Rigid local systems with monodromy group the Conway group Co3

Journal of Number Theory 206 (2020) 1–23 Contents lists available at ScienceDirect Journal of Number Theory www.elsevier.com/locate/jnt JNT Prime Rigid local systems with monodromy group the Conway group Co3✩ Nicholas M. Katz a, Antonio Rojas-León b,∗, Pham Huu Tiep c aDepartment of Mathematics, Princeton University, Princeton, NJ 08544, USA bDepartamento de Álgebra, Universidad de Sevilla, c/Tarfia s/n, 41012 Sevilla, Spain cDepartment of Mathematics, Rutgers University, Piscataway, NJ 08854, USA a r t i c l e i n f o a b s t r a c t Article history: Received 23 October 2018 Received in revised form 4 June 2019 Accepted 7 June 2019 Available online 30 July 2019 Communicated by A. Pal Keywords: Monodromy Exponential sums Representation theory Sporadic groups We first develop some basic facts about certain sorts of rigid local systems on the affine line in characteristic p >0. We then apply them to exhibit a number of rigid local systems of rank 23 on the affine line in characteristic p =3whose arithmetic and geometric monodromy groups are the Conway group Co3 in its orthogonal irreducible representation of degree 23. © 2019 Elsevier Inc. All rights reserved. Contents 0. Introduction ....................................................... 2 1. The basic set up, and general results ...................................... 2 2. Criteria for finite monodromy ........................................... 6 3. Theorems of finite monodromy........................................... 18 4. Determination of the monodromy groups.................................... 20 ✩The second author was partially supported by MTM2016-75027-P (Ministerio de Economía y Competitividad) and FEDER. The third author gratefully acknowledges the support of the NSF (grant DMS-1840702) and the Joshua Barlaz Chair in Mathematics. *Corresponding author. E-mail addresses: [email protected] (N.M. Katz), aro[email protected] (A. Rojas-León), [email protected] (P.H. Tiep). https://doi.org/10.1016/j.jnt.2019.06.016 0022-314X/© 2019 Elsevier Inc. All rights reserved. 2N.M. Katz et al. / Journal of Number Theory 206 (2020) 1–23 References ............................................................. 23 0. Introduction In the first section, we recall the general set up, and some basic results. In the second section, we generalize the criteria of [R-L, Thm. 1] and [Ka-RLSA, 5.1] for finite (arithmetic and geometric) monodromy to more general local systems. In the third section, we apply these criteria to show that certain local systems have finite (arithmetic and geometric) monodromy groups. In the fourth section, we show that the finite monodromy groups in question are the Conway group Co3in its 23-dimensional irreducible orthogonal representation. 1. The basic set up, and general results We fix a prime number p, a prime number  =p, and a nontrivial Q ×-valued additive character ψof Fp. For k/Fpa finite extension, we denote by ψkthe nontrivial additive character of kgiven by ψk:= ψ◦Tracek/Fp. In perhaps more down to earth terms, we fix a nontrivial Q(μp)×-valued additive character ψ, of Fp, and a field embedding of Q(μp) into Qfor some  =p. Given an integer D≥3which is prime to p, a finite extension k/Fp, and a polynomial f(x) ∈k[x]of degree D, we form the local system Fp,D,f,1on A1/k whose trace function is given as follows: for L/k a finite extension, at L-valued points t ∈A1(L) =L, is given by t→− x∈L ψL(f(x)+tx). This is a geometrically irreducible rigid local system on A1, being the Fourier Transform of the rank one local system Lψ(f(x)). It has rank D−1, it is totally wild at ∞, and each of its D−1I(∞)-slopes is D/(D−1). It