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Simple zeros of $\mathrm{GL}(2)$ $L$-functions

Alexandre de Faveri

TL;DR

The paper advances the quantitative understanding of simple zeros for GL(2) L-functions by removing the level-parity restriction and proving a power-saving lower bound $N_f^s(T)=\Omega(T^\delta)$ for any $\delta<2/27$ when the level is nontrivial. It introduces a unified pole-detection mechanism for additive twists to produce poles of an auxiliary object $H_{f,\alpha}$ inside the critical strip, and then leverages zero-density results for twists to preclude pole cancellations, thereby obtaining the power bound; for level 1 it achieves an even stronger result, $N_f^s(T)=\Omega(T^{1/5-\varepsilon})$, via a sixth-moment input. The work thereby extends the scope of Booker–Milinovich–Ng-type arguments to general level and refines classical bounds in the level-1 case, with potential implications for understanding the distribution of simple zeros and their role in automorphic $L$-functions. The methods combine inverse Mellin transforms, additive/multiplicative twists, and zero-density techniques to connect analytic properties of twists with arithmetic information about zeros, offering a framework that could extend to Maass forms or higher rank cases.

Abstract

Let $f \in S_k(Γ_1(N))$ be a primitive holomorphic form of arbitrary weight $k$ and level $N$. We show that the completed $L$-function of $f$ has $Ω\left(T^δ\right)$ simple zeros with imaginary part in $\left[-T, T\right]$, for any $δ< \frac{2}{27}$. This is the first power bound in this problem for $f$ of non-trivial level, where previously the best results were $Ω(\log\log\log{T})$ for $N$ odd, due to Booker, Milinovich, and Ng, and infinitely many simple zeros for $N$ even, due to Booker. In addition, for $f$ of trivial level ($N=1$), we also improve an old result of Conrey and Ghosh on the number of simple zeros.

Simple zeros of $\mathrm{GL}(2)$ $L$-functions

TL;DR

The paper advances the quantitative understanding of simple zeros for GL(2) L-functions by removing the level-parity restriction and proving a power-saving lower bound for any when the level is nontrivial. It introduces a unified pole-detection mechanism for additive twists to produce poles of an auxiliary object inside the critical strip, and then leverages zero-density results for twists to preclude pole cancellations, thereby obtaining the power bound; for level 1 it achieves an even stronger result, , via a sixth-moment input. The work thereby extends the scope of Booker–Milinovich–Ng-type arguments to general level and refines classical bounds in the level-1 case, with potential implications for understanding the distribution of simple zeros and their role in automorphic -functions. The methods combine inverse Mellin transforms, additive/multiplicative twists, and zero-density techniques to connect analytic properties of twists with arithmetic information about zeros, offering a framework that could extend to Maass forms or higher rank cases.

Abstract

Let be a primitive holomorphic form of arbitrary weight and level . We show that the completed -function of has simple zeros with imaginary part in , for any . This is the first power bound in this problem for of non-trivial level, where previously the best results were for odd, due to Booker, Milinovich, and Ng, and infinitely many simple zeros for even, due to Booker. In addition, for of trivial level (), we also improve an old result of Conrey and Ghosh on the number of simple zeros.

Paper Structure

This paper contains 19 sections, 22 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. Discussion
  3. Results
  4. Sketch of the argument
  5. The setup
  6. Definitions and background
  7. Functional equations
  8. The detection mechanism for simple zeros
  9. Existence of poles of H f, alpha
  10. Outline of the method
  11. Implementation
  12. Preliminary results
  13. Producing poles
  14. Location of poles of H f, alpha
  15. From poles of H f, alpha to simple zeros of Lambda f
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $f \in S_k(\Gamma_0(N), \xi)$ be a primitive holomorphic modular form of arbitrary weight $k$, level $N$, and nebentypus $\xi$. Then for any $\delta < \frac{2}{27}$.

Theorems & Definitions (48)

  • Theorem 1.1: Power bound for arbitrary level
  • Theorem 1.2: Improved exponent for full level
  • Proposition 2.1: Functional equation for $\Delta_{f, a, q}$ BMN19
  • Proposition 2.2: Detecting poles of $\Delta_{f, a, q}$ via further additive twists BMN19
  • Proposition 3.1: Ruling out complete cancellation of poles
  • Remark 3.2
  • Proposition 3.3: Existence of poles with large real part
  • proof
  • Lemma 3.4: Inverse Mellin transform computations
  • proof
  • ...and 38 more