BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Daniel Katz (California State University\, Northridge) DTSTART:20230809T173000Z DTEND:20230809T183000Z DTSTAMP:20260422T053750Z UID:SFUQNTAG/92 DESCRIPTION:Title: Rationality of four-valued families of binomial Weil sums\nby Daniel Katz (California State University\, Northridge) as part of SFU NT-AG semi nar\n\nLecture held in K9509.\n\nAbstract\nConsider the Weil sum $W_{F\,d} (a)=\\sum_{x \\in F} \\psi(x^d-a x)$\, where \n$F$ is a finite field\, $\\ psi$ is the canonical additive character of \n$F$\, the coefficient $a$ is a nonzero element of $F$\, and $d$ is a \npositive integer such that $\\g cd(d\,|F|-1)=1$. This last condition makes \n$x\\mapsto x^d$ a power perm utation of $F$\, that is\, a power map that \npermutes $F$. These Weil su ms include Kloosterman sums as the special \ncase when one sets $d=|F|-2$ and deducts $1$ from the Weil sum to obtain \nthe Kloosterman sum. The We il spectrum for $F$ and $d$ records the \nvalues $W_{F\,d}(a)$ as $a$ runs through $F^*$. Weil sums in which the \nargument of the character is a b inomial of the form $x^d-a x$ are used \nto count points on varieties over finite fields\, and have multiple \napplications to cryptography and comm unications. Since one sums roots \nof unity in the complex plane to obtai n the Weil spectrum values\, these \nare always algebraic integers. A rat ional Weil spectrum is one whose \nvalues are all rational integers. If o ne sets aside degenerate cases\, \nHelleseth showed that Weil spectra have at least three distinct values. \nIt has been shown that all spectra with exactly three distinct values \nare rational. In this talk\, we show tha t\, with one exception\, Weil \nspectra with exactly four distinct values are also always rational. \nThis is joint work with Allison E.\\ Wong\n LOCATION:https://researchseminars.org/talk/SFUQNTAG/92/ END:VEVENT END:VCALENDAR