is pure of weight one. [When f(x)is xD, it is the local system F(Fp, ψ, 1, D)of [Ka-RLSA].] Suppose in addition we are given anontrivial character χof k×. For L/k a finite extension, we obtain a nontrivial character χLof L×by defining χL:= χ ◦NormL/k. We then form the local system Fp,D,f,χ on A1/k whose trace function is given as follows: for L/k a finite extension, at L-valued points t ∈A1(L) =L, is given by t→− x∈L ψL(f(x)+tx)χL(x). This too is a geometrically irreducible rigid local system, being the Fourier Transform of the rank one local system Lψ(f(x)) ⊗Lχ(x). It has rank D. Its I(∞) representation is the direct sum of the tame character Lχ(x)with a totally wild representation of rank D−1, N.M. Katz et al. / Journal of Number Theory 206 (2020) 1–23 3 each of whose D−1I(∞)-slopes is D/(D−1) [Lau, Thm. 2.4.3]. It is pure of weight one. [When f(x)is xD, it is the local system F(Fp, ψ, χ, D)of [Ka-RLSA].] Lemma 1.1. (Primitivity Lemma) We have the following results. (i) If both Dand D−1are prime to p, the local system Fp,D,f,1is not geometrically induced, i.e., there is no triple (U, π, H)consisting of a connected smooth curve U/k, a finite etale map f:U→A1/k of degree d ≥2, and a local system Hon Usuch that there exists an isomorphism of πHwith (the pullback to A1/k of) Fp,D,f,1. (ii) When Dis prime to p, then for any nontrivial χ, the local system Fp,D,f,χ is not geometrically induced. Proof. In case (i), our local system is an Airy sheaf (the Fourier transform of a nonconstant lisse sheaf on A1of rank one). By a result [Such, 11.1] if an Airy sheaf is induced, it is Artin-Schreier induced, so has rank divisible by p. For (ii), we argue as follows. If such a triple exists, then we have an equality of Euler characteristics EP(U, H)=EP(A1/k, πH)=EP(A1/k,Fp,D,f,χ). Denote by Xthe complete nonsingular model of U, and by gXits genus. Then πextends to a finite flat map of Xto P1, and the Euler-Poincaré formula gives EP(U, H) = rank(H)(2 −2gX−#(π−1(∞))) − w∈π−1−(∞) Swanw(H), EP(A1/k,Fp,D,f,χ) = rank(Fp,D,f,χ)−Swan∞(Fp,D,f,χ)=D−D=0. So we have the equality 0 = rank(H)(2 −2gX−#(π−1(∞))) − w∈π−1(∞) Swanw(H). We first bound the genus gX. We must have gX≤0, otherwise the factor 2 −2gX− #(π−1(∞)) is ≤−1, and the right hand side is strictly negative. Thus gX=0, and we have 0 = rank(H)(2 −#(π−1(∞))) − w∈π−1(∞) Swanw(H). If #(π−1(∞)) =1, then Uis P1\(one point) ∼ =A1, and so πis a finite etale map of A1to itself of degree >1. But any such map has degree divisible by p. Indeed, when the map is given by the polynomial F(x), the hypothesis is that for every t ∈k, the two equations F(x) =t, F(x) =0have no common solution. If Fhad a zero, say a, then 4N.M. Katz et al. / Journal of Number Theory 206 (2020) 1–23 awould be a solution of F(x) =F(a), F(x) =0. Thus Fhas no zeroes, so is some nonzero constant A, and hence F(x)is of the form G(x)p+Ax. We cannot have #(π−1(∞)) ≥3, otherwise the factor 2 −#(π−1(∞)) is strictly negative, and the right side is then strictly negative. It remains to treat the case when #(π−1(∞)) =2(and gX=0). Throwing the two points to 0and ∞, we have a finite etale map π:Gm→A1. The equality of EP’s now gives 0=Swan0(H)+Swan∞(H). Thus His lisse on Gmand everywhere tame, so a successive extension of lisse, everywhere tame sheaves of rank one. But πHis irreducible, so Hmust itself be irreducible, hence of rank one, and either Qor an Lρ. [It cannot be Q, because πQis not irreducible when πhas degree >1; by adjunction πQcontains Q.] Now consider the maps induced by πon punctured formal neighborhoods π(0) : Gm(0)→A1(∞),π(∞):Gm(∞)→A1(∞). The I(∞)-representation of Fp,D,f,χ is then the direct sum π(0)Lρ⊕π(∞)Lρ. Denote by d0and d∞their degrees. For any tame LΛ, we have π(0)LΛ∼ =LΛd0,π(∞)LΛ∼ =LΛd∞. Since the tame character group is divisible, there exist Λ0with Λd0 0=ρ(in fact, as many as the prime to ppart n0of d0=n0×(a power of p)), and there exist Λ∞with Λd∞ ∞=ρ (in fact, as many as the prime to ppart n∞of d∞=n∞×(a power of p)). Thus if Fp,D,f,χ were induced, its I(∞) representation would contain at least two tame characters. 