Independence-of-ell for attributes of Tannakian monodromy groups of perverse sheaves on commutative algebraic groupsMPIM
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MPIM, Vivatsgasse, 7 - Lecture Hall
Max Planck Institute for Mathematics
Number Theory Lunch Seminar
In ongoing work with Chris Hall, we start with a connected,
commutative algebraic group G over a finite field Fq along with nice
perverse sheaves M1 and M2 on G with respectively ell1-adic and ell2-adic
coefficients with essentially the same Frobenius eigenvalues at stalks. M1
and M2 each generate (arithmetic and geometric) neutral tannakian
categories, associated to which are tannakian groups G1 and G2, which are
affine algebraic groups over Qell1bar and Qell2bar. Our goal is to prove
that attributes, such as the group of components, of G1 and G2 are equal to
each other. I will talk about progress that we have made towards such
results; in particular, we expect to show that the groups of connected
components of the geometric versions of the tannakian groups are
isomorphic. Our work fundamentally uses Katz's geometric ideas, originally
used to prove equidistribution theorems of exponential sums of traces of
Frobenius on complexes of ell-adic sheaves on tori over finite fields, and
work of Forey-Fresán-Kowalski, which generalized Gabber and Loeser's
tannakian formalism of perverse sheaves from tori to general connected
commutative algebraic groups over finite fields to generalize Katz's
equidistribution theorems.