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The integral monodromy of hyperelliptic and trielliptic curves

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IntroductionModuli spaces of curves with Z/d-actionStable Z/d-covers of a genus zero curveModuli spacesDegenerationThe case of hyperelliptic curves: d=2The case of trielliptic curves: d=3Components of the trielliptic locusDegeneration of the trielliptic locusMonodromy groupsDefinition of monodromyDegeneration and monodromyMonodromy of hyperelliptic curvesMonodromy of trielliptic curvesUnitary groupsCalculation of trielliptic monodromyThe integral monodromy of hyperelliptic and triellipticcurvesJeffrey D. Achter & Rachel Pries∗AbstractWe compute the Z/` and Z`monodromy of every irreducible component of the mod-uli spaces of hyperelliptic and trielliptic curves. In particular, we provide a proof thatthe Z/` monodromy of the moduli space of hyperelliptic curves of genus g is the sym-plectic group Sp2g(Z/`). We prove that the Z/` monodromy of the moduli space oftrielliptic curves with signature (r, s) is the special unitary group SU(r,s)(Z/` ⊗ Z[ζ3]).MSC 11G18, 14D05, 14H40keywords monodromy, hyperelliptic, trigonal, moduli, Jacobian1 IntroductionIf C → S is a relative smooth proper curve of genus g ≥ 1 over an irreducible base,then the `-torsion of the relative Jacobian of C encodes important information about thefamily. Suppose ` is invertible on S, and let s ∈ S be a geometric point. The fundamentalgroup π1(S, s) acts linearly on the fiber P ic0(C)[`]s∼=(Z/`)2g, and one can consider themod-` monodromy representation associated to C:ρC→S,`: π1(S, s)-Aut(Pic0(C)[`]s)∼=GL2g(Z/`).Let M`(C → S), or simply M`(S), be the image of this representation. If a primitive `throot of unity is defined globally on S, then Pic0(C)[`]sis equipped with a skew-symmetricform h·, ·i and M`(C → S) ⊆ Sp(Pic0(C)[`]s, h·, ·i)∼=Sp2g(Z/`). If C → S is a sufficientlygeneral family of curves, then M`(C → S)∼=Sp2g(Z/`) [8].In this paper, we compute M`(S) when S is an irreducible component of the mod-uli space of hyperelliptic or trielliptic curves and C → S is the tautological curve. Thefirst result implies that there is no restriction on the monodromy group in the hyperel-liptic case other than that it preserve the symplectic pairing. As a trielliptic curve is aZ/3-cover of a genus zero curve, the Z/3-action constrains the monodromy group to liein a unitary group associated to Z[ζ3]. The second result implies that this is the onlyadditional restriction in the trielliptic case.Theorem 3.4 Let ` be an odd prime, and let k be an algebraically closed field in which 2` isinvertible. For g ≥ 1 , M`(Hg⊗k)∼=Sp2g(Z/`).∗The second author was partially supported by NSF grant DMS-04-00461.1Theorem 3.8 Let ` ≥ 5 be prime, and let k be an algebraically closed field in which 3` isinvertible. Let Tγbe any component of the moduli space of trielliptic curves of genus g ≥ 3.Then M`(Tγ⊗k)∼=SG(rγ,sγ)(Z/`) (where the latter is a unitary group defined in (3.4)).We also prove that the `-adic monodromy group is Sp2g(Z`) in the situation of Theo-rem 3.4 and is SG(rγ,sγ)(Z`) in the situation of Theorem 3.8.Theorem 3.4 is an unpublished result of J.K. Yu and has already been used multipletimes in the literature. In [7], Chavdarov assumes this result to show that the numeratorof the zeta function of the typical hyperelliptic curve over a finite field is irreducible.Kowalski also uses this result in a similar fashion [21]. The first author used Theorem 3.4to prove a conjecture of Friedman and Washington on class groups of quadratic functionfields [2].There are other results in the literature which are similar to Theorem 3.4 but whichare not quite strong enough for the applications above. A’Campo [1, Th. 1] computesthe topological monodromy of Hg⊗C. On the arithmetic side, the Q`, as opposed to Z`,monodromy of Hgis computed in [17, 10.1.16]. For purely group-theoretic reasons, onecan deduce that M`(Hg) is maximal for all sufficiently large ` [25]; see also [23]. However,the exceptional set of primes is difficult to calculate and depends on g.There are results on Q`-monodromy of cyclic covers of the projective line of arbitrarydegree, e.g., [16, Sec. 7.9]. Also, in [12, 5.5], the authors prove that the projective rep-resentation PρC→S,`is surjective for many families of cyclic covers of the projective line.Due to a combinatorial hypothesis, their theorem does not apply to Hgand applies to atmost one component of the moduli space of trielliptic curves for each genus, see Remark2.8. See also work of Zarhin, e.g., [31].As an application, for all p ≥ 5, we show using [6] that there exist hyperelliptic andtrielliptic curves of every genus (and signature) defined over Fpwhose Jacobians areabsolutely simple. In contrast with the applications above, these corollaries do not usethe full strength of our results. Related work can be found in [15] where the authorsproduce curves with absolutely simple Jacobians over Fpunder the restriction g ≤ 3.Corollary 3.6 Let p 6= 2 and let g ∈ N. Then there exists a smooth hyperelliptic curve of genusg defined over Fpwhose Jacobian is absolutely simple.Corollary 3.11 Let p 6= 3. Let g ≥ 3 and let (r, s) be a trielliptic signature for g (Definition2.10). Then there exists a smooth trielliptic curve defined over Fpwith genus g and signature(r, s) whose Jacobian is absolutely simple.Our proofs proceed by induction on the genus. The base cases for the hyperellipticfamily rely on the fact that every curve of genus g = 1, 2 is hyperelliptic; the claim onmonodromy follows from the analogous assertion about the monodromy of Mg. Thebase case g = 3 for the trielliptic family involves a comparison with a Shimura varietyof PEL type, namely, the Picard modular variety. An important step is to show that themonodromy group does not change in the base cases when one adds a labeling of theramification points to the moduli problem.The inductive step is similar to the method used in [9] and uses the fact that familiesof smooth hyperelliptic (trielliptic) curves degenerate to trees of hyperelliptic (triellip-2tic) curves of lower genus. The combinatorics of admissible degenerations require us tocompute the monodromy exactly for the inductive step rather than up to isomorphism.The inductive strategy using admissible degeneration developed here should workfor other families of curves, especially for more general cyclic covers of the projectiveline. The difficulty is in the direct calculation of monodromy for the necessary base cases.We thank C.-L. Chai, R.


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