Rationality of four-valued families of binomial Weil sums

Abstract: Consider the Weil sum $W_{F,d}(a)=\sum_{x \in F} \psi(x^d-a x)$, where $F$ is a finite field, $\psi$ is the canonical additive character of $F$, the coefficient $a$ is a nonzero element of $F$, and $d$ is a positive integer such that $\gcd(d,|F|-1)=1$. This last condition makes $x\mapsto x^d$ a power permutation of $F$, that is, a power map that permutes $F$. These Weil sums include Kloosterman sums as the special case when one sets $d=|F|-2$ and deducts $1$ from the Weil sum to obtain the Kloosterman sum. The Weil spectrum for $F$ and $d$ records the values $W_{F,d}(a)$ as $a$ runs through $F^*$. Weil sums in which the argument of the character is a binomial of the form $x^d-a x$ are used to count points on varieties over finite fields, and have multiple applications to cryptography and communications. Since one sums roots of unity in the complex plane to obtain the Weil spectrum values, these are always algebraic integers. A rational Weil spectrum is one whose values are all rational integers. If one sets aside degenerate cases, Helleseth showed that Weil spectra have at least three distinct values. It has been shown that all spectra with exactly three distinct values are rational. In this talk, we show that, with one exception, Weil spectra with exactly four distinct values are also always rational. This is joint work with Allison E.\ Wong

algebraic geometrynumber theory

Audience: researchers in the discipline

SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

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