM fields. The results provide both theoretical insight and practical tools for further advances in the arithmetic of automorphic forms and $p$-adic $L$-functions.