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SUMMARY:Arul Shankar (University of Toronto)
DTSTART:20201202T200000Z
DTEND:20201202T210000Z
DTSTAMP:20260422T205533Z
UID:HarvardNT/9
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/HarvardNT/9/
 ">The 2-torsion subgroups of the class groups in families of cubic fields<
 /a>\nby Arul Shankar (University of Toronto) as part of Harvard number the
 ory seminar\n\n\nAbstract\nThe Cohen--Lenstra--Martinet conjectures have b
 een verified in\nonly two cases. Davenport--Heilbronn compute the average 
 size of the\n3-torsion subgroups in the class group of quadratic fields an
 d Bhargava\ncomputes the average size of the 2-torsion subgroups in the cl
 ass groups of\ncubic fields. The values computed in the above two results 
 are remarkably\nstable. In particular\, work of Bhargava--Varma shows that
  they do not\nchange if one instead averages over the family of quadratic 
 or cubic fields\nsatisfying any finite set of splitting conditions.\n\nHow
 ever for certain "thin" families of cubic fields\, namely\, families of\nm
 onogenic and n-monogenic cubic fields\, the story is very different. In\nt
 his talk\, we will determine the average size of the 2-torsion subgroups o
 f\nthe class groups of fields in these thin families. Surprisingly\, these
 \nvalues differ from the Cohen--Lenstra--Martinet predictions! We will als
 o\nprovide an explanation for this difference in terms of the Tamagawa num
 bers\nof naturally arising reductive groups. This is joint work with Manju
 l\nBhargava and Jon Hanke.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/9/
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