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One-level densities in families of Grössencharakters associated to CM elliptic curves

Chantal David, Lucile Devin, Ezra Waxman

TL;DR

The paper investigates the low-lying zeros of L-functions attached to Grössencharakters ξ_{d,k} arising from CM elliptic curves E_d, revealing that the natural family F_d splits into symplectic and orthogonal subfamilies according to k mod 8. By computing the one-level density with explicit lower-order terms in powers of 1/log(KM_{d,α}), the authors classify the symmetry type of each subfamily as symplectic (even α) or orthogonal (odd α), and they provide detailed expressions for the non-universal lower-order constants. The work also delivers refined expressions for inert-prime contributions, extends density results to larger test-function supports, and yields conditional non-vanishing results: a solid 75% non-vanishing rate in even subfamilies, a 25% bound in certain odd subfamilies, and precise behavior when central values vanish with prescribed multiplicities. These findings illuminate how root-number fluctuations and arithmetic features of Grössencharakters influence local-zero statistics beyond the leading Katz–Sarnak term, with implications for refined universality in families of L-functions with CM structure.

Abstract

We study the low-lying zeros of a family of $L$-functions attached to the CM elliptic curve $E_d \;:\; y^2 = x^3 - dx$, for each odd and square-free integer $d$. Specifically, upon writing the $L$-function of $E_d$ as $L(s-\frac12, ξ_d)$ for the appropriate Grössencharakter $ξ_d$ of conductor $\mathfrak{f}_d$, we consider the collection $\mathcal{F}_d$ of $L$-functions attached to $ξ_{d,k}$, $k \geq 1$, where for each integer $k$, $ξ_{d, k}$ denotes the primitive character inducing $ξ_d^k$. We observe that $25\%$ of the $L$-functions in $\mathcal{F}_d$ have negative root number. $\mathcal{F}_d$ is thus not one of the essentially homogeneous families of the Universality Conjecture of Sarnak, Shin and Templier, with unitary, symplectic or orthogonal (odd or even) symmetry type. By computing the one-level density in the family of $L$-functions in $\mathcal{F}_{d}$ with conductor at most $K^2 \mathrm N (\mathfrak{f}_d)$, we find that $\mathcal{F}_d$ naturally decomposes into subfamilies: more specifically, a collection of symplectic ($L(s, ξ_{d,k})$ for $k \equiv α\bmod 8$, $α$ even) and orthogonal ($L(s, ξ_{d,k})$ for $k \equiv α\bmod 8$, $α$ odd) subfamilies. For each such subfamily, we moreover compute explicit lower order terms in decreasing powers of $\log (K^2 \mathrm N(\mathfrak{f}_d))$.

One-level densities in families of Grössencharakters associated to CM elliptic curves

TL;DR

The paper investigates the low-lying zeros of L-functions attached to Grössencharakters ξ_{d,k} arising from CM elliptic curves E_d, revealing that the natural family F_d splits into symplectic and orthogonal subfamilies according to k mod 8. By computing the one-level density with explicit lower-order terms in powers of 1/log(KM_{d,α}), the authors classify the symmetry type of each subfamily as symplectic (even α) or orthogonal (odd α), and they provide detailed expressions for the non-universal lower-order constants. The work also delivers refined expressions for inert-prime contributions, extends density results to larger test-function supports, and yields conditional non-vanishing results: a solid 75% non-vanishing rate in even subfamilies, a 25% bound in certain odd subfamilies, and precise behavior when central values vanish with prescribed multiplicities. These findings illuminate how root-number fluctuations and arithmetic features of Grössencharakters influence local-zero statistics beyond the leading Katz–Sarnak term, with implications for refined universality in families of L-functions with CM structure.

Abstract

We study the low-lying zeros of a family of -functions attached to the CM elliptic curve , for each odd and square-free integer . Specifically, upon writing the -function of as for the appropriate Grössencharakter of conductor , we consider the collection of -functions attached to , , where for each integer , denotes the primitive character inducing . We observe that of the -functions in have negative root number. is thus not one of the essentially homogeneous families of the Universality Conjecture of Sarnak, Shin and Templier, with unitary, symplectic or orthogonal (odd or even) symmetry type. By computing the one-level density in the family of -functions in with conductor at most , we find that naturally decomposes into subfamilies: more specifically, a collection of symplectic ( for , even) and orthogonal ( for , odd) subfamilies. For each such subfamily, we moreover compute explicit lower order terms in decreasing powers of .
Paper Structure (14 sections, 18 theorems, 278 equations, 2 figures, 3 tables)

This paper contains 14 sections, 18 theorems, 278 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Quartic Residue Symbol
  4. $L$-functions of Elliptic Curves
  5. Grössencharakters
  6. Grössencharakters
  7. Computing Conductors
  8. Computing Root Numbers
  9. Computing $\mathcal{D}(\phi, \xi_{d,k})$
  10. An Alternative Expression for $c_{j,\textnormal{inert}}(k)$
  11. Computing $\mathcal{D}(K; \phi, \mathcal{F}^{\alpha}_d)$
  12. Proof of Theorem \ref{['thm-main']}
  13. Non-vanishing results
  14. Evaluating the $c_j$ constants numerically

Key Result

Theorem 1.1

Let $d$ be an odd square-free integer, $\phi$ be an even Schwartz function with $\mathrm{supp}(\widehat{\phi}) \subset (-1, 1)$ and fix $J \in \mathbb{N}$ and $\alpha \in \{ 1, 2, \dots, 8 \}$. Let $M_{d, \alpha} := {\mathrm N}( \mathfrak{f}_{d,\alpha})^\frac{1}{2}$, where $\mathfrak{f}_{d,\alpha}$ and when $K \rightarrow \infty$ and $\alpha$ is odd, where the $C_{m}(d,\alpha, \phi)$ are given b

Figures (2)

  • Figure 2: Values of $(f'/f)^{(j)}(1)$, $(L'/L)^{(j)}(1,\chi_{4})$, and $B_{j}$, for $j=0$ and $j=2$.
  • Figure 3: Numerical approximation of $c_{0}$ and $c_{2}$.

Theorems & Definitions (37)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Proposition 2.1: Law of Quartic Reciprocity
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • ...and 27 more