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The Quarterly Journal of Mathematics
, Volume Advance Article (2) – Feb 8, 2018

22 pages

/lp/ou_press/quotients-of-hyperelliptic-curves-and-tale-cohomology-Vr1Amsuj9F

- Publisher
- Oxford University Press
- Copyright
- © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com
- ISSN
- 0033-5606
- eISSN
- 1464-3847
- D.O.I.
- 10.1093/qmath/hax053
- Publisher site
- See Article on Publisher Site

Abstract We study hyperelliptic curves C with an action of an affine group of automorphisms G. We establish a closed form expression for the quotient curve C/G and for the first étale cohomology group of C as a representation of G. The motivation comes from the arithmetic of hyperelliptic curves over local fields, specifically their local Galois representations and the associated invariants. 1. Introduction This paper studies hyperelliptic curves C/k with a group of affine automorphisms G. We show that both the equation of the quotient curve C/G and the G-representation Hét1(Ck¯,Ql) admit simple closed form descriptions in terms of the defining equation of C. The application we have in mind is to Galois representations of hyperelliptic curves over local fields [4]. Throughout the paper, C will be a non-singular projective hyperelliptic curve over a field k with chark≠2. It will be given by an (affine) equation C:Y2=f(X)with f(X)∈k[X] a squarefree polynomial of degree ≥1 that factors over k- as f(X)=c∏r∈R(X−r),R⊂k-.Let G⊂AutkC be an affine group of automorphisms. Thus, g∈G acts as g(X)=α(g)X+β(g),g(Y)=γ(g)Yfor some α(g),β(g),γ(g)∈k. In particular, G acts naturally on the set of roots R through the X-coordinate. The punchline is that both the quotient curve C/G and the étale cohomology group Hét1(Ck¯,Ql) with G-action can be very simply expressed in terms of γ and the G-action on R. Explicitly, writing R/G for the set of orbits of G on R, and C[R] for the permutation representation, we prove: Theorem 1.1 (Quotient curve). If Gcontains the hyperelliptic involution, then C/G≃Pk1. If Gacts trivially on the Y-coordinate, then C/G:y2=c(−1)∣R∣−∣R/G∣∏O∈R/G(x−∏r∈Or). Otherwise, C/G:y2=c(−1)(m−1)(∣R/G∣−1)(x−μ)∏O∈R/GO≠U(x−∏r∈Or),where Uis the set of a∈kthat are fixed points of some non-trivial g∈Gacting on kthrough the X-coordinate, m=∣G∣/∣U∣and μ=∏a∈Uam. It is easy to see that in case (3), ∣G∣=m, ∣U∣=1 when chark=0, and ∣G∣=mpj, ∣U∣=pj, p∤m when chark=p>0 (Lemma 2.1). Theorem 1.2 (Étale cohomology). For every prime l≠chark, and every embedding Ql↪C, Hét1(Ck¯,Ql)⊗C≃V⊖ϵas a complex representation of G, where V=γ˜⊗(C[R]⊖1),ϵ={0if∣R∣isodd,detVif∣R∣iseven,and γ˜:G→C×is any one-dimensional representation with kerγ˜=kerγ. The representation 1⊕ϵis the permutation action of Gon the points at infinity of C. The assumptions chark≠2 and l≠chark are certainly necessary. When chark=2, the curve C can be given y2+y=f(x), but neither the set of roots of f nor any other G-set will compute Hét1(Ck¯,Ql) as in the theorem: for example, if C/F-2 is an elliptic curve with G=Q8 or G=SL2(F3), then Hét1(Ck¯,Ql) is not a 1-dimensional twist of a representation realizable over Q. Similarly, Hét1(Ck¯,Ql) behaves differently when l=chark: its dimension drops, and it is not necessarily a rationally traced representation (for example, for an ordinary elliptic curve over Fp with G=C4). Our motivation comes from the arithmetic of hyperelliptic curves over local fields, specifically their Hét1 as a Galois representation and the associated invariants: the conductor, local polynomial, root number, etc. Suppose for simplicity that C/K is such a curve, and that it acquires good reduction over a finite Galois extension F/K. The inertia group G=IF/K and Frobenius element ϕ∈Gal(K-/K) act naturally on the reduced curve C- over the residue field k of K, and G acts as a group of affine automorphisms. It turns out that Hét1(CK¯,Ql) as a Gal(K-/K)-module is the same as Hét1(C-k¯,Ql) with its G- and ϕ-action, and Theorem 1.2 describes the G-action explicitly. To complement this, we need to describe the action of Frobenius-like maps on the quotient curve C/G: Theorem 1.3 Suppose chark=p>2, Gdoes not contain the hyperelliptic involution, and Φ:C→Cis a morphism of the form ΦX=aXq+b,ΦY=dYq(qpowerofp)that normalizes G (meaning ΦG = GΦ as sets). Then Φdescends to a morphism Ψ:C/G→C/Ggiven by Ψx=a∣G∣xq+∏g∈G(α(g)b+β(g)),Ψy={dyqifγ=1,a⌊m/2⌋∣G∣/mdyqifγ≠1,where mis the prime-to- ppart of ∣G∣, and the model for C/Gis that of Theorem1.1. See Section 6 for an example of how this theorem can be used over local fields to determine the local Galois representation of a hyperelliptic curve (and [4] for the general theory). 