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.

In 1980 Carleson posed a question on the minimal regularity of an initial data function in a Sobolev space H s (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 PDE's. We develop a flexible new method to approach such problems, and prove that for any integer k ≥ 2, if a degree k generalization of the Schrödinger maximal operator is bounded fromIn dimensions n ≥ 2, for every degree k ≥ 3, this is the first result that exceeds a long-standing barrier at 1/4. Our methods are numbertheoretic, and in particular apply the Weil bound, a consequence of the truth of the Riemann Hypothesis over finite fields.

Rizzo showed that the family W t : y 2 = x 3 + tx 2 − (t + 3)x + 1, a well-known example of Washington, is such that W (W t ) = −1 for all t ∈ Z [Riz03]. In this paper, we fully determinate the non-isotrivial 1-parameter families of elliptic curves E t with coefficients (in parameter t) of small degree such that the root number is the same for all E t with t ∈ Z.(1) Chowla's conjecture predicts the behavior of λ(n), the parity of the number of prime factors of n, when n varies through the values of a polynomial:The conjecture is known to be true for deg f ≤ 1, which makes Corollary 3 unconditional relatively to this conjecture for families with at most one (linear) factor of the discriminant ∆ not dividing the c 4 -invariant.(2) The Squarefree conjecture estimates the proportion of squarefree values of a polynomial:This conjecture is known to hold when every factors of f has degree at most 3. Thus the only case where Corollary 3 is conditional relatively to this conjecture is the case where the c 4 -coefficient of a family is irreducible over Q(T ). Families with coefficients of bounded degreesLet E be a family of elliptic curves given for t ∈ Q by the Weierstrass equationwhere deg a i ≤ 2 for i = 2, 4, 6. Suppose that there are no factor of ∆ that divides the c 4 -invariant (or in other words there is no place of multiplicative reduction: the family is potentially parity-biaised ), then Bettin, David and Delauney prove in recent work [BDD18, Theorem 7 and 8] that there are essentially 6 different classes of non-isotrivial such famillies, namely:

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