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SUMMARY:Andrew Sutherland (MIT)
DTSTART:20200703T160000Z
DTEND:20200703T163000Z
DTSTAMP:20260423T200616Z
UID:ANTS14/23
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/ANTS14/23/">
 Counting points on superelliptic curves in average polynomial time</a>\nby
  Andrew Sutherland (MIT) as part of Algorithmic Number Theory Symposium (A
 NTS XIV)\n\n\nAbstract\nWe describe the practical implementation of an ave
 rage polynomial-time algorithm for counting points on superelliptic curves
  defined over $\\Q$ that is substantially faster than previous approaches.
   Our algorithm takes as input a superelliptic curves $y^m=f(x)$ with $m\\
 ge 2$ and $f\\in \\Z[x]$ any squarefree polynomial of degree $d\\ge 3$\, a
 long with a positive integer $N$.  It can compute $\\#X(\\Fp)$ for all $p\
 \le N$ not dividing $m\\lc(f)\\disc(f)$ in time $O(md^3 N\\log^3 N\\log\\l
 og N)$.  It achieves this by computing the trace of the Cartier-Manin matr
 ix of reductions of $X$.  We can also compute the Cartier--Manin matrix it
 self\, which determines the $p$-rank of the Jacobian of $X$ and the numera
 tor of its zeta function modulo~$p$.\n\nThe slides used in the pre-recorde
 d video can be found <a href="https://math.mit.edu/~drew/ANTSXIV/Superelli
 pticPointCountingSlides.pdf">here</a>.\n\nChairs: Marco Streng and David K
 ohel\n
LOCATION:https://researchseminars.org/talk/ANTS14/23/
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