1.1. Outline Section 2 concerns affine groups of automorphisms of hyperelliptic curves, the permutation action on the set of roots R and the character γ. The main theorem here is Theorem 2.4, which is Theorem 1.1 with an explicit description of the quotient map. Section 3 proves some general facts about Hét1 for quotient curves, in particular showing that Hét1(Ck¯,Ql) is the unique representation V of G with rational character for which dimVH=2genus(C/H) for every subgroup H<G. Then Section 4 applies this to hyperelliptic curves, proving Theorem 1.2 (= 4.1), with a slightly cumbersome representation-theoretic computation. An alternative approach would be to use the Weil–Serre description of the G-action on étale cohomology in terms of an equivariant Riemann–Hurwitz formula [11], [9, Chapter VI], but that seems to be an equally long computation. Section 5 proves Theorem 1.3 (= 5.1). Remark 1.4 To obtain an explicit description for Hét1(Ck¯,Ql), we only use that it is the unique representation V of G with rational character for which dimVH=2genus(C/H) for every subgroup H<G (Theorem 3.3). Thus Theorem 1.2 would also hold for any other cohomology theory with these properties. 1.2. Notation Throughout the paper, we use the following notation. C/k hyperelliptic curve Y2=c∏r∈R(X−r) with the right-hand side in k[x], R⊂k-, chark≠2; genus ≤1 is allowed G a finite group of affine automorphisms of C/k, acting by g(X)=α(g)X+β(g),g(Y)=γ(g)Y for g∈G H1(C) =Hét1(Ck¯,Ql)⊗C (choosing some embedding Ql↪C) 1,⊕,⊖ trivial representation, direct sum and direct difference ( V⊖W is well-defined up to isomorphism because the category of C[G]-modules is semisimple) ⌊·⌋ the floor function ζm primitive mth root of unity C/k hyperelliptic curve Y2=c∏r∈R(X−r) with the right-hand side in k[x], R⊂k-, chark≠2; genus ≤1 is allowed G a finite group of affine automorphisms of C/k, acting by g(X)=α(g)X+β(g),g(Y)=γ(g)Y for g∈G H1(C) =Hét1(Ck¯,Ql)⊗C (choosing some embedding Ql↪C) 1,⊕,⊖ trivial representation, direct sum and direct difference ( V⊖W is well-defined up to isomorphism because the category of C[G]-modules is semisimple) ⌊·⌋ the floor function ζm primitive mth root of unity C/k hyperelliptic curve Y2=c∏r∈R(X−r) with the right-hand side in k[x], R⊂k-, chark≠2; genus ≤1 is allowed G a finite group of affine automorphisms of C/k, acting by g(X)=α(g)X+β(g),g(Y)=γ(g)Y for g∈G H1(C) =Hét1(Ck¯,Ql)⊗C (choosing some embedding Ql↪C) 1,⊕,⊖ trivial representation, direct sum and direct difference ( V⊖W is well-defined up to isomorphism because the category of C[G]-modules is semisimple) ⌊·⌋ the floor function ζm primitive mth root of unity C/k hyperelliptic curve Y2=c∏r∈R(X−r) with the right-hand side in k[x], R⊂k-, chark≠2; genus ≤1 is allowed G a finite group of affine automorphisms of C/k, acting by g(X)=α(g)X+β(g),g(Y)=γ(g)Y for g∈G H1(C) =Hét1(Ck¯,Ql)⊗C (choosing some embedding Ql↪C) 1,⊕,⊖ trivial representation, direct sum and direct difference ( V⊖W is well-defined up to isomorphism because the category of C[G]-modules is semisimple) ⌊·⌋ the floor function ζm primitive mth root of unity 2. Quotients of hyperelliptic curves We begin with affine actions and invariant functions on A1, and then move to hyperelliptic curves. We refer the reader interested in arbitrary (non-affine) automorphism groups of hyperelliptic curves to [1, 10]. Lemma 2.1 Let kbe a field, and G⊂AutAk1a finite group of affine linear transformations X↦g(X)=α(g)X+β(g).Then G≃T⋊Cm, where Tis the subgroup of translations. If chark=0, then Tis trivial. If chark=p>0, then p∤m, and Tis an elementary abelian p-group. Moreover, Tis an Fp(ζm)-vector space, with Cmacting on Tby multiplication by mth roots of unity. Every g∈G⧹Thas a unique fixed point a∈k-; moreover, a∈k. If m=1, then every G-orbit on A1(k-)is regular (recall that an action of a finite group G on a set X is regular if X≃G as a G-set, equivalently the action is transitive and non-trivial elements of G have no fixed points). If m≠1, then there is a unique non-regular G-orbit on A1(k-). It is a regular T-orbit, and consists of the fixed points of all g∈G⧹T. The field of invariant rational functions k(X)Gis generated by I=∏g∈G(α(g)X+β(g)),the unique G-invariant polynomial of degree ∣G∣with constant coefficient 0 and leading coefficient (−1)∣G∣−1. If O⊂k-=A1(k-)is a regular G-orbit, then ∏r∈O(X−r)=(−1)∣G∣−1(I−∏r∈Or). If U⊂k-=A1(k-)is a non-regular G-orbit, let IT=∏g∈Tg(X)and λ=∏r∈Ur. Then ∏r∈U(X−r)=IT−λ,the unique monic polynomial of degree ∣G∣/mthat is G-invariant up to scalars. Proof (1) The elements of G with α(g)=1 form a group of translations T, which is naturally an additive subgroup of k. As G is finite, T is trivial if chark=0. The map g↦α(g) embeds A=G/T↪k×, so it is a cyclic group, A≃Cm. Suppose chark=p>0. Then p∤m. As ∣T∣ and ∣A∣ are coprime, the extension of A by T is split. Take a generator g of A=Cm with α(g)=ζm. It conjugates a translation x↦x+v to x↦x+ζmv. This makes T into an Fp(ζm)-vector space, with the asserted conjugation action by Cm. (2) If α(g)≠1, then