Number Theory Seminar

We are interested in Weil sums of binomials of the form
\[
W_{k,d}(a)=\sum_{x\in k}\psi(x^d+ax),
\]
where $k$ is a finite field of characteristic $p$, $\psi$ is the canonical additive character of $k$, and the exponent $d$ is relatively prime to $|k^*|$ and nondegenerate, that is, not a power of $p$ modulo $|k^*|$.
Fix $q$ and $d$ and consider the set of values obtained as $a$ runs through $k^*$.
At least three distinct values must appear, and we discuss recent results about the case where precisely three appear.

One conjecture about three-valued Weil sums that has been attacked repeatedly in the past two decades was posed by Helleseth in 1976.
The conjecture states that it is impossible for $W_{k,d}$ to be three-valued if $k$ is obtained from the prime field ${\mathbf F}_p$ via a tower of quadratic extensions, that is, if $[k:{\mathbf F}_p]$ is a power of $2$.
We sketch the history of the proof of the conjecture when $p=2$, beginning with the foundations laid by Calderbank-McGuire-Poonen-Rubinstein (1996), the critical insight of Feng (2012), and our final step which completed the proof.
We show how an additional insight proves the conjecture when $p=3$.
Then we discuss partial results, obtained jointly with Yves Aubry and Philippe Langevin, for $p \geq 5$.