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We introduce a new class of generalised quadratic forms over totally real number fields, which is rich enough to capture the arithmetic of arbitrary systems of quadrics over the rational numbers. We explore this connection through a version of the Hardy–Littlewood circle method over number fields. 

Abstract 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 Schrödinger equation. After progress by many authors, this was recently resolved (up to the endpoint) by Bourgain, whose counterexample construction for the Schrödinger 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 partial differential equations. 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 Schrödinger 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. 

Abstract This paper provides a rigorous derivation of a counterexample of Bourgain, related to a well-known question of pointwise a.e. convergence for the solution of the linear Schrödinger equation, for initial data in a Sobolev space. This counterexample combines ideas from analysis and number theory, and the present paper demonstrates how to build such counterexamples from first principles, and then optimize them.