Counting rank two local systems with at most one, unipotent, monodromy

The number of rank two ℚ_l-local systems, or ℚ_l-smooth sheaves, on (X — {u}) ⊗𝔽_q 𝔽, where X is a smooth projective absolutely irreducible curve over 𝔽_q, 𝔽 an algebraic closure of 𝔽_q and u is a closed point of X, with principal unipotent monodromy at u, and fixed by Gal(𝔽/𝔽_q), is computed. It is expressed as the trace of the Frobenius on the virtual ℚ_l-smooth sheaf found in the author’s work with Deligne on the moduli stack of curves with etale 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}) ⊗_𝔽q 𝔽) invariant under the Frobenius Fr_q with principal unipotent monodromy at u, or cuspidal representations of GL(2) over the function field 𝔽 = 𝔽_q(X) of X over 𝔽_q 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 f_uΠ_v≠uχK_v, with an Iwahori component f_u = χ_Iu/\Iu\, hence also the pseudo-coefficient χI_u/\Iu\ — 2 χK_u 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 ℚ_l-local systems, or ℚ_l-smooth sheaves, on X ⊗_Fq 𝔽, fixed by Fr_q, namely ℚ_l-representations of the absolute fundamental group π(X ⊗_𝔽q 𝔽) invariant under the Frobenius, by counting the nowhere ramified cuspidal representations of GL(2) trivial at a fixed idele α of degree 1. This number is expressed as the trace of the Frobenius of a virtual ℚ_l-smooth sheaf on a moduli stack. This number is obtained on evaluating the trace formula at the characteristic function Π_v χK_V of the maximal compact subgroup, with volume normalized by |K_v| = 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.