2 Let kbe a finite extension of Fp,  =p, U/k a smooth, geometrically connected k-scheme of relative dimension ≥0, and Ga Qlocal system on Uof rank d ≥1. Viewing Gas a representation of π1(U), say ρG:π1(U)→GLd(Q), we get its arithmetic monodromy group Garith, defined to be the Zariski closure of the image of π1(U). Inside π1(U)we have its normal subgroup πgeom 1(U) := π1(U⊗kk). N.M. Katz et al. / Journal of Number Theory 206 (2020) 1–23 5 The Zariski closure of the image of πgeom 1(U)is the geometric monodromy group Ggeom. Thus we have Ggeom Garith ⊂GLd(Q). When we apply this general machine to the local system Fp,D,f,1on A1/k, we get its Ggeom Garith ⊂GLD−1(Q). Similarly, for any nontrivial χ, when we apply the general machine to the local system Fp,D,f,χ on A1/k, we get its Ggeom Garith ⊂GLD(Q). Lemma 1.2. (p-subgroups of Ggeom) Suppose both D≥3and D−1are prime to p. Denote by fthe multiplicative order of pin (Z/(D−1)Z)×, so that Fpfis the extension Fp(μD−1)of Fpobtained by adjoining the D−1roots of unity. Then for the particular local systems Fp,D,xD,1or Fp,D,xD,χ with any nontrivial χ, the image in Ggeom of the wild inertia group P(∞)is isomorphic to (the additive group of) Fpf. Proof. In all cases, the local system, restricted to Gm, descends through the D’th power map. For Fp,D,xD,1, the descent is given explicitly in terms of trace functions as t→− x∈k ψk(xD/t +x). For Fp,D,xD,χ, we must first choose a character Λwith ΛD=χ. Then the descent is given explicitly interms of trace function as t→− x∈L× ΛL(t)ψk(xD/t +x)χL(x). In fancier terms, the first descent is to a Kloosterman sheaf of rank D−1, the second is to a hypergeometric sheaf of type (D, 1), cf. [Ka-RLSA, 2.1]. In all cases, the wild part Wof the I(∞) representation has rank D−1and all slopes 1/(D−1). Because D−1 ≥2, one knows [Ka-ESDE, 8.6.3] that Wis unique up to tensoring with a tame character and performing a multiplicative translation. Thus the underlying P(∞)-representation is unique up to a multiplicative translation, which does not change its image in Ggeom. Because D−1is prime to p, we obtain one such Wby forming the direct image by D−1power W:= [D−1]Lψ(x). Because D−1is prime to p, the image of P(∞)does not change if we pass to the pullback 6N.M. Katz et al. / Journal of Number Theory 206 (2020) 1–23 [D−1]W=[D−1][D−1]Lψ(x)∼ = ζ∈μD−1Lψ(ζx). In other words, the image of P(∞)is the abelian group whose character group consists of all monomials ⊗ζ∈μD−1L⊗nζ ψ(ζx)=Lψ(ζ∈μD−1nζζx), as each nζruns over Z/pZ. This character group is thus the subring Fp[μD−1] ⊂Fpf consisting of all Fp-linear combinations of elements of μD−1. But this subring, being a finite integral domain, is itself a field. It lies in Fpf, and contains μD−1, so it is Fpf. One knows that (by the trace), Fpfis its own Pontrayagin dual. 2 2. Criteria for finite monodromy We first recall the basic underlying result, cf. [Ka-ESDE, 8.14]. [It is stated there for a local system on an open curve, but the “on a curve” hypothesis never enters the proof. Also, the exact rule of the hypothesis of geometric irreducibility is less clear than it might be, cf. Remark 2.2 below.] Let kbe a finite extension of Fp,  =p, U/k a smooth, geometrically connected k-scheme of relative dimension ≥0, and Ga Qlocal system on Uof rank d ≥1. We have