the map X↦α(g)X+β(g) has a unique fixed point a=β(g)(1−α(g)) on A1, and it is k-rational. (3) Clear, since translations have no fixed points. (4) Let g be a generator of Cm and a∈k its fixed point. The stabilizer of a is precisely Cm (as it meets T trivially), so a has orbit of length ∣T∣; it is a regular T-orbit. Every other point in this orbit has a conjugate stabilizer that meets Cm trivially, by the uniqueness of fixed points. Thus, these stabilizers cover (m−1)∣T∣ elements of G, which must be the whole of G\T. Hence, the orbit of a accounts for all fixed points of elements of G, in other words every other G-orbit is regular. (5) I(X) is clearly G-invariant, and has the right degree ∣G∣. Its constant coefficient is 0, and the leading coefficient is ∏g∈Gα(g)=(∏i=0m−1ζmi)∣T∣={(−1)∣T∣=−1,if2∣m1∣T∣=1,if2∤m=(−1)∣G∣−1. (6) The right-hand side is G-invariant. So is the left-hand side, since every g∈G permutes its roots and α(g)∣G∣=1. The claim follows from (5), by comparing the leading and the constant terms. (7) Because U is a G-orbit, the polynomial ∏x∈U(X−r) is G-invariant up to scalars (and it is a unique such polynomial of this degree, because a non-regular orbit is unique if it exists). By part (6) with G=T, ∏r∈U(X−r)=IT−λ.□ Proposition 2.2 Let kbe a field with chark≠2, and C/ka hyperelliptic curve C:Y2=c∏r∈R(X−r),R⊂k-.Let G⊂AutkCbe a subgroup of affine automorphisms g(X)=α(g)X+β(g),g(Y)=γ(g)Y(g∈G).Write Kand G-for the kernel and the image of the map X↦g(X)from Gto AutkA1. Then Gis a central extension of G-by K, and G=G-⇔∣K∣=1⇔hyperellipticinvolution∉G.G≠G-⇔∣K∣=2⇔hyperellipticinvolution∈G. G-≃T⋊αCmwith T=kerαthe subgroup of translations of G-. If chark=0, then Tis trivial. If chark=p>0, then p∤m, and Tis an elementary abelian p-group. Moreover, Tis an Fp(ζm)-vector space, with Cmacting on Tby multiplication by mth roots of unity. αis a primitive character of G-/T≃Cm, and γ2=α∣R∣. The group G-acts naturally on R, and either ∣R∣≡0modmand γ2=1; as a G--set, Ris a union of regular orbits; or ∣R∣≡1modmand γ2=α≠1; as a G--set, Ris a union of regular orbits and one non-regular orbit R0≃G-/Cm, which is the set of fixed points of non-trivial elements of G-. If ∣R∣is even, then γ(g)/α(g)∣R∣/2=±1for g∈G, and it is −1if and only if gswaps the two points at infinity of C. Proof (1) Clear. (2)–(4) follow directly from Lemma 2.1. (5) Since automorphisms preserve the equation of C up to a constant, G- permutes the elements of R, and γ2=α∣R∣. The rest follows from the following lemma and Lemma 2.1 (2) and (3). (6) The coordinates of the chart at infinity for C are u=1/X and v=Y/X∣R∣/2, and the two points at infinity are u=0,v=±c on this chart, when ∣R∣ is even. The claim follows.□ Lemma 2.3 Suppose Gis a group of the form G=Fpr⋊Cm(p∤m)such that no non-trivial elements g∈Cm and h∈Fprcommute. Let Rbe a faithful G-set on which elements id≠g∈Fprhave no fixed points, and elements id≠g∈Cmhave at most one fixed point. Then pr≡1modm; Ris a union of G-regular orbits plus at most one non-regular orbit R0≃G/Cm, consisting of fixed points of non-trivial elements of Gon R. Proof (1) From the non-commutativity assumption, it follows that the orbits of Cm on Fpr⧹{id} have length m. (2) Since non-trivial elements of Fpr have no fixed points, R is a union of regular Fpr-orbits. Now suppose that R has a non-regular orbit of G. It is a union of m/d regular Fpr-orbits for some d>1, which Cm permutes transitively. As d and ∣Fpr∣ are coprime, elements of Ck must have fixed points in each one of them. But they have at most one fixed point in total, and so d=m, such an orbit is ≃G/Cm, and it is necessarily unique (and clearly consists of fixed points of non-trivial elements of G). Hence, R≃G⨿⋯⨿GorR≃G⨿⋯⨿G⨿G/Cm.In the first case, ∣R∣≡0modm, and, in the second case, ∣R∣≡pr≡1modm.□ Theorem 2.4 Let Cbe a hyperelliptic curve over a field kwith chark≠2, C:Y2=cf(X);f(X)=∏r∈R(X−r),and G⊂AutkCa group of automorphisms acting via g(X)=α(g)X+β(g),g(Y)=γ(g)Y(g∈G).Let I=∏g∈Gg(X), and write R/Gfor the set of orbits of Gon the roots of fthrough the X-coordinate action: If Gcontains the hyperelliptic involution, then C/G≃Pk1with field of rational functions k(C/G)=k(I). If Gacts trivially on the Y-coordinate, then C/G:y2=c(−1)∣R∣−∣R/G∣∏O∈R/G(x−∏r∈Or)with x=Iand y=Y. Otherwise, C/G:y2=c(−1)δ(x−λm)∏O∈R/GO≠U(x−∏r∈Or)with x=Iand y=(IT−λ)⌊m/2⌋Y. Here Uis the set of a∈kthat are fixed points of some non-trivial g∈Gacting through the X-coordinate, m=∣G∣∣U∣,λ=∏a∈Ua,δ=(m−1)(∣R/G∣−1),IT=∏g∈kerαg(X). Proof (1) Write ι for the hyperelliptic involution. We clearly have C/⟨ι⟩≃Pk1, and k(C/⟨ι⟩)=k(X). Its quotient by G/⟨ι⟩ is again Pk1, and its field of rational functions is k(I) by Lemma 2.1(5). (2) The polynomials Y and I are G-invariant, and k(C/G)=k(C)G=k(I,Y)by degree considerations. To find the relation between I and Y, first note that