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SUMMARY:James Upton (UCSD)
DTSTART:20201112T220000Z
DTEND:20201112T230000Z
DTSTAMP:20260423T005834Z
UID:UCSD_NTS/15
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/UCSD_NTS/15/
 ">Newton Slopes in $\\mathbb{Z}_p$-Towers of Curves</a>\nby James Upton (U
 CSD) as part of UCSD number theory seminar\n\nLecture held in normally APM
  7321\, currently online.\n\nAbstract\nLet $X/\\mathbb{F}_q$ be a smooth a
 ffine curve over a finite field of characteristic $p > 2$. In this talk we
  discuss the $p$-adic variation of zeta functions $Z(X_n\,s)$ in a pro-cov
 ering $X_\\infty:\\cdots \\to X_1 \\to X_0 = X$ with total Galois group $\
 \mathbb{Z}_p$. For certain ``monodromy stable'' coverings over an ordinary
  curve $X$\, we prove that the $q$-adic Newton slopes of $Z(X_n\,s)/Z(X\,s
 )$ approach a uniform distribution in the interval $[0\,1]$\, confirming a
  conjecture of Daqing Wan. We also prove a ``Riemann hypothesis'' for a fa
 mily of Galois representations associated to $X_\\infty/X$\, analogous to 
 the Riemann hypothesis for equicharacteristic  $L$-series as posed by Davi
 d Goss. This is joint work with Joe Kramer-Miller.\n\npre-talk at 1:30\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/15/
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