its geometric and arithmetic monodromy groups Ggeom Garith ⊂GLd(Q). Proposition 2.1. Suppose we have (k, , U, G)as above. Suppose further that Gis pure of weight zero for all embeddings of Qinto C. Consider the following four conditions. (a) Garith is finite. (b) All traces of Gare algebraic integers. More precisely, for every finite extension L/k, and for every point u ∈U(L), Trace(FrobL,u|G)is an algebraic integer. (c) Ggeom is finite. (d) det(G)is arithmetically of finite order. Then we have the implications (a) =⇒(b) =⇒(c),(b) =⇒(d), and if Gis geometrically irreducible, we have (a) ⇐⇒ (b) ⇐⇒ (c). Proof. The implications (a) =⇒(b) and (a) =⇒(c) are both obvious. We next show that (b) =⇒(c) and (b) =⇒(d). If (b) holds, then all the eigenvalues of each Frobenius are algebraic integers which, by purity, have absolute value 1at all N.M. Katz et al. / Journal of Number Theory 206 (2020) 1–23 7 archimedean places, hence are roots of unity. Because Gis realizable over some finite extension Eλof Q, each of these roots of unity lies in a extension of Eλof degree at most the rank of G. As there are only finitely many such extensions inside Q, all eigenvalues are roots of unity in a fixed finite extension of Q, so are all N’th roots of unity for some N. Applying this same argument to the rank one local system det(G), we see that det(G)⊗Nis trivial, i.e. we see that (b) =⇒(d). By Chebotarev and Zariski density, every γ∈Garith has γNunipotent. In particular, every element in Ggeom, and hence every element in the identity component G0 geom has N’th power unipotent. By Deligne [De-Weil II, 1.3.8 and 3.4.1 (iii)], G0 geom is a semisimple algebraic group. Looking at elements of a maximal torus, we see that G0 geom has rank 0, hence G0 geom is trivial and thus Ggeom is finite. When Gis geometrically irreducible, the implication (c) =⇒(a), using (d), is proven in [Ka-ESDE, 8.14.3.1]. 2 Remark 2.2. Here is an example to show that geometric irreducibility is needed to prove that (b) =⇒(a). Take any U/k, and take on it the pullback from Spec(k)of the geometrically constant local system βdeg for βthe upper unipotent matrix 11 01 . Then all Frobenius eigenvalues are 1, Ggeom is trivial, but Garith is the upper unipotent subgroup 1 01 of SL(2). On the other hand, suppose (b) holds. If we pass from Gviewed as a representation of πarith 1, to its semisimplification Gss, (which has the same trace function as G), then Garith,Gss is reductive. Then the fact that every element in this group has N’power unipotent shows that the identity component G0 arith,Gss is trivial (look at a maximal torus), and hence Garith,Gss is finite. We now define the sort of multi-parameter local systems it will be convenient to work with. We fix an integer D≥3 prime to p, and a sequence of integers of length r≥1, 1=d1<d 2<...<d r<D, each of which is itself prime to p. [Because of Artin-Schreier reduction, requiring the di to be prime to pis no loss of generality.] We form the local system F(p, D, d1, ..., dr, 1) on Ar−1whose trace function is as follows: for K/Fpa finite extension, and (t1, ..., tr) ∈ Ar(K), the trace function is (t1,...,t r)→− x∈K ψK(xD+ i tixdi). When pis odd, we also form the local system F(p, D, d1, ..., dr, χ2)on Ar−1whose trace function is as follows: for K/Fpa finite extension, and (t1, ..., tr) ∈Ar(K), the trace function is 8N.M. Katz et al. / Journal of Number Theory 206 (2020) 1–23 (t1,...,t r)→−  x∈K× ψK(xD+ i tixdi)χ2,K(x). Because d1=1, these local systems are geometrically irreducible (indeed, for r=1 these are F(p, D, xD, 1)and F(p, D, xD, χ2)). If r≥2, their pullbacks to A1by freezing t2=... =tr=0are F(p, D, xD, 1)and F(p, D, xD, χ2). They are pure of