as γ2=1, R is a union of regular G--orbits by Proposition 2.2. Because Y2/c=f(X)∈k(X) is G-invariant, by Lemma 2.1(6), f(X)=∏r∈R(X−r)=∏O∈R/G(−1)∣G∣−1(I−∏r∈Or)=(−1)∣R∣−∣R/G∣∏O∈R/G(I−∏r∈Or),which gives the required identity between x=I and y=Y. (3) Write G=T⋊Cm as in Proposition 2.2(1) and (2). First, suppose that T is trivial, so that G=Cm with m>1. We have two cases: Case ∣T∣=1, modd. Since γ≠1, we are in case (5b) of Proposition 2.2. So γ=α−(m−1)/2 (the unique character that squares to γ2=α) and there is a unique G-invariant root λ∈R. Hence, elements g∈G must act on the curve C as g(X)=α(g)(X−λ)+λ,g(Y)=α(g)−(m−1)/2Y,and the polynomials I=(X−λ)m+λm,J=(X−λ)(m−1)/2Yare easily seen to be G-invariant (and the first one is I by Lemma 2.1(5)). As [k(X):k(I)]≤m and k(X,Y)=k(X,J), the polynomials I and J generate k(C)G by degree considerations. They satisfy the relation J2=Y2(X−λ)m−1=cf(X)I−λmX−λ=c(I−λm)f(X)X−λ.Finally, since f(X) has all roots but λ coming in G-regular orbits, by Lemma 2.1(6), f(X)X−λ=∏O∈R/GO≠{λ}(I−∏r∈Or),which gives the required equation J2=c(I−λm)∏O∈R/GO≠{λ}(I−∏r∈Or). Case ∣T∣=1, meven. If we were in case (5b) of Proposition 2.2, then α would have order m and γ order 2m; in that case, kerα≠{id} and G contains the hyperelliptic involution—a contradiction. Hence, we must be in case (5a), in other words, R is a union of regular G--orbits. Note that γ=αm/2 since γ2=1 and γ≠1. By Lemma 2.1(3), there is a unique fixed point λ∈k of G=Cm, with λ∉R by the above. The polynomials I=−(X−λ)m+λm,J=(X−λ)m/2Yare G-invariant (and the first one is I by Lemma 2.1(5)), and generate k(C)G by degree considerations. Using Lemma 2.1(6) as before, we get the required relation J2=(X−λ)mcf(x)=c(−I+λm)∏O∈R/G−(I−∏r∈Or)=(−1)1+∣R/G∣c(I−λm)∏O∈R/G(I−∏r∈Or). Case ∣T∣≠1. Finally, suppose that T is non-trivial. First, we can compute C/T by using (2) and then C/G=(C/T)/Cm using the ∣T∣=1 cases. We have k(C/T)=k(C)T=k(Y,IT),IT=∏g∈Tg(X),and the equation for C/T is C/T:Y2=c∏O∈R/T(IT−∏r∈Or).The group Cm acts on this curve by affine transformations of the form g(IT)=αT(g)IT+βT(g),g(Y)=γ(g)Y(g∈Cm),and αT(g)=α(g)∣T∣=α(g),since ∣T∣≡1modm by Lemma 2.3(1) and Proposition 2.2(5). In particular, Cm acts non-trivially on the Y-coordinate of C/T and does not contain the hyperelliptic involution of C/T. By Lemma 2.1 (7), IT−λ is the unique monic polynomial of degree ∣G∣/m that is G-invariant up to scalars. In other words, on the IT-coordinate, λ is the unique fixed point for the Cm-action. Hence, by the ∣T∣=1 cases, we find the quotient (C/T)/Cm:y2=c(−1)δ(x−λm)∏O∈(R/T)/CmO≠{λ}(x−∏r∈Or),where δ=(m−1)(∣(R/T)/Cm∣−1)=(m−1)(∣R/G∣−1),x=∏g∈Cmg(IT)=∏g∈Cmg(∏t∈Tt(X))=I,y=(IT−λ)⌊m/2⌋Y,using that R is a union of regular T-orbits by Proposition 2.2(5) with G=T.□ Corollary 2.5 Suppose Cand Gare as in Theorem2.4, and let rbe the number of regular orbits of Gon R. Then genus(C/G)={0ifι∈G,⌊r2−12⌋=⌊∣R∣2∣G∣−12⌋ifγ=1,⌊r2⌋=⌊∣R∣2∣G∣⌋ifι∉G,γ≠1. Proof Apply Theorem 2.4, and recall that the genus of a hyperelliptic curve given by a polynomial of degree n is ⌊n−12⌋.□ 3. Étale cohomology of quotient curves In this section, C/k is any non-singular projective curve with a finite group of automorphisms G<AutkC. We will show that the G-action on the étale cohomology group Hét1(Ck¯,Ql) is determined by the genera of the quotients of C by subgroups of G. Recall that for every prime l, there are canonical isomorphisms Hét1(Ck¯,Ql)≃Hét1(Jac(Ck¯),Ql)≃(VlJac(C))*,where, as usual, VlJac(C)=(lim⟵Jac(C)[ln])⊗ZlQlis the vector space associated to the Tate module, and * denotes Ql-linear dual. If G<AutkC is a group of automorphisms, these become naturally QlG-representations, and the isomorphisms respect this structure. Theorem 3.1 Let π:C→Dbe a Galois cover of non-singular projective curves over a field k, with Galois group G, and let lbe a prime number: If l≠chark, then Hét1(Ck¯,Ql)is a QlG-representation with rational character. There is an isomorphism of Ql-vector spaces Hét1(Ck¯,Ql)G≃Hét1(Dk¯,Ql). If Φ:C→Cis a morphism of curves that commutes with π, then the isomorphism in (2) commutes with the action on Φ. If l≠chark, then genus(D)=12dimHét1(Ck¯,Ql)G. Proof (1) By the Lefschetz fixed-point formula [7, Theorem 25.1], the trace of any σ∈G on H1 is an integer, namely 2 minus the number of fixed points of σ on C. (2) The projection π induces pushforward and pullback on divisors π*:Div(C)⟶Div(D),π*:Div(D)⟶Div(C).The composition π*π* is multiplication by ∣G∣, while π*π* is the trace map Tr:P↦∑g∈GPg. These maps descend to Pic0(C)=Jac(C) and its ln-torsion Jac(C)[ln]. The image of Tr on VlJac(C) is the group of G-invariants, and (VlJac(C))G≃Tr(VlJac(C))≃π*(VlJac(C))≃VlJac(D),as multiplication by ∣G∣ is an isomorphism on Vl, and hence π* is injective and π* is surjective. (3) Clear from the construction in (2). (4) Clear from (2) and the genus formula dimHét1(Dk¯,Ql)=2genus(D).