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Klein Quartic Curve and its Modularity

Paresh Singh Arora

TL;DR

This work computes the local zeta function of the Klein Quartic curve by realizing it as a quotient of the degree-$7$ Fermat curve and analyzes the induced Galois action to relate its Hasse–Weil $L$-function to three CM weight-2 newforms of level $49$ with CM by $\mathbb{Q}(\sqrt{-7})$. The method combines explicit point counting via Gauss and Jacobi sums, the Weil conjectures, and Hecke character theory, culminating in a Langlands base-change argument that identifies the Klein quartic’s $L$-function as the product of the three CM modular forms’ $L$-functions. The paper provides an explicit formula for the local factors $Z_p(KQ;T)$ for $p\neq7$ and details the Jacobi-sum structure that governs these factors, thereby establishing the modularity and analytic continuation of the Hasse–Weil $L$-function. This illustrates a concrete instance where arithmetic geometry, CM theory, and automorphic forms converge to describe the zeta data of a highly symmetric algebraic curve.

Abstract

The local Zeta function of a variety encodes important information about the variety. From the works of Weil, Deligne, Dwork, and others, many things are known about the local Zeta function of a smooth projective variety. In this article, we find the local Zeta function for the Klein Quartic curve, $x^3y+y^3z+z^3x=0$, by realizing it as a quotient of degree 7 Fermat curve. We conclude by giving the associated modular forms via Galois representations.

Klein Quartic Curve and its Modularity

TL;DR

This work computes the local zeta function of the Klein Quartic curve by realizing it as a quotient of the degree- Fermat curve and analyzes the induced Galois action to relate its Hasse–Weil -function to three CM weight-2 newforms of level with CM by . The method combines explicit point counting via Gauss and Jacobi sums, the Weil conjectures, and Hecke character theory, culminating in a Langlands base-change argument that identifies the Klein quartic’s -function as the product of the three CM modular forms’ -functions. The paper provides an explicit formula for the local factors for and details the Jacobi-sum structure that governs these factors, thereby establishing the modularity and analytic continuation of the Hasse–Weil -function. This illustrates a concrete instance where arithmetic geometry, CM theory, and automorphic forms converge to describe the zeta data of a highly symmetric algebraic curve.

Abstract

The local Zeta function of a variety encodes important information about the variety. From the works of Weil, Deligne, Dwork, and others, many things are known about the local Zeta function of a smooth projective variety. In this article, we find the local Zeta function for the Klein Quartic curve, , by realizing it as a quotient of degree 7 Fermat curve. We conclude by giving the associated modular forms via Galois representations.

Paper Structure

This paper contains 14 sections, 9 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Gauss and Jacobi sums
  4. Point counting on $x^n+y^n=1$ in $\mathbb{F}_q$
  5. Remark:
  6. Zeta function of a variety
  7. Weil conjectures
  8. Hecke Characters and $L$-functions
  9. Galois representations
  10. Klein Quartic curve
  11. Number of points modulo a prime $p$ on Klein Quartic curve over finite fields of characteristic $p$.
  12. Zeta function of the Klein Quartic curve
  13. Modular forms in the background
  14. Proof of Theorem \ref{['theorem 1']}

Key Result

Theorem 1

Let $S=$ {newforms of level 49, weight 2 and admitting complex multiplication by the field $\mathbb{Q}(\sqrt{-7})\}$, where $|S|=3$. For $f\in S$, let $\chi_f$ be its corresponding character, then for a prime $p\neq 7$, Here, characters $\chi_f$ are Dirichlet characters modulo $49$ of order $1$ or $3$.

Theorems & Definitions (12)

  • Theorem 1
  • Proposition 1
  • proof
  • Proposition 2
  • Theorem 2: Weil Conjectures
  • Proposition 3
  • Theorem 3
  • Theorem 4
  • Lemma 5
  • proof
  • ...and 2 more