BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Rahul Dalal (Johns Hopkins) DTSTART:20211028T210000Z DTEND:20211028T220000Z DTSTAMP:20260423T024725Z UID:UCSD_NTS/45 DESCRIPTION:Title: Counting level-1\, quaternionic automorphic representations on $G_2$ \nby Rahul Dalal (Johns Hopkins) as part of UCSD number theory seminar\n\n Lecture held in APM 7321 and online.\n\nAbstract\nQuaternionic automorphic representations are one attempt to generalize to other groups the special place holomorphic modular forms have among automorphic representations of $GL_2$. Like holomorphic modular forms\, they are defined by having their real component be one of a particularly nice class (in this case\, called quaternionic discrete series). We count quaternionic automorphic represen tations on the exceptional group $G_2$ by developing a $G_2$ version of th e classical Eichler-Selberg trace formula for holomorphic modular forms. \ n\nThere are two main technical difficulties. First\, quaternionic discret e series come in L-packets with non-quaternionic members and standard inva riant trace formula techniques cannot easily distinguish between discrete series with real component in the same L-packet. Using the more modern sta ble trace formula resolves this issue. Second\, quaternionic discrete seri es do not satisfy a technical condition of being "regular"\, so the trace formula can a priori pick up unwanted contributions from automorphic repre sentations with non-tempered components at infinity. Applying some computa tions of Mundy\, this miraculously does not happen for our specific case o f quaternionic representations on $G_2$. \n\nFinally\, we are only studyin g level-1 forms\, so we can apply some tricks of Chenevier and Taïbi to r educe the problem to counting representations on the compact form of $G_2$ and certain pairs of modular forms. This avoids involved computations on the geometric side of the trace formula.\n\n30 min pre-talk before\n LOCATION:https://researchseminars.org/talk/UCSD_NTS/45/ END:VEVENT END:VCALENDAR