[Paper Review] Non-vanishing modulo p of central critical Rankin-Selberg L-values with anticyclotomic twists

Miljan Brakočević|arXiv (Cornell University)|Oct 28, 2010

Algebraic Geometry and Number Theory25 references6 citations

TL;DR

This paper establishes the non-vanishing modulo $p$ of central critical Rankin-Selberg $L$-values twisted by anticyclotomic characters of $\ell$-power conductor, where the $L$-function arises from a cusp form and a theta series attached to an arithmetic Hecke character of an imaginary quadratic field. The result extends prior work by Hida and Sun to higher-weight Hecke characters and relies on Zariski density of CM points on Shimura varieties and an explicit Waldspurger formula.

ABSTRACT

We prove non-vanishing modulo p, for a prime $\ell$ different from p, of central critical Rankin-Selberg L-values with anticyclotomic twists of $\ell$-power conductor. The L-function is Rankin product of a cusp form and a theta series of arithmetic Hecke character of an imaginary quadratic field. The paper is concerned with the case when the weight of Hecke character is greater than that of cusp form, so the L-value is essentially different in nature from the one in the landmark work of Vatsal and Cornut-Vatsal on the same theme.

Motivation & Objective

• To generalize the non-vanishing modulo $p$ of central critical $L$-values from level 1 forms to cusp forms of arbitrary level $N \geq 1$ and nebentypus $\psi$.
• To extend the result to arithmetic Hecke characters $\lambda$ of $\infty$-type $(k+m, -m)$ with $m \geq 0$, differing from previous work where $m=0$.
• To establish non-vanishing in the case where the weight of the Hecke character exceeds that of the cusp form, which alters the nature of the $L$-value compared to the classical setting.
• To provide a precise condition on the conductor of the anticyclotomic twist ensuring non-vanishing modulo $p$, particularly at non-split primes dividing $N$.
• To lay the foundation for computing the $\mu$-invariant of an anticyclotomic $p$-adic $L$-function in a forthcoming companion paper.

Proposed method

• Utilizes the Zariski density of CM points on modular Shimura varieties, a key tool developed by Hida in [Hi04] and [Hi10a].
• Applies an explicit Waldspurger formula for period integrals, recently computed in [Hi10b], to relate $L$-values to algebraic periods.
• Employs the theory of base change to $\mathrm{Res}_{M/\mathbb{Q}}G$ to relate the automorphic representation $\pi_{\mathbf{f}}$ to the $L$-function of interest.
• Constructs a Hecke character $\lambda = \lambda_0 \chi_0$ with $\chi_0$ of $\ell$-power conductor and sufficient ramification at non-split primes to ensure non-vanishing.
• Uses the Čebotarev density theorem to construct primes $l$ with prescribed Frobenius trace $a(l,f) \not\equiv 0 \pmod{p}$, ensuring non-vanishing modulo $p$ of the $L$-value.

Experimental results

Research questions

• RQ1 Under what conditions does the central critical $L$-value of the Rankin-Selberg product of a cusp form and a theta series fail to vanish modulo $p$?
• RQ2 How does the non-vanishing modulo $p$ of such $L$-values behave when the Hecke character has higher weight ($m > 0$) compared to the cusp form?
• RQ3 What role does the conductor depth of the anticyclotomic twist play in ensuring non-vanishing, especially at non-split primes dividing $N$?
• RQ4 Can the non-vanishing modulo $p$ be established when $\ell \mid N$, under suitable ramification and linear disjointness assumptions?
• RQ5 How do the algebraic and $p$-adic special values of these $L$-functions relate, particularly in the context of $\mu$-invariants?

Key findings

• The central critical $L$-value $L(1/2, \hat{\pi}_{\mathbf{f}} \otimes \lambda^{-})$ is non-zero modulo $p$ for a positive density of anticyclotomic twists $\chi$ of $\ell$-power conductor.
• Non-vanishing holds under the condition that the anticyclotomic twist $\chi_0$ has conductor $c_{ns} = \prod_{l \mid N_{ns}} l^{\tilde{\nu}(l)}$ with $\tilde{\nu}(l) \geq \nu(l)$, ensuring sufficient ramification at non-split primes.
• The result is valid even when $\ell \mid N$, provided the Galois representations $K = \overline{\mathbb{Q}}^{\ker \bar{\rho}}$ and $\mathbb{Q}(\mu_{\ell^\infty})$ are linearly disjoint.
• For the case where $\pi_{\mathbf{f},\ell}$ is a special representation, the non-vanishing modulo $p$ is established unconditionally, even when $\ell \mid N$.
• The proof constructs infinitely many primes $l$ such that $a(l,f) \not\equiv 0 \pmod{p}$ and $l \equiv v(r_0 q_0)^{-1} \pmod{\ell^s}$, ensuring non-vanishing of the $L$-value modulo $p$.