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:20260416T144531Z 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 BEGIN:VEVENT SUMMARY:Jackson Morrow (Emory University) DTSTART:20200425T190000Z DTEND:20200425T195000Z DTSTAMP:20260416T144531Z UID:FRNTD/2 DESCRIPTION:Title: Al gebraic points on curves\nby Jackson Morrow (Emory University) as part of Front Range Number Theory Day\n\n\nAbstract\nTo begin\, I will introdu ce rational and degree $d>1$ points on curves\, describe how\ntheir behavi or differs\, and define what it means for a degree $d>1$ point to be\n``un expected/sporadic''. Then\, I will talk about joint with with J.~Gunther w here we prove\, under\na technical assumption\, that for each positive int eger $d>1$\, there exists a number $B_d$ such\nthat for each $g > d$\, a p ositive proportion of odd hyperelliptic curves of genus $g$ over\n$\\mathb b{Q}$ have at most $B_d$ ``unexpected'' points of degree $d$\; furthermore \, I will briefly\nsay how one may take $B_2 = 24$ and $B_3 = 114$. After this\, I will discuss joint work with\nA.~Etropolski\, M.~Derickx\, M.~van Hoeij\, and D.~Zureick-Brown where we use the explicit\ndetermination of ``unexpected/sporadic" cubic points on modular curves to classify torsion\ nsubgroups of elliptic curves over cubic number fields.\n LOCATION:https://researchseminars.org/talk/FRNTD/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Kirsten Eisenträger (Penn State University) DTSTART:20200425T201000Z DTEND:20200425T210000Z DTSTAMP:20260416T144531Z UID:FRNTD/3 DESCRIPTION:Title: Co mputing endomorphism rings of supersingular elliptic curves\nby Kirste n Eisenträger (Penn State University) as part of Front Range Number Theor y Day\n\n\nAbstract\nComputing endomorphism rings of supersingular ellipti c curves is an important\nproblem in computational number theory\, and it is also closely connected to the security of\nsome of the recently propose d isogeny-based cryptosystems. In this talk we give a new\nalgorithm for c omputing the endomorphism ring of a supersingular elliptic curve. The algo rithm\nworks by first computing two cycles in the l-isogeny graph that cre ate an order in the\nendomorphism ring of the curve E. Then we determine w hich maximal order containing this\norder is the endomorphism ring of E.\n This is joint work with Hallgren\, Leonardi\, Morrison and Park.\n LOCATION:https://researchseminars.org/talk/FRNTD/3/ END:VEVENT END:VCALENDAR