weight one, for all embeddings of Qinto C. Moreover, their traces lie in Z[ζp]. For each of them, we now fix a version of a half Tate twist as follows: we choose an algebraic integer αwhich is (some root of unity) √p, typically some fourth root of p2. Using our chosen α, we then define G(p, D, d1,...,d r,1):=F(p, D, d1,...,d r,1)⊗(1/α)deg, and, when pis odd, G(p, D, d1,...,d r,χ 2):=F(p, D, d1,...,d r,χ 2)⊗(1/α)deg. The local systems Gp,D,f,1and, when pis odd, Gp,D,f,χ2, are pure of weight zero. Their determinants are arithmetically of finite order, cf. the proof of [Ka-RL, Lemma 1.1]. For later use, we record the following facts about autoduality. Lemma 2.3. Suppose that Dand all the diare odd. Then there is a preferred choice of α, as follows. (1) For αeither choice of √p, G(p, D, d1, ..., dr, 1)has Ggeom Garith ⊂SpD−1(Q). (2) If pis odd, write D=2δ+1. Then for α:= −χ2((−1)δD)g(ψ, χ2)(g(ψ, χ2)the quadratic Gauss sum over Fp), G(p, D, d1, ..., dr, χ2)has Ggeom Garith ⊂SOD(Q). (3) Denote by Q(ζp)+the real subfield of Q(ζp)+. In case (1), the traces attained lie in Q(ζp)+(√p). In case (2), they lie in Q(ζp)+. Proof. Assertion (1) is Poincaré duality. Assertion (2) is proved in [Ka-NG2, 1.7]. Assertion (3) is obvious from applying complex conjugation to the formulas for the traces. 2 The local system G(p, D, d1, ..., dr, 1)on Ar/Fphas its Ggeom Garith ⊂GLD−1(Q). N.M. Katz et al. / Journal of Number Theory 206 (2020) 1–23 9 Similarly, when pis odd, the local system G(p, D, d1, ..., dr, χ2)on Ar/Fphas its Ggeom Garith ⊂GLD(Q). We now apply 2.1 to these local systems. Proposition 2.4. The following conditions are equivalent. (a) G(p, D, d1, ..., dr, 1)has finite Ggeom. (a-bis) F(p, D, d1, ..., dr, 1)has finite Ggeom. (b) G(p, D, d1, ..., dr, 1)has finite Garith. (c) All traces of G(p, D, d1, ..., dr, 1)are algebraic integers. When pis odd, we have these same equivalences for G(p, D, d1, ..., dr, χ2). We now perform two successive reduction steps. The first is to pull back these local systems from Arto Gr m, i.e., requiring the coefficients tito all be invertible. Lemma 2.5. The following conditions are equivalent. (a) G(p, D, d1, ..., dr, 1)has finite Garith. (b) The pullback of G(p, D, d1, ..., dr, 1)to Gr mhas finite Garith. If pis invertible, the same equivalence holds for G(p, D, d1, ..., dr, χ2). Proof. That (a) implies (b) is obvious. Because Gr mis a dense open set of Ar, π1(Gr m) maps onto π1(Ar), hence (b) implies (a). 2 We now form local systems Gbig(p, D, d1, ..., dr, 1) and, if pis odd, Gbig(p, D, d1, ..., dr, χ2)on Gr+1 m, by letting the coefficient of xDalso vary over invertible scalars. Thus the trace function of Gbig(p, D, d1, ..., dr, 1)is (t1,...,t r+1)∈Gm(K)r+1 →  x∈K ψK(tr+1xD+ i tixdi)/αdeg(K/Fp). When pis odd, the trace function of Gbig(p, D, d1, ..., dr, χ2)is (t1,...,t r+1)∈Gm(K)r+1 →  x∈K× ψK(tr+1xD+ i tixdi)χ2,K(x)/αdeg(K/Fp). Lemma 2.6. The following conditions are equivalent. (a) Gbig(p, D, d1, ..., dr, 1)on Gr+1 mhas finite Garith. (b) The pullback of G(p, D, d1, ..., dr, 1)to Gr mhas finite Garith. 16 N.M. Katz et al. / Journal of Number Theory 206 (2020) 1–23 We also have one further criterion that involves the simpler function [x]p:= sum of the p-adic digits of x. We first prove the following Lemma 2.10 (Hasse-Davenport relation). Let f, kbe positive integers and x ∈Z. Then we have pfk −1 pf−1xp,fk =k·[x]p,f and pfk −1 pf−1xp,fk,− =k·[x]p,f,− Proof. If x ≡y(mod pf−1) then pfk−1 pf−1x ≡pfk−1 pf−1y(mod pfk −1), so it suffices to prove it for 0 ≤x <p f−1(so pfk−1 pf−1x <p fk −1). But then the result is clear since the p-adic expansion of pfk−1 pf−1xis the concatenation of kcopies of the p-adic expansion of x(filled with leading 0’s so that it has exactly fdigits). 