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Sign changes along geodesics of modular forms

Dubi Kelmer, Alex Kontorovich, Christopher Lutsko

TL;DR

This work addresses sign changes of real-valued automorphic functions along cuspidal geodesics on the modular surface, focusing on Eisenstein series and Maass cusp forms. The authors develop a framework that uses Littlewood’s sign-change lemma together with mean-square and $L$-function bounds to convert control of the integral averages along a geodesic into quantitative lower bounds on the number of sign changes, $K^{β}$. They obtain an unconditional sharp lower bound for Eisenstein series along a full density set of spectral parameters and show how to extend the result to cusp forms under an $L^2$-type mean bound for the associated $L$-functions (Conjecture \ref{con:L2}); the Eisenstein case can be handled unconditionally on the imaginary axis, while cusp forms require conditional input. The strategy hinges on replacing the Lindelöf hypothesis with second-moment bounds (or $L^2$-type bounds) for $L$-functions, connecting to the fourth moment of $\zeta$ and related mean-square estimates, and leveraging recent mean-square bounds from $[KKL24]$ alongside the refinements of Ki (removing Lindelöf) to push sign-change results into new regimes. Collectively, the results advance understanding of nodal structures and sign patterns for modular forms, with implications for nodal domain counts and quantum chaos on the modular surface.

Abstract

Given a compact segment, $β$, of a cuspidal geodesic on the modular surface, we study the number of sign changes of cusp forms and Eisenstein series along $β$. We prove unconditionally a sharp lower bound for Eisenstein series along a full density set of spectral parameters. Conditioned on certain moment bounds, we extend this to all spectral parameters, and prove similar theorems for cusp forms. The arguments rely in part on the authors' mean square bounds [KKL24], and on removing the assumption of the Lindelöf hypothesis from recent work of Ki [Ki23].

Sign changes along geodesics of modular forms

TL;DR

This work addresses sign changes of real-valued automorphic functions along cuspidal geodesics on the modular surface, focusing on Eisenstein series and Maass cusp forms. The authors develop a framework that uses Littlewood’s sign-change lemma together with mean-square and -function bounds to convert control of the integral averages along a geodesic into quantitative lower bounds on the number of sign changes, . They obtain an unconditional sharp lower bound for Eisenstein series along a full density set of spectral parameters and show how to extend the result to cusp forms under an -type mean bound for the associated -functions (Conjecture \ref{con:L2}); the Eisenstein case can be handled unconditionally on the imaginary axis, while cusp forms require conditional input. The strategy hinges on replacing the Lindelöf hypothesis with second-moment bounds (or -type bounds) for -functions, connecting to the fourth moment of and related mean-square estimates, and leveraging recent mean-square bounds from alongside the refinements of Ki (removing Lindelöf) to push sign-change results into new regimes. Collectively, the results advance understanding of nodal structures and sign patterns for modular forms, with implications for nodal domain counts and quantum chaos on the modular surface.

Abstract

Given a compact segment, , of a cuspidal geodesic on the modular surface, we study the number of sign changes of cusp forms and Eisenstein series along . We prove unconditionally a sharp lower bound for Eisenstein series along a full density set of spectral parameters. Conditioned on certain moment bounds, we extend this to all spectral parameters, and prove similar theorems for cusp forms. The arguments rely in part on the authors' mean square bounds [KKL24], and on removing the assumption of the Lindelöf hypothesis from recent work of Ki [Ki23].
Paper Structure (12 sections, 8 theorems, 77 equations, 1 figure)

This paper contains 12 sections, 8 theorems, 77 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof strategy
  4. Preliminaries
  5. Littlewood's sign changes lemma
  6. Preparation for Eisenstein series
  7. Preparation for Maass forms
  8. Proof for Eisenstein series
  9. Upper bound on $J$
  10. Proof of Theorem \ref{['thm:cond Eis']}
  11. Proof for cusp forms
  12. Upper bound on $J$

Key Result

Theorem 1.3

Let $\beta=i[a,b]$ be a compact segment of the imaginary line. Let $t_j\to\infty$ and suppose that there is some $p>2$ such that for all $\varepsilon>0$, Then for any $\varepsilon>0$,

Figures (1)

  • Figure 1: An example of the Eisenstein series with $t=14$. The red dots denote positive regions while the blue dots denote negative regions. Note the sign change visible along the imaginary axis.

Theorems & Definitions (15)

  • Theorem 1.3
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.10
  • Lemma 2.1
  • proof
  • Proposition 2.2: Young2018
  • Theorem 2.3: HeathBrown1979
  • Proposition 2.5: GhoshReznikovSarnak2013
  • Conjecture 2.7
  • ...and 5 more