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SUMMARY:Arul Shankar (University of Toronto)
DTSTART:20210527T120000Z
DTEND:20210527T130000Z
DTSTAMP:20260423T005801Z
UID:AGNTISTA/38
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/AGNTISTA/38/
 ">Nonvanishing at the critical point of the Dedekind zeta functions of cub
 ic $S_3$-fields</a>\nby Arul Shankar (University of Toronto) as part of Al
 gebraic Geometry and Number Theory seminar - ISTA\n\n\nAbstract\nLet $K$ b
 e a number field\, and denote the Dedekind zeta function of $K$ by $\\zeta
 _K(s)$. A classical question in number theory is: Can this zeta function v
 anish at the critical point $s=1/2$?  In successive works\, Armitage\, and
  then Frohlich\, gave examples of number fields  which satisfy $\\zeta_K(s
 )=0$. Conversely\, it is believed that certain conditions on $K$ can guara
 ntee the nonvanishing of $\\zeta_K(s)$ at the critical point. For example\
 , it is believed that $\\zeta_K(s)$ is never $0$ when $K$ is an $S_n$-numb
 er field for any $n\\geq 1$.\nWhen $n=1$\, $\\zeta_K(s)$ is simply the Rie
 mann zeta function\, and Riemann himself established the non vanishing of 
 $\\zeta(1/2)$.\nWhen $n=2$\, there has been amazing progress towards under
 standing the statistics of $\\zeta_K(1/2)$. Jutila first proved that infin
 itely many quadratic fields $K$ satisfy $\\zeta_K(1/2)\\neq 0$\, and Sound
 ararajan establishes that this is in fact true for at least $87.5\\%$ of f
 ields $K$ in families of quadratic fields. \nIn this talk\, I will discuss
  joint work with Anders Södergren and Nicolas Templier\, in which we stud
 y the statistics of $\\zeta_K(1/2)$ in families of $S_3$-cubic fields. In 
 particular\, we will prove that the Dedekind zeta functions of infinitely 
 many such fields have nonvanishing critical value.\n
LOCATION:https://researchseminars.org/talk/AGNTISTA/38/
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