2 For use below, we recall the following result from [Ka-RL, Prop. 2.2], whose inequalities are used in the proof of Theorem 2.12. Proposition 2.11. For strictly positive integers xand y, and any f≥1, we have: (i) [x +y]p≤[x]p+[y]p; (ii) [x]p,f ≤[x]p; (iii) [px]p=[x]p. Theorem 2.12. We have the following results. (i) Gbig(p, D, d1, ..., dr, 1)has finite Garith if and only if there exists some real A ≥0 such that for every positive integer fand every r-tuple of integers 0 ≤x2, ..., xr+1 < pf−1which are not all 0, we have the inequality r+1  i=2 dixip≤ r+1  i=2 [xi]p+f(p−1) 2+A. (2.12.1) (ii) If p is odd, then Gbig(p, D, d1, ..., dr, χ2)has finite Garith if and only if there exists some real A ≥0such that for every positive integer fand every r-tuple of integers 0 ≤x2, ..., xr+1 <p f−1, we have the inequality r+1  i=2 dixi+pf−1 2p≤ r+1  i=2 [xi]p+f(p−1) 2+A. (2.12.2) N.M. Katz et al. / Journal of Number Theory 206 (2020) 1–23 17 Proof. We will prove that the hypotheses of this theorem are equivalent to those of Theorem 2.9. Suppose that there exists A ≥0such that (2.12.1)holds for every f≥1and every 0 ≤x2, ..., xr+1 <p f−1which are not all zero. Then r+1 i=2 dixi>0, and r+1  i=2 dixip,f ≤r+1  i=2 dixip≤ r+1  i=2 [xi]p+f(p−1) 2+A = r+1  i=2 [xi]p,f,−+f(p−1) 2+A. In particular, for every positive integer k, r+1  i=2 di pfk −1 pf−1xip,fk ≤ r+1  i=2 pfk −1 pf−1xip,fk,− +fk(p−1) 2+A. By Lemma 2.10, dividing by kwe get r+1  i=2 dixip,f ≤ r+1  i=2 [xi]p,f,−+f(p−1) 2+A/k, and taking k→∞gives us (2.9.1). The proof for case (ii) is similar. Conversely, suppose that (2.9.1)holds for every f≥1and every 0 ≤x2, ..., xr+1 < pf−1. Let lbe an integer such that idi<p l. Then, if 0 ≤x2, ..., xr+1 <p f−1, idixi<p f+l−1, so r+1  i=2 dixip =r+1  i=2 dixip,f+l≤ r+1  i=2 [xi]p,f+l,−+(f+l)(p−1) 2 = r+1  i=2 [xi]p+f(p−1) 2+l(p−1) 2, and (2.12.1)holds with A =l(p −1)/2. Finally, suppose that (2.9.2)holds for every f≥1and every 0 ≤x2, ..., xr+1 <p f−1. Let lbe an integer such that 2 idi<p l. Then idixi+(pf+l−1)/2 <p f+l−1, so 1+r+1  i=2 dixi+pf−1 2p =pf+l+ r+1  i=2 dixi+pf−1 2p =pfpl+1 2+ r+1  i=2 dixi+pf+l−1 2p 18 N.M. Katz et al. / Journal of Number Theory 206 (2020) 1–23 ≤pl+1 2p +r+1  i=2 dixi+pf+l−1 2p =1+l(p−1) 2+r+1  i=2 dixi+pf+l−1 2p,f+l ≤1+l(p−1) 2+ r+1  i=2 [xi]p,f+l,−+(f+l)(p−1) 2 =1+ r+1  i=2 [xi]p+f(p−1) 2+l(p−1), and (2.12.2)holds with A =l(p −1). 2 3. Theorems of finite monodromy From known results of Kubert, explained in [Ka-RLSA, 4.1,4.2,4.3], and the result [G-K-T, Thm. 3.1], we know that Ggeom and Garith for Fp,D,xD,χ2⊗α−deg are finite when qis a power of an odd prime pand Dis any of q+1 2,qn+1 q+1 with odd n, 2q−1. We will refer to these as the known cases. We stumbled upon the empirical fact that F3,23,x23,χ2⊗α−deg seemed to have finite (arithmetic and geometric) monodromy, although it was not one of the known cases. As we will prove below, the monodromy is in fact finite. A computer search for each of p =3, 5, 7, 11 and each 2 ≤D≤106found no other cases than this one and the known cases with finite monodromy. It is not clear whether there should be infinitely many (p, D) other that the known ones with finite monodromy, or finitely many, or just this one. In this section, we prove that F3,23,x23,χ2⊗α−deg has finite arithmetic and geometric monodromy groups. More generally, we prove that the two-parameter family G(3,23,1,5,χ 2), whose traces at points (s, t) ∈A2(K), for K/F3a finite extension, are the sums (s, t)→−  x∈K× ψK(x23 +sx5+tx)χ2,K(x)/α−deg(K/F3), has finite arithmetic and geometric monodromy groups. We will do so by applying the criterion from Theorem 2.12. N.M. Katz et al. / Journal of Number