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Let $T_t^{P_2}f(x)$ denote the solution to the linear Schr\"odinger equationat time $t$, with initial value function $f$, where $P_2 (\xi) = |\xi|^2$. In1980, Carleson asked for the minimal regularity of $f$ that is required for thepointwise a.e. convergence of $T_t^{P_2} f(x)$ to $f(x)$ as $t \rightarrow 0.$This was recently resolved by work of Bourgain, and Du and Zhang. This paperconsiders more general dispersive equations, and constructs counterexamples topointwise a.e. convergence for a new class of real polynomial symbols $P$ ofarbitrary degree, motivated by a broad question: what occurs for symbols lyingin a generic class? We construct the counterexamples using number-theoreticmethods, in particular the Weil bound for exponential sums, and the theory ofDwork-regular forms. This is the first case in which counterexamples areconstructed for indecomposable forms, moving beyond special regimes where $P$has some diagonal structure.

Generalizations of the Schr\"odinger maximal operator: building arithmetic counterexamples