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The arithmetic of critical values I: equicritical quartic polynomials

Francesco Naccarato

TL;DR

This work develops a rigorous framework for the arithmetic of critical values of polynomials, focusing on equicritical quartics over number fields and revealing a deep link to elliptic curves. By placing Morse quartics inside the Hurwitz-space paradigm, the author shows that the reduced Hurwitz space for quartics is a genus-0 modular curve, identified with $X_0(3)$, and constructs an explicit parametrization of equicritical pairs via a modular interpretation tied to elliptic 2-torsion. A principal outcome is a quasi-basis for the set of equicritical quartics $EC_4(K)$, parametrized by $\mathbb{P}^1(K)$, along with a complete description of exceptional points corresponding to special $j$-invariants and cusps; these results illuminate how critical values encode elliptic-curve data. The work also connects to applications in exponential sums: the equicritical quartic pairs yield Weyl-sum identities modulo $p^2$, offering concrete examples and a framework for further arithmetic and analytic investigations.

Abstract

A polynomial $f$ of degree $d$ and coefficients in an algebraically closed field $k$ defines a morphism $f:\mathbb{P}^1_k\longrightarrow\mathbb{P}^1_k$ which, if char$(k)\nmid d$, is unramified outside a finite set of points in the image: the critical values of $f$. In this work we establish a rigorous framework for the study of their arithmetic, which we carry out for $d=4$ and $k=\overline{\mathbb{Q}}$, uncovering a connection to the arithmetic of elliptic curves. Recent progress in the theory of Weyl sums has sparked some interest in finding pairs of polynomials having the same critical values for "nontrivial" reasons: building on our analysis, we provide a complete classification of such pairs in the case of quartics over number fields.

The arithmetic of critical values I: equicritical quartic polynomials

TL;DR

This work develops a rigorous framework for the arithmetic of critical values of polynomials, focusing on equicritical quartics over number fields and revealing a deep link to elliptic curves. By placing Morse quartics inside the Hurwitz-space paradigm, the author shows that the reduced Hurwitz space for quartics is a genus-0 modular curve, identified with , and constructs an explicit parametrization of equicritical pairs via a modular interpretation tied to elliptic 2-torsion. A principal outcome is a quasi-basis for the set of equicritical quartics , parametrized by , along with a complete description of exceptional points corresponding to special -invariants and cusps; these results illuminate how critical values encode elliptic-curve data. The work also connects to applications in exponential sums: the equicritical quartic pairs yield Weyl-sum identities modulo , offering concrete examples and a framework for further arithmetic and analytic investigations.

Abstract

A polynomial of degree and coefficients in an algebraically closed field defines a morphism which, if char, is unramified outside a finite set of points in the image: the critical values of . In this work we establish a rigorous framework for the study of their arithmetic, which we carry out for and , uncovering a connection to the arithmetic of elliptic curves. Recent progress in the theory of Weyl sums has sparked some interest in finding pairs of polynomials having the same critical values for "nontrivial" reasons: building on our analysis, we provide a complete classification of such pairs in the case of quartics over number fields.
Paper Structure (13 sections, 20 theorems, 35 equations)

This paper contains 13 sections, 20 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Critical values of polynomials: arithmetic and geometry
  4. The Hurwitz space of Morse polynomials
  5. Moduli spaces of $G$-covers
  6. Lifting points on Hurwitz spaces, and the case of polynomials
  7. The case of quartics: modularity
  8. The Hurwitz space of quartics as a modular curve
  9. A direct method
  10. Equicritical quartics and elliptic curves
  11. Elliptic curves with constant mod $3$ representation
  12. Elliptic fibers and cusps
  13. Application to Weyl sums mod $p^2$, and one example

Key Result

Theorem 1

Let $K$ be a number field and, for $t\in K"$, let $v=\frac{t^4(t-1)^3}{t+2}$. Then, the collection $\mathcal{E}_K=(\mathcal{c}_t=(f_t,g_t), \ t\in K'\cup\{\infty\})$ given by: and is a basis for $\text{EC}_4(K)$. Moreover, there exist quadruples of equicritical, pairwise inequivalent quartics over $K$ if and only if $\omega\in K$, in which case the above parametrization for $t\in K"$ produces th

Theorems & Definitions (49)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.3.1
  • Theorem 1
  • Remark 1.1
  • Theorem 2
  • Remark 2.1
  • Proposition 2.1
  • Remark 2.1.1
  • ...and 39 more