Theory 206 (2020) 1–23 19 Theorem 3.1. The two-parameter family G(3,23,1,5,χ 2) has finite arithmetic monodromy. Proof. We will prove that for every positive integer fand every pair of integers 0 ≤ x, y<3fwe have the inequality 23x+5y+3f−1 23≤[x]3+[y]3+f+2. The result follows then from Theorem 2.12. We proceed by induction on f: for f≤4one checks it by hand. Let f≥5and 0 ≤x, y<3f. Case 1: x ≡0(mod 3). Write x =3a, y=3c +dwith a, c < 3f−1,d=0,1,2. Then [5d +1] 3≤[d]3+1(check by hand), so 23x+5y+3f−1 23 =323a+5c+3f−1−1 2+5d+1 3 ≤23a+5c+3f−1−1 23 +[5d+1] 3 ≤[a]3+[c]3+(f+1)+[d]3+1 =[x]3+[y]3+f+2 by induction. Case 2: x ≡1(mod 3). Write x =9a +b, y=9c +dwith a, c < 3f−2,b∈{1,4,7},d<9. Then [23b +5d +4] 3≤[b]3+[d]3+2(check by hand), so 23x+5y+3f−1 23 =923a+5c+3f−2−1 2+23b+5d+4 3 ≤23a+5c+3f−2−1 23 +[23b+5d+4] 3 ≤[a]3+[c]3+f+[b]3+[d]3+2 =[x]3+[y]3+f+2 20 N.M. Katz et al. / Journal of Number Theory 206 (2020) 1–23 by induction. Case 3: x ≡2(mod 3) but x ≡ 8, 17 or 20 (mod 27). Write x =27a +b, y=27c +dwith a, c < 3f−3,b∈{2,5,11,14,23,26},d<27. Then [23b +5d + 13]3≤[b]3+[d]3+3(check by hand), so 23x+5y+3f−1 23 =27 23a+5c+3f−3−1 2+23b+5d+13 3 ≤23a+5c+3f−3−1 23 +[23b+5d+ 13]3 ≤[a]3+[c]3+(f−1) + [b]3+[d]3+3 =[x]3+[y]3+f+2 by induction. Case 4: x ≡8, 17 or 20 (mod 27). Write x =81a +b, y=81c +dwith a, c < 3f−4,b∈{8,17,20,35,44,47,62,71,74},d<81. Then [23b +5d + 40]3≤[b]3+[d]3+4(check by hand), so 23x+5y+3f−1 23 =81 23a+5c+3f−4−1 2+23b+5d+40 3 ≤23a+5c+3f−4−1 23 +[23b+5d+ 40]3 ≤[a]3+[c]3+(f−2) + [b]3+[d]3+4 =[x]3+[y]3+f+2 by induction. 2 4. Determination of the monodromy groups In this section, we will show that with the correct choice [Ka-NG2, 1.7] of α, namely α:= −g(ψ,χ2), with g(ψ, χ2)the quadratic Gauss sum over F3, we can determine the monodromy of F3,23,x23,χ2⊗α−deg, and of some other related local systems, exactly. Recall from Lemma 2.3 that we have N.M. Katz et al. / Journal of Number Theory 206 (2020) 1–23 21 Ggeom Garith <SO23(Q). Thus Ggeom is an irreducible, primitive (by Lemma 1.1) finite subgroup of SO23(Q). The larger group Garith is a fortiori also an irreducible, primitive finite subgroup of SO23(Q). The traces attained by F3,23,x23,χ2⊗α−deg, i.e., the traces of elements of Garith, are all integers (being algebraic integers in Q(ζ3)+=Q). Over the field F81, the traces obtained are, by direct calculation, {−2, −1, 0, 1, 2, 3}. Over F243, the traces attained are, by direct calculation, {−5, −2, −1, 0, 1, 2}. Finally, we recall that from Lemma 1.2 that the image of the wild inertia group is the additive group of F35, the least extension of F3containing the 22’nd roots of unity. First we prove the following theorem on finite subgroups of SL23(C): Theorem 4.1. Let V=C23 and let G <SL(V)be a finite irreducible subgroup. Let χ denote the character of Gafforded by V, and suppose that all the following conditions hold: (i) χis real-valued; (ii) χis primitive; (iii) χ(g) <−1for some g∈G; (iv) The 3-rank of Gis at least 5. Then G ∼ =Co3in its unique (orthogonal) irreducible representation of degree 23. Proof. By the assumption, the G-module Vis irreducible and primitive; furthermore, it is tensor indecomposable and not tensor induced since dim V=23is prime. Next, we observe by Schur’s Lemma that condition (i) implies Z(G) =1. Now we can apply [G-T, Proposition 2.8] (noting that the subgroup Hin its proof is just Gsince G <SL(V)) and arrive at one of the following two cases. (a) Extraspecial case: PGfor some extraspecial 23-group of order 233that acts irreducibly on V. But in this case, χ|Pcannot be real-valued (in fact, Q(χ|P)would be Q(exp(2πi/23)), violating (i). (b) Almost simple case: SG ≤Aut(S)for some finite non-abelian simple group S. In