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SUMMARY:Counting Galois representations with Steinberg monodromy - Yuval F
 licker (Ohio)
DTSTART:20090324T143000Z
DTEND:20090324T153000Z
UID:TALK16703@talks.cam.ac.uk
CONTACT:Tom Fisher
DESCRIPTION:Let _X_ be a smooth projective absolutely irreducible curve ov
 er _*F*<sub>q</sub>_ . Let _S_ be a finite set of closed points of _X_ of 
 cardinality _N > 1_ . Put _X^S^ = X - S_ . Let _pi<sub>1</sub>(X^S^)_ be t
 he arithmetic fundamental group of the affine curve _X^S^_ . It is _Gal(F^
 S^/F)_ \, where _F_ is the function field of _X_ over _*F*<sub>q</sub>_ an
 d _F^S^_ is the maximal extension of _F_ unramified at each closed point o
 f _X^S^_ inside a fixed separable closure \\ov{F} of _F_. We compute\, in 
 terms of the zeta function of _X_\, the number of equivalence classes of i
 rreducible _n_-dimensional ell-adic representations of _pi<sub>1</sub>(X^S
 ^)_\, whose local monodromy at each point of _S_ is a single Jordan block 
 of rank _n_\, assuming _N_ is even if _n = 2_\, that _n_ is a prime and (_
 n\,q_) = _1_. This number is reduced to that of the nowhere ramified cuspi
 dal automorphic representations of the multiplicative group of a division 
 algebra of degree _n_ over _F_\, which we compute using the trace formula.
 \n
LOCATION:MR13
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