□ Lemma 3.2 Let Gbe a finite group. A representation Vof Gwith rational character is uniquely determined by dimVHfor all cyclic subgroups H<G. Proof A suitable multiple of V is realizable over Q ([8, Chapter 12, Proposition 34]), and for such representations the claim is proved in [8, Chapter 13, Corollary to Theorem 30′].□ Theorem 3.3 Let C/kbe a curve, and l≠charka prime number. Suppose Gis a finite group acting as automorphisms of C/k (not necessarily faithfully). Then Hét1(Ck¯,Ql)is the unique Ql-representation Vof Gsuch that all character values of Gon Vare rational, and dimVH=2genus(C/H)for every subgroup H<G. Proof Uniqueness follows from Lemma 3.2. Theorem 3.1(1,4) shows that V=Hét1(Ck¯,Ql) satisfies (a) and (b) when the action is faithful. Therefore, if q:G→AutC is any action, then dimVH=dimVq(H)=2genus(C/q(H))=2genus(C/H),and q(g) has rational trace on V for every g∈G.□ Remark 3.4 In Theorem 3.3(b), ‘subgroup’ may be replaced by ‘cyclic subgroup’, since these are sufficient for Lemma 3.2. Note also that the assertion concerns only the character of V, so the result remains true after any extension of the coefficient field (for example to Q-l or C). 4. Étale cohomology of hyperelliptic curves In this section, we prove the following theorem that describes the action of automorphisms on the first étale cohomology group of a hyperelliptic curve. Theorem 4.1 Let kbe a field of characteristic ≠2, and C/ka hyperelliptic curve given by Y2=c∏r∈R(X−r),R⊂k-.Let G⊂AutkCbe an affine group of automorphisms acting as g(X)=α(g)X+β(g),g(Y)=γ(g)Y(g∈G).Then for every prime l≠charkand every embedding Ql↪C, Hét1(Ck¯,Ql)⊗C≃V⊖ϵas a complex representation of G, where V=γ˜⊗(C[R]⊖1),ϵ={0if∣R∣isodd,detVif∣R∣iseven,and γ˜:G→C×is any one-dimensional representation with kerγ˜=kerγ. The representation 1⊕ϵis the permutation action of Gon the points at infinity of C. Proof When ∣R∣=1, the cohomology group Hét1(Ck¯,Ql) is zero, and the theorem is true. Assume ∣R∣≥2, so that ker(G→AutAk1)=ker(G→AutR).By Theorem 3.3 (and Remark 3.4), it suffices to prove that V⊖ϵ has rational character and that dim(V⊖ϵ)H=2genus(C/H) for every H<G. This is a purely representation-theoretic statement, proved below in Theorem 4.2. By Lemma 2.1 and Proposition 2.2, Theorem 4.2 applies here with G, R, γ˜ and with κ the hyperelliptic involution if it is in G and κ=id otherwise. Theorem 4.2 shows that V⊖ϵ has rational character, and that dim(V⊖ϵ)H={0ifhyperellipticinvolution∈H2⌊∣R∣2∣H∣−12⌋ifγ(H)=12⌊∣R∣2∣H∣⌋otherwise.By Corollary 2.5, this is precisely 2genus(C/H). The claim about 1⊕ϵ is clear when ∣R∣ is odd (as there is one point at infinity, fixed by G), and follows from Proposition 2.2 (6) and Lemma 4.9 when ∣R∣ is even.□ Theorem 4.2 Let Gbe a finite group acting on a set R, and γ:G→C×a one-dimensional representation, satisfying the following: The kernel of the action of Gon Ris generated by an element κof order 1 or 2. Write G-=G/⟨κ⟩. Then G-≃T⋊Cm, where Tis an Fp(ζm)-vector space with p∤m, and some generator of Cmacts on Tby v↦ζmv. Every g∈T⧹{id}has no fixed points on R, and every g∈G⧹{id,κ}has at most one fixed point. If κ≠idthen γ(κ)=−1. γis trivial on T, and γ2is an ∣R∣th power of some one-dimensional representation αof Cm=G-/Tof exact order m.For H<Gdefine F(H)={0ifid≠κ∈H(CaseI)2⌊∣R∣2∣H∣−12⌋ifγ(H)=1(CaseII)2⌊∣R∣2∣H∣⌋otherwise(CaseIII),and let V=γ⊗(C[R]⊖1),ϵ={0if∣R∣isodd,detVif∣R∣iseven.Then V⊖ϵis the unique representation with rational character for which dim(V⊖ϵ)H=F(H)forallH<G. (*) Proof Lemma 3.2 proves uniqueness. By Lemma 4.5(2), V⊖ϵ has rational character, and it remains to prove (*). When H=G, this is shown in Lemmas 4.6, 4.7, 4.8, in Cases I, II and III, respectively. To establish (*) for general H<G, we can invoke the lemmas with H in place for G and with R and γ restricted to H (observe that every H<G satisfies the conditions of the theorem).□ In the remainder of this section, we prove the ingredients of Theorem 4.2. Notation 4.3 Let G, G-=T⋊Cm, R, γ, κ, F, V and ϵ be as in Theorem 4.2. Write r for the number of regular orbits of G- on R, and r˜=0 if r is even and 1 if r is odd (that is the last binary digit of r). Lemma 4.4 We have (C[G-/Cm]⊖1)⊕m≃C[G-]⊖C[G-/T]as representations of G-. The representations C[G-]and C[G-/Cm]⊖1of G-are invariant under twisting by one-dimensional representations of G-/T≃Cm. For every one-dimensional representation ψof G/T, the G-representations ψ⊗C[G-]and ψ⊗(C[G-/Cm]⊖1)have rational characters. Proof (1) The group G-/T≃Cm acts on 1-dimensional characters of T by conjugation, and from the action of Cm on T, we see that every non-trivial character has trivial stabilizer. Hence, by Clifford theory (or see [8, Section 8.2]), every irreducible representation of G- is either a lift of a 1-dimensional one of G-/T, or is m-dimensional and is induced from a 1-dimensional