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- Title
COUNTING RANK TWO LOCAL SYSTEMS WITH AT MOST ONE, UNIPOTENT, MONODROMY.
- Authors
FLICKER, YUVAL Z.
- Abstract
The number of rank two ...ℓ-local systems, or (...ℓ-smooth sheaves, on (X - {u}) ... 픽, where X is a smooth projective absolutely irreducible curve over 픽q, 픽 an algebraic closure of Fq and u is a closed point of X, with principal unipotent monodromy at u, and fixed by Gal(픽/ 픽q), is computed. It is expressed as the trace of the Frobenius on the virtual (...ℓ-smooth sheaf found in the author's work with Deligne on the moduli stack of curves with étale divisors of degree M ≥ 1. This completes the work with Deligne in rank two. This number is the same as that of representations of the fundamental group π1((X - {u}) ... 픽) invariant under the Frobenius Frq with principal unipotent monodromy at u, or cuspidal representations of GL(2) over the function field F = 픽q(X) of X over 픽q with Steinberg component twisted by an unramified character at u and unramified elsewhere, trivial at the fixed idèle α of degree 1. This number is computed in Theorem 4.1 using the trace formula evaluated at fu ∏v≠u χKv, with an Iwahori component fu = χIu/|Iu|, hence also the pseudo-coefficient ... of the Steinberg representation twisted by any unramified character, at u. Theorem 2.1 records the trace formula for GL(2) over the function field F. The proof of the trace formula of Theorem 2.1 recently appeared elsewhere. Theorem 3.1 computes, following Drinfeld, the number of ...ℓ-local systems, or ...ℓ-smoofh sheaves, on ... 픽, fixed by Frq, namely (...ℓ-representations of the absolute fundamental group π(X ... 픽) invariant under the Frobenius, by counting the nowhere ramified cuspidal representations of GL(2) trivial at a fixed idèle α of degree 1. This number is expressed as the trace of the Frobenius of a virtual ...ℓ-smooth sheaf on a moduli stack. This number is obtained on evaluating the trace formula at the characteristic function ∏v χKv of the maximal compact subgroup, with volume normalized by |Kv| = 1. Section 5, based on a letter of P. Deligne to the author dated August 8, 2012, computes the number of such objects with any unipotent monodromy, principal or trivial, in our rank two case. Surprisingly, this number depends only on X and deg(S), and not on the degrees of the points in S1.
- Subjects
MONODROMY groups; ALGEBRA; FROBENIUS algebras; MATHEMATICAL research; GROUP theory
- Publication
American Journal of Mathematics, 2015, Vol 137, Issue 3, p739
- ISSN
0002-9327
- Publication type
Article
- DOI
10.1353/ajm.2015.0016