  In 1980 Carleson posed a question on the minimal regularity of an initial
data function in a Sobolev space $H^s(\mathbb{R}^n)$ that implies pointwise
convergence for the solution of the linear Schr\"odinger equation. After
progress by many authors, this was recently resolved (up to the endpoint) by
Bourgain, whose counterexample construction for the Schr\"odinger maximal
operator proved a necessary condition on the regularity, and Du and Zhang, who
proved a sufficient condition. Analogues of Carleson's question remain open for
many other dispersive PDE's. We develop a flexible new method to approach such
problems, and prove that for any integer $k\geq 2$, if a degree $k$
generalization of the Schr\"odinger maximal operator is bounded from
$H^s(\mathbb{R}^n)$ to $L^1(B_n(0,1))$, then $s \geq \frac{1}{4} +
\frac{n-1}{4((k-1)n+1)}.$ In dimensions $n \geq 2$, for every degree $k \geq
3$, this is the first result that exceeds a long-standing barrier at $1/4$. Our
methods are number-theoretic, and in particular apply the Weil bound, a
consequence of the truth of the Riemann Hypothesis over finite fields.