representation of T. In particular, C[G-]⊖C[G-/T] is the sum of the m-dimensional irreducibles, each with multiplicity m. Since G-≃T⋊Cm, G-/Cm≃TasT-sets,and so the restriction ResT(C[G-/Cm]⊖1) contains every non-trivial 1-dimensional representation of T. So, by Frobenius reciprocity, C[G-/Cm]⊖1 contains every m-dimensional irreducible of G-. By comparing dimensions, each occurs with multiplicity one, and the claim follows. (2) The twist-invariance is clear for C[G-] and C[G-/T], and so follows for C[G-/Cm]⊖1 from (1). (3) Generally, for every rational character ρ of G which is invariant under Cm-twists, ψ⊗ρ is again rational. Indeed, ψ2 clearly kills κ and factors through Cm, and so ψ(g)2ρ(g)=ρ(g)forallg∈G.Thus, either ψ(g)ρ(g)=0 or ψ(g)=±1, and ψ(g)ρ(g) is rational in both cases.□ Lemma 4.5 We also have Either ∣R∣≡0modmand γ2=1; as a G--set, Ris a union of regular orbits; or ∣R∣≡1modmand γ2=α≠1; as a G--set, Ris a union of regular orbits and one non-regular orbit ≃G-/Cm. The representations Vand V⊖ϵhave rational characters. Proof (1) The G--structure of R follows from Lemma 2.3. By assumption, γ2=α∣R∣, and this is 1 or α, respectively. (2) In Case 1(a), the representation C[R]⊖1 is realizable over Q and γ has order ≤2, so V=γ⊗(C[R]⊖1) is also realizable over Q. In Case 1(b), V=γ⊗(C[R]⊖1) has rational character by Lemma 4.4(3). Finally, ϵ=0 or ϵ=detV, each of which is also rational.□ Lemma 4.6 (Case I). If κ≠id, then dim(γ⊗C[R])G=dimγG=dimϵG=0.In particular, F(G)=dim(V⊖ϵ)Gin this case. Proof Because κ≠id, γ(κ)=−1. Since κ acts trivially on C[R], both γ and γ⊗C[R] have trivial G-invariants, as does ϵ=γ∣R∣−1detC[R] when ∣R∣ is even.□ Lemma 4.7 (Case II). If κ=idand γ=1, then F(G)=r−2+r˜,dim(γ⊗C[R])G=r,dimγG=1anddimϵG=1−r˜.In particular, F(G)=dim(V⊖ϵ)Gin this case. Proof First of all, G=G- since κ=id. Next, as γ=1, R is a union of G-regular orbits by Lemma 4.5(1). Now, F(G)=2⌊r∣G∣2∣G∣−12⌋=2⌊r−12⌋={r−2,if2∣rr−1,if2∤r=r−2+r˜,dim(γ⊗C[R])G=dimC[R]G=∣R/G∣=r,dimγG=1.If ∣R∣ is even, then ϵ=detV=γ∣R∣−1detC[R]=detC[R]=(detC[G])r.Recall that for any group detC[G] is the trivial character unless G has a non-trivial cyclic 2-Sylow subgroup, in which case detC[G] is of order 2. Hence, ϵ is non-trivial if and only if r is odd (in which case ∣G∣ is even as ∣R∣ is even). So dimϵG=1−r˜, as claimed. On the other hand, if ∣R∣ is odd, then ϵ=0. As R is a union of regular G-orbits, there must be an odd number of them, so that r˜=1 and once again dimϵG=1−r˜.□ Lemma 4.8 (Case III). Suppose κ=idand γ≠1. Then F(G)=r−r˜. Ris a union of regular G-orbits if and only if ∣R∣and mare both even. We have dim(γ⊗C[R])G=r,dimγG=0anddimϵG=r˜.In particular, F(G)=dim(V⊖ϵ)Gin this case. Proof First of all, G=G-=T⋊Cm, since κ=id, and m>1 as γ≠1. By Lemma 4.5(1), R decomposes as a G-set as R=G⨿rorR=G⨿r⨿G/Cm. (1) By definition of F, F(G)=2⌊∣R∣2∣G∣⌋=2⌊r∣G∣+δ2∣G∣⌋,with δ=0 or δ=∣G/Cm∣. As δ<∣G∣, we have F(G)=2⌊r2⌋=r−r˜. (2) If m is odd, then γ≠1⇒γ2≠1, so R has an irregular orbit by Lemma 4.5(1b). If m is even, then every regular orbit has even size while ∣G/Cm∣=∣T∣ is odd, so the parity of ∣R∣ is determined by whether there is an irregular orbit. (3) Clearly dimγG=0. By Lemma 4.5, γ⊗C[G]≃C[G]andγ⊗(C[G/Cm]⊖1)=C[G/Cm]⊖1,and it follows that either γ⊗C[R]=C[G]rorγ⊗C[R]=C[G]r⊕C[G/Cm]⊕γ⊖1,depending on whether R is a union of regular orbits or not. Each C[G] summand has 1-dimensional G-invariants, and their dimensions add up to r, while dim(C[G/Cm])G+dimγG−dim1G=1+0−1=0.This proves the first claim, and it remains to show that dimϵG=r˜. Suppose ∣R∣ is even, so that ϵ=detV. If m is odd, then G has odd order while detV has rational character by Lemma 4.5(2), so ϵ=detV=1. On the other hand, R has an irregular orbit by (2), and all orbits are of odd size, so r is odd. Hence r˜=1=dimϵG. If m is even, then R is a union of regular orbits by (2), and detC[G]=η, the non-trivial character of Cm of order 2. Moreover, γ=η because γ≠1 but γ2=1 by Lemma 4.5(1). Therefore, ϵ=detV=det(γ⊗(C[R]⊖1))=γ−1⊗det(γ⊗C[G]⊕r)=γ−1⊗det(C[G]⊕r)=ηr−1,and so dimϵG=r˜. Finally suppose ∣R∣ is odd, so that ϵ=0 and we need to show that r is even. By (2), R has an irregular orbit, so γ2=α by Lemma 4.5(1), which has order m. Hence m must be odd, as G has no 1-dimensional representation of order 2m. Thus, every G-orbit has odd size, and r≡∣R∣−1mod2 is even.□ Lemma 4.9 If Ris even, then ϵ=α∣R∣/2⊗γ−1. Proof We need show that γ∣R∣−1det(C[R])=α∣R∣/2⊗γ−1. As R is even and γ2=α∣R∣ by assumption (cf. Theorem 4.2), this is equivalent to det(C[R])=α(∣R∣−1)∣R∣/2.Both sides are rational characters (that is of order 1 or 2) of G-; this is clear for det(C[R]), and follows from the fact that ∣R∣≡0,1modm for the right-hand side (Lemma 4.5), and m is the order α. Moreover, if ∣R∣≡1modm then m is odd as R is even, so both characters are trivial. Therefore we may assume that R=C[G-]⊕r, a union of r regular orbits (Lemma 4.5 again). If r is even, then the left-hand side is trivial, and so is the right-hand side, as ∣R∣/2 is a multiple of m. Finally, suppose r is odd and ∣R∣=rm∣T∣, in particular, m is even. Let η=αm/2 be the non-trivial character of order 2 of G-. Both det(C[R]) and α(∣R∣−1)∣R∣/2 are odd powers of η in this case, and the claim follows.