The arithmetic of critical values I: equicritical quartic polynomials

A polynomial $f$ of degree $d$ and coefficients in an algebraically closed field $k$ defines a morphism $f:{\mathbb{P}}_k^1⟶{\mathbb{P}}_k^1$ 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 $k=\bar{\mathbb{Q}}$, 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.