Counting Rank Two Local Systems with at Most One, Unipotent, Monodromy

COUNTING RANK TWO LOCAL SYSTEMS WITH AT MOST ONE, UNIPOTENT, MONODROMY By YUVAL Z. FLICKER Abstract. The number of rank two Qℓ-local systems, or Qℓ-smooth sheaves, on (X u ) F, −{ } ⊗Fq where X is a smooth projective absolutely irreducible curve over Fq, F an algebraic closure of Fq and u is a closed point of X,withprincipalunipotentmonodromyatu,andfixedbyGal(F/Fq),is computed. It is expressed as the trace of the Frobenius on the virtual Qℓ-smooth sheaf found in the author’s work with Deligne on the moduli stack of curves withetale ´ 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 ) F) invariant under the Frobenius Frq with principal −{ } ⊗Fq unipotent monodromy at u,orcuspidalrepresentationsofGL(2) over the function field F = Fq(X) of X over Fq with Steinberg component twistedbyanunramifiedcharacteratu 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 χ ,withanIwahoricomponentf = χ / I ,hencealso u v=u Kv u Iu | u| the pseudo-coefficient χ / I ̸ 2χ of the Steinberg representation twisted by any unramified Iu | u!|− Ku character, at u.Theorem2.1recordsthetraceformulaforGL(2) over the function field F .Theproof of the trace formula of Theorem 2.1 recently appeared elsewhere. Theorem 3.1 computes, following Drinfeld, the number of Qℓ-local systems, or Qℓ-smooth sheaves, on X F,fixedbyFrq,namely ⊗Fq Qℓ-representations of the absolute fundamental group π(X F) invariant under the Frobenius, by ⊗Fq 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 Qℓ-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 K = 1. Section 5, based on a letter | v| ! 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),andnotonthedegreesofthepointsinS1. 1. Introduction. Let X (denoted by X1 in [DF13]) be a smooth projective absolutely irreducible curve over the finite field Fq of cardinality q and characteristic p.Fixaprimeℓ = p. ̸ Drinfeld [D81] computed the number of isomorphism classes of two dimensional irreducible ℓ-adic representations of the absolute fundamental group π1(X q F) which are invariant under Gal(F/Fq) in terms of the zeta function of ⊗F the curve X. To understand this, we computed in [DF13] the number isomorphism classes of n-dimensional irreducible ℓ-adic representations of the geometric fundamental group π1((X S) q F) which are Gal(F/Fq)-invariant and with a principal − ⊗F unipotent (rank n)monodromyateachpointofS.HereS is anetale ´ divisor of X, Manuscript received December 26, 2012; revised March 14, 2014. Research supported in part by a Simons Foundation grant (#267097 to Yuval Flicker). American Journal of Mathematics 137 (2015), 739–763. c 2015 by Johns Hopkins University Press. ⃝ 739 740 Y. Z . F L I C K E R consisting of N>1closedpoints,andZ n>1. In [DF13], N, S are denoted by ∋ N1, S1. When n = 2theresultof[DF13]assertsthatthenumberofisomorphism classes of the irreducible rank two Qℓ-local systems (= Qℓ-smooth sheaves) on (X S) q F,invariantundertheFrobenius,withunipotentmonodromyateach − ⊗F point of S,whereN = S > 1, is equal to the trace of the Frobenius Fr on the | | q virtual Qℓ-smooth sheaf i j+2k i j ( 1) ( 1) Q( j) c . − H ⊗ ⎛ − − H ⎞ ! $ 0 "i # "j> "k>0 # ⎝ ⎠ 1 1 j Here c is the local system of H (X S), of H (X),andQ( j)=Q( 1) , H c − H − − where Q( 1) is the Tate local system, on the moduli stack g;[M ,M ] over Z of − M 1 2 triples (X;S1,S2) consisting of a curve X of genus g and disjoint sets S1, S2 of M1, M2 > 0geometricpointsonX, S = S1 S2.RecallthatM = deg(S ) is ∪ i i v S deg(v),wheredeg(v) is the residual exponent of v.IfF = Fq(X) is the ∈ i function field of X over Fq, Fv is its completion at v,theresiduefieldFqv has ) deg(v) cardinality qv = q . The first aim of this work is to extend this result from the covering Mg;[M1,M2] to the moduli stack g;[M] over Z of pairs (X;S) consisting of a curve X of M genus g and anetale ´ divisor S of M 1geometricpointsonX.Thusweshow ≥ that the number T of the isomorphism classes of the two-dimensional irreducible ℓ-adic representations ρ of π1((X S) q F),thegeometricfundamentalgroup − ⊗F of the affine curve X S,invariantundertheFrobenius,thatisunderGal(F/Fq), − 