□ 5. Descending morphisms In this section, we describe how certain morphisms descend to quotients of hyperelliptic curves. Our motivation comes from the arithmetic of hyperelliptic curves over finite and local fields, and the question of how the Frobenius automorphism acts on the quotient curve. See Section 6 for an example. Let k be a field of characteristic p>2, and let C/k be a hyperelliptic curve. Let G⊂AutkC be an affine group of automorphisms, given by g(X)=α(g)X+β(g),g(Y)=γ(g)Y(g∈G)as before, and C/G be the quotient curve given explicitly in Theorem 2.4. We say that a morphism Φ:C→Cnormalizes G if for every g∈G there is g′∈G for which gΦ=Φg′. Theorem 5.1 Suppose chark=p>2, Gdoes not contain the hyperelliptic involution, and Φ:C→Cis a morphism of the form ΦX=aXq+b,ΦY=dYqthat normalizes G, with qa power of p. Then Φ descendsto a morphism Ψ:C/G→C/G (that is πΦ=Ψπ where π:C→C/G is the quotient map) givenby Ψx=a∣G∣xq+∏g∈G(α(g)b+β(g)),Ψy={dyqifγ=1,a⌊m/2⌋∣G∣/mdyqifγ≠1,where mis the prime-to- ppart of ∣G∣, and x, yand the model for C/Gare those of Theorem2.4. Proof We may assume that k is algebraically closed. The quotient map C→C/G corresponds to a field inclusion k(C/G)=k(x,y)↪k(X,Y)=k(C). The morphism Φ preserves k(C/G)=k(C)G as it normalizes G, so it descends to a morphism Ψ:C/G→C/G. On the level of functions, Ψ is just the restriction of Φ to k(C/G). We now describe the action of Ψ on the generators x and y explicitly. Note that for every polynomial h(X), Φ·h(X)=h(aXq+b)=h((aqX+bq)q)=(h(1/q)(aqX+bq))q, (†)where h(1/q)(X) denotes the polynomial obtained from h(X) by raising every coefficient to the power 1/q. Action on x=I(X). Recall from Theorem 2.4 that x=I(X)=∏g∈Gg(X). Since it is G-invariant, so is Φ·I(X)=(I(1/q)(aqX+bq))q.This has a unique qth root, namely I(1/q)(aqX+bq), which must therefore be G-invariant as well. As it has the same degree as I(X), by Lemma 2.1(5), I(1/q)(aqX+bq)=uI(X)+vfor some u,v∈k. Comparing the leading and the constant coefficients, we see that u=a∣G∣/q and v=I(1/q)(bq). Thus, Ψ(x)=(ux+v)q=a∣G∣xq+I(b). Action on y. We have two cases: Case γ=1. Here y=Y, and so Ψ(y)=dyq. Case γ≠1. In this case, y=(IT(X)−λ)⌊m/2⌋Y, where IT(X)−λ is the unique monic polynomial of degree ∣G∣/m that is G-invariant up to scalars (see Lemma 2.1 (7)). Because Φ normalizes G, g·Φ·(IT(X)−λ)=Φ·g′·(IT(X)−λ)=scalar·Φ·(IT(X)−λ),so Φ·(IT(X)−λ) is also G-invariant up to scalars. By (†), it is a qth power, Φ·(IT(X)−λ)=(IT(1/q)(aqX+bq)−λq)q,and IT(1/q)(aqX+bq)−λq must be G-invariant up to scalars as well. But it has the same degree as IT(X), so by uniqueness, we must have Φ·(IT(X)−λ)=constant·(IT(X)−λ)q.Comparing the leading terms, we see that the constant is adegIT(X)=a∣G∣/m. Hence, Ψ·y=a⌊m/2⌋∣G∣/m(IT(X)−λ)q⌊m/2⌋dYq=a⌊m/2⌋∣G∣/mdyq.□ 6. An example To illustrate the results of this article, let us identify the local Galois representation attached to a specific hyperelliptic curve over a local field. This requires a compatibility statement for étale cohomology (for curves whose Jacobian has potentially good reduction this is (‡) below); it appears to be well known, but seems not to have been phrased in terms of explicit actions on points ([2, Section 2.3] or [3, Section 2.10], paragraph ‘Naïvely,…’). We will explain this compatibility in a forthcoming work [5]. The actual result (Section 6.2) can alternatively be obtained as in [6, Example 3.1], which bypasses (‡). 6.1. Setting Let Z/Q3 be the hyperelliptic curve of genus 3 given by Z:y2=x8+34.Let ζ be a primitive 8th root of 1 and let α=3ζ be a root of x8+34, so that the other roots are ζiα for i=1,…,7. Let F=Q3(ζ,α) be the splitting field of x8+34. It is a C4-extension of Q3 with ramification and residue degrees 2, so that in particular Q3(3)nr=Fnr, the maximal unramified extension of F. Finally, let ϕ∈Gal(Fnr/Q3(3)) be the (arithmetic) Frobenius element and let τ∈Gal(Fnr/Q3nr) be the element of order 2. Thus, ϕ gives a Frobenius element of Fnr/Q3 and τ generates its inertia group. 