BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Robert Lemke Oliver (Tufts University) DTSTART:20200425T160000Z DTEND:20200425T165000Z DTSTAMP:20260416T180735Z UID:FRNTD/1 DESCRIPTION:Title: An effective Chebotarev density theorem for fibers\nby Robert Lemke Oliv er (Tufts University) as part of Front Range Number Theory Day\n\n\nAbstra ct\nThe Chebotarev density theorem asserts that\, in a normal extension K/ k\, the number of\nprimes of k with norm at most x whose Frobenius element in Gal(K/k) lies in a specified\nconjugacy class C is proportional to the size of C. Lagarias and Odlyzko made this theorem\neffective by determini ng a lower bound on x for which it’s guaranteed there are many primes\np of k with Frobp ∈ C and norm at most x. Unfortunately\, this lower boun d requires x\nto be larger than any fixed power of the discriminant DK of K\, and for this reason\, in\nmany applications where the size of the prim es in question matter\, one must appeal to much\nstronger results that are only available under the assumption of the generalized Riemann\nhypothesi s.\nIn a recent breakthrough\, Pierce\, Turnage-Butterbaugh\, and Wood sho wed that nearly\nGRH-quality results are available for “almost all” ex tensions K/k in a family\, provided that\nthe Artin conjecture is known fo r every field in the family\; the dependence on the Artin\nconjecture was subsequently removed by Thorner and Zaman. However\, an obstacle in this\n work is the possible presence of intermediate normal extensions\, and for this reason it is\nalso conditional upon progress toward the so-called “ discriminant multiplicity conjecture”\nand imposes restrictions on the r amification of K/k. In forthcoming joint work with Jesse\nThorner\, we pro ve an unconditional result that allows these obstacles to be bypassed in\n many cases of interest. For example\, we show that almost all degree n Sn- extensions have\nGRH-quality bounds on the `-torsion subgroups of their cl ass groups\, and we determine a\nlower bound on the extremal order of the class number of degree n extensions that agrees\nwith GRH-quality upper bo unds.\n LOCATION:https://researchseminars.org/talk/FRNTD/1/ END:VEVENT END:VCALENDAR