01 with a single Jordan block (of rank two, namely 00 )monodromyateachplace of S [equivalently: the number of isomorphism classes of -smooth irreducible * + Qℓ sheaves of rank two on (X S) q F with principal unipotent local monodromy − ⊗F at each place of S and fixed by the Frobenius, or equivalently: the number of Qℓ- smooth irreducible sheaves of rank two on X S,withprincipalunipotentlocal − monodromy at each place of S,uptoisomorphismandtwistingbyacharacterof Gal(F/Fq)], is equal to the trace of the Frobenius on the virtual Qℓ-smooth sheaf displayed above, on the moduli stack g;[M] over Z. M The new case, compared with [DF13] where the case N>1(denotedbyN1 in [DF13]) is dealt with, is that of N = 1. Here N is the number of Frq-orbits in S,namelythenumberofclosedpointsinS.InthiscaseS consists of a single closed point u of degree d = du = deg(u).Ourresultinthiscase,recordedas Theorem 4.1 below, is expressed in terms of the ζ-function of the curve X.Recall 1 deg(v) that ζ(X,t)= v X (1 tv)− , tv = t , X denotes the set of closed points in ∈| | − | | h1(t) i X,andζ(X,t)=, (1 q)(1 qt) where h1(t)= 0 i 2g ait , ai Z, g = genus(X). ≤ ≤ ∈ The result is T = h −b ,where− h = a = h (1) and 1 u 1 i i )1 ) COUNTING RANK TWO LOCAL SYSTEMS 741 j j b = h1(q) [(d 1 j)/2]q +[d/2] a q 1 + 1 u − − i ⎛ − ⎞ 0 j d 3 0 i 2g 0 j<i ≤"≤ − ≤"≤ "≤ ⎝ ⎠ i 1 2j +(d 2[d/2]) a q − − a0 . − ⎛ i − ⎞ 1 i 2g 0 j<(i 1)/2 ≤"≤ ≤ "− ⎝ ⎠ The proof is based on an explicit evaluation of the trace formula of GL(2) over the function field F = Fq(X) of X over Fq at a suitable test function vfv. ⊗ Here fv is the characteristic function χKv of the maximal compact subgroup Kv = GL(2,O ) at each place v = u of F (here O denotes the ring of integers in v ̸ v the completion Fv of the field F at the place v), and a pseudo coefficient fu = χ / I 2χ =(q + 1)χ 2χ of the Steinberg representation twisted by Iu | u|− Ku u Iu − Ku any unramified character, at u.ThestatementofthetraceformulaisTheorem2.1. The proof, which follows work of Drinfeld, is developed in [F14]. Drinfeld computed the trace formula for GL(2) in order to count in [D81] the Frobenius fixed ℓ-adic local systems, namely smooth sheaves, namely representations of the fundamental group π1(X q F).WeconstructavirtualQℓ-smooth sheaf to express his ⊗F result as follows. Let u(X) denote the number of the isomorphism classes of the two- dimensional irreducible ℓ-adic representations ρ of the geometric fundamental group π1(X q F) of the curve X,invariantundertheFrobenius,thatisunder ⊗F Gal(F/Fq),[equivalently:thenumberofisomorphismclassesofQℓ-smooth irreducible sheaves of rank two on X q F fixed by the Frobenius, or equivalently: ⊗F the number of Qℓ-smooth irreducible sheaves of rank two on X up to isomorphism and twisting by a character of Gal(F/Fq)], is equal to the trace of the Frobenius i i on the virtual Qℓ-smooth sheaf ( 1) tensored with i − H i ) - i ( 1) [(i j 1)/2]Q( j) − H⊗ − − − 0 i 2g .0 j i 3 ≤"≤ # ≤"≤ − +(g i) Q( j)+(1 i)Q Q − − − − 0<j<g i / "− where is the local system of H1(X),andQ( j)=Q( 1)j,onthemodulistack H − − g over Z of the curves X of genus g.WededuceDrinfeld’soriginalresult[D81], M that u(X)=bh1 with h1 = i ai and j ) j b = a q [(i j 1)/2]+ a (g i) q +(1 g)h1 1, i − − i − − − 0 j<i 2g 0 i 2g 0 j<g i ≤"≤ ≤"≤ ≤"− in Theorem 3.1 from the trace formula of Theorem 2.1 upon evaluation at the test function f = f , f being the characteristic function of GL(2,O ) for all v,as ⊗v v v v Drinfeld did. 742 Y. Z . F L I C K E R As is clear from [F14], the trace formula is a heavy analytic machine. The point of [DF13] is that considering objects with suitable ramification at least at two points, and we take at [DF13] principal unipotent ramification, one can transfer the question to an anisotropic form of the group, where the compact quotient trace formula is easy to compute. It will of course be of much interest to compute the trace formula for GL(n) and a test function fu v=ufv, fv = χKv for all v = u ⊗⊗ ̸ ̸ and fu pseudo coefficient of the Steinberg representation twisted by any unramified character, and extend Theorem 4.1 to all n 2.

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