6.2. Result We claim that the Galois action on H1(Z)=Hét1(ZQ¯3,Ql)⊗C for l≠3 factors through Fnr/Q3 and that, with respect to a suitable basis ϕ−1↦(−3000000−−30000001+20000001−2000000−1+2000000−1−2),τ↦(10000001000000−1000000−1000000−1000000−1).In particular, the representation is tamely ramified with conductor exponent 4 and local polynomial 1+3T2, so that the Euler factor at p=3 of the L-series of Z/Q is 11+31−2s. 6.3. Galois action on the semistable model The curve Z acquires good reduction over F, since the substitution s(x)=αx,s(y)=α4yclearly transforms it to the model C/OF:y2=x8−1, which has the 8th roots of unity as roots of the right-hand side. We will write C for its special fibre C:y2=x8−1overk=F9. The group Gal(Fnr/Q3)=⟨τ,ϕ⟩≃C2×Zˆacts naturally on C(k-) (see [5]). For an element σ∈Gal(Fnr/Q3), this action is given by the composition C(k-)⟶liftC(OFnr)⟶s−1Z(Fnr)⟶σZ(Fnr)⟶sC(OFnr)⟶reduceC(k-).In our example, σ:(x,y)⟼(ασαxσ,(ασα)4yσ)modm,where m is the maximal ideal of Fnr. Writing ζ16 for the 16th root of unity =α3, we clearly have ατ=−3ζ16=−α and αϕ=ζ1633=ζα, from the definitions of τ and ϕ. In particular, τ:(x,y)↦(−x,y)andϕ:(x,y)↦(ζ-−1x3,−y3),where ζ- denotes the image of ζ in F9. Define morphisms g,Φ:C→C by the above formulae for τ,ϕ on points. By the Néron–Ogg–Shafarevich criterion, the natural Galois action on the étale cohomology group H1(Z) factors through Gal(Fnr/Q3). By [5], there is an isomorphism of Ql-vector spaces H1(Z)≃H1(C), (‡)under which the action of τ and ϕ on H1(Z) translates to the natural geometric action of g and Φ on H1(C). We are now in a position to apply the results of Sections 4 and 5 to explicitly determine the representation H1(Z). 6.4. H1(Z) as inertia representation To determine the action of the inertia group on H1(Z), we apply Theorem 4.1 to the curve C with the automorphism group G=⟨g⟩≃C2 (with the action described above) and γ˜=1. Write η for the non-trivial 1-dimensional representation of G. The roots of x8−1 come in four regular G-orbits {1,−1},{ζ-,−ζ-},{ζ-2,−ζ-2},{ζ-3,−ζ-3},so the theorem shows that, as a G-module, H1(Z)≃H1(C)≃C[G]⊕4⊖1⊖1≃1⊕2⊕η⊕4. In other words, g has eigenvalues 1 and −1 with multiplicities 2 and 4, respectively, in its action on H1(Z), as claimed. 6.5. Counting fixed points To describe H1(Z) as a full Gal(Q-3/Q3)-module, we will exploit the identifications H1(Z)≃H1(C)andH1(Z)C2≃H1(C)C2≃H1(C/C2).To be precise, first note that as Φ commutes with g, it preserves the 1- and the η-isotypical components of C2=⟨g⟩, and that H1(C) is completely determined by the eigenvalues of Φ on them (the action of Φ is known to be semisimple, although in our case this will be clear as its eigenvalues will turn out to be distinct). By (‡), the eigenvalues of ϕ on the inertia invariants H1(Z)⟨τ⟩ agree with those of Φ on H1(C)C2. These are, by Theorem 3.1(2) and (3), the eigenvalues of Ψ on H1(C/C2), where Ψ is the induced morphism on C/C2. By Theorem 2.4, the quotient C/C2 is the genus 1 curve C/C2:y2=(x+1)(x+ζ-2)(x+ζ-4)(x+ζ-6)=x4−1and, by Theorem 5.1, Φ descends to Ψ:C/C2⟶C/C2(x,y)⟼(−ζ-2x3,−y3).From the Lefschetz fixed point formula, the inverse characteristic polynomial of Ψ on H1(C/C2) is det(1−Ψ−1T∣H1(C/C2))=1−aT+3T2for some a∈Z, and its value at T=1 is the number of fixed points of Ψ on the curve. To find it explicitly, first count F-3-solutions to the system y2=x4−1,x=−ζ-2x3,y=−y3.Starting from the last equation, y=0⟹x4=1,x=−ζ-2x3⟹nosolutions;y=±ζ-2⟹x4=ζ-4+1=0,x=−ζ-2x3⟹x=0,y=±ζ-2.Finally, to see the action on the points at infinity ∞±, let s=1x,t=yx2. The equation of the curve becomes t2=1−s4,∞±=(0,±1),and the transformation Ψ:(x,y)↦(−ζ-2x3,−y3) on this chart is s=1x↦−1ζ-2x3=ζ-2s3,t=yx2↦−y3(−ζ-2x3)2=y3x6=t3.It fixes both ∞+ and ∞−. Overall, Ψ has four fixed points, and its inverse characteristic polynomial on H1(C/C2) is, therefore, 1+3T2. Hence the eigenvalues of Φ on this subspace are ±1−3, as claimed. Similarly, counting ai=the number of fixed points of Φi on C itself, we find the sequence to be (4,20,28,92,244,692,…). Thus, by the Lefschetz fixed point formula again, the inverse characteristic polynomial of Φ on the full space H1(C) is exp(∑i≥1aiiTi)(1−T)(1−3T)=(1+3T2)(1+2T2+9T4).The first factor lives, as we have seen, on the 1-component of C2=⟨g⟩, and so the second factor lives on the η-component. In other words, the eigenvalues of Φ on the −1-eigenspace of g on H1(C) are ±11±2, as claimed. Funding This research is partially supported by EPSRC Grants EP/M016838/1 and EP/M016846/1. All relevant data are contained within the article; no supporting data are held elsewhere. The second author is supported by a Royal Society University Research Fellowship. References 1 R. Brandt and H. Stichtenoth, Die Automorphismgruppen hyperelliptischer Kurven, Manuscripta Math. 55 ( 1986), 83– 92. Google Scholar CrossRef Search ADS 2 I. Bouw and S. 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Madan, A Hauptsatz of L. E. Dickson and Artin–Schreier extensions, J. Reine und Angew. Math. (Crelle) 318 ( 1980), 156– 177. 11 A. Weil, Sur les courbes algébriques et les variétés qui s’en déduisent , Hermann, Paris, 1948. © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

The Quarterly Journal of Mathematics – Oxford University Press

**Published: ** Feb 8, 2018

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