this case, we can apply the main result of [H-M]and arrive at one of the following possibilities for S. •S=A24, M24, or PSL2(23). Correspondingly, we have that G =A24 or S24, M24, and PSL2(23) or PGL2(23). In all of these possibilities, χ(x) ≥−1for x ∈Gby [ATLAS], violating (iii). •S=PSL 2(47). This is ruled out since Q(χ|S)would be Q(√−47), violating (i). •S=Co2. In this case, G =S, and a Sylow 3-subgroup Pof Ghas a normal extraspecial 3-subgroup Q ∼ =31+4 +of index 3, see [ATLAS]. Since Ghas 3-rank ≥5 by (iv), Pcontains an elementary abelian 3-subgroup of order 35, whence Qcontains a 22 N.M. Katz et al. / Journal of Number Theory 206 (2020) 1–23 subgroup R∼ =C4 3. Identifying Z(Q)with F3, we see that the commutator map induces a non-degenerate symplectic form on Q/Z(Q) ∼ =F4 3. As Ris abelian, this form is totally isotropic on Z(Q)R/Z(Q)which has order at least 33. But this is a contradiction, since any isotropic subspace in F4 3has dimension at most 2. •S=Co3. In this case G =Co3, as stated. 2 Theorem 4.2. We have the following results. (i) For F3,23,x23,χ2⊗α−deg, we have Ggeom =Garith =Co3, the Conway group Co3, in its irreducible orthogonal representation of dimension 23. (ii) For the two-parameter family G(3, 23, 1, 5, χ2), we have Ggeom =Garith =Co3. (iii) The local system on Gm×A2/F3whose trace is given as follows: for any finite extension K/F3, and any (r=0, s, t) ∈Gm(K) ×A2(K), (r, s, t)→−χ2,K(r) x∈K× ψK(rx23 +sx5+tx)χ2,K (x)/α−deg(K/Fp), has Ggeom =Garith =Co3. (iv) For any finite extension K/F3, and any s0∈K, the local system on A1/K whose trace function, at points t ∈L, La finite extension of K, is given by t→− x∈L× ψL(x23 +s0x5+tx)χ2,L(x)/α−deg(L/Fp), has Ggeom =Garith =Co3. (v) The local system on A1/F3whose trace function, at points t ∈K, Ka finite extension of F3, is given by t→−  x∈K× ψL(x23 +tx5)χ2,K(x)/α−deg(K/Fp), has Ggeom =Garith =Co3. (vi) For any finite extension Kof F3, and any (s0, t0) ∈A2(K)other than (0, 0), the local system on Gm/K whose trace function at points r∈L×, La finite extension of K, is given by r→−χ2,L(r) x∈L× ψL(rx23 +s0x5+t0x)χ2,L(x)/α−deg(L/Fp), has Ggeom =Garith =Co3. Proof. We first note that among finite irreducible subgroups of SO23(C), the Conway group Co3is both maximal and minimal. Indeed, the minimality is clear from the [ATLAS] list of maximal subgroups of Co3and their character tables. The maximality is clear from Theorem 4.1. N.M. Katz et al. / Journal of Number Theory 206 (2020) 1–23 23 The local system in (iv) has finite monodromy because its restriction to the dense open set (Gm)3/F3has finite monodromy, being the Gbig partner, in the sense of Lemma 2.6 of two-parameter local system of (ii). The χ2(t)term in front keeps its Garith in SO23(C), by [Ka-NG2, 1.7]. It is geometrically irreducible because this is so already after pullback to the (1, 0, t)t-line, where it is F3,23,x23 ,χ2⊗α−deg. We remark that the local system in (v) is geometrically irreducible, because it is the Fourier Transform of [5](Lψ(x23)⊗L χ2(x)), a middle extension sheaf (cf. [Ka-TLFM, proof of 3.3.1]) which is geometrically irreducible. Indeed it is I(∞)-irreducible, because its five I(∞)-slopes are each 23/5, with exact denominator 5. We also remark that the local system in (vi) is geometrically irreducible, because it is Lχ2(t)tensored with the Fourier Transform of [23](Lψ(s0x5+r0x)⊗Lχ2(x)), a middle extension sheaf (cf. [Ka-TLFM, proof of 3.3.1]) which is geometrically irreducible. Indeed it is I(∞)-irreducible, because its 23 I(∞)-slopes are each either 5/23, with exact denominator 23, if s0=0, or each 1/23 if s0= 0 but r0=0. Thus it suffices to prove (i). 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