BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Oussama Hamza (University of Western Ontario) DTSTART:20230130T190000Z DTEND:20230130T200000Z DTSTAMP:20260423T021136Z UID:NTC/17 DESCRIPTION:Title: Fil trations\, arithmetic and explicit examples in an equivariant context\ nby Oussama Hamza (University of Western Ontario) as part of Lethbridge nu mber theory and combinatorics seminar\n\nLecture held in M1040 (Markin Hal l).\n\nAbstract\nPro-$p$ groups arise naturally in number theory as quotie nts of absolute Galois groups over number fields. These groups are quite m ysterious. During the 60's\, Koch gave a presentation of some of these quo tients. Furthermore\, around the same period\, Jennings\, Golod\, Shafarev ich and Lazard introduced two integer sequences $(a_n)$ and $(c_n)$\, clos ely related to a special filtration of a finitely generated pro-p group $G $\, called the Zassenhaus filtration. These sequences give the cardinality of $G$\, and characterize its topology. For instance\, we have the well-k nown Gocha's alternative (Golod and Shafarevich): There exists an integer $n$ such that $a_n=0$ (or $c_n$ has a polynomial growth) if and only if $G $ is a Lie group over $p$-adic fields.\n\nIn 2016\, Minac\, Rogelstad and Tan inferred an explicit relation between $a_n$ and $c_n$. Recently (2022) \, considering geometrical ideas of Filip and Stix\, Hamza got more precis e relations in an equivariant context: when the automorphism group of $G$ admits a subgroup of order a prime $q$ dividing $p-1$.\n\nIn this talk\, w e present equivariant relations inferred by Hamza (2022) and give explicit examples in an arithmetical context.\n LOCATION:https://researchseminars.org/talk/NTC/17/ END:VEVENT END:VCALENDAR