5. Implementations
5.2. Computing the absolute Frobenius action in larger characteristic
We have implemented Algorithm 2 as a package for Magma [BCP97]. It is available under a GPL license at github.com/mminzlaff/superelliptic. The package provides the intrinsicsAbsoluteFrobeniusAction(a,h,N)and ZetaFunction(a,h). The former corresponds to Algorithm 2, while the latter uses the absolute Frobenius action to compute the zeta function according to (the proof of) Corollary 4.1. The input consists of an integer a and a squarefree polynomialh that together define the superelliptic curveya−h(x) = 0. The desired precision is entered via the positive integerN. In case of ZetaFunction(a,h), a sufficient precision to compute the zeta function is deduced automatically. Here is one full example for how to use the package when Magma is started from the directory where the package resides:
1 > A t t a c h S p e c (" s u p e r e l l i p t i c ");
2 > F := F i n i t e F i e l d ( 4 0 9 9 ) ;
3 > FPol < x > := P o l y n o m i a l R i n g ( F );
4 > a := 4;
5 > h := x ^ 5 + 1 ;
6 > Z e t a F u n c t i o n ( a , h );
7 ( 4 7 4 3 1 5 7 1 0 6 2 0 3 3 3 2 1 2 5 4 0 1 * t ^12 + 6 9 4 2 8 9 8 9 1 1 2 5 1 5 2 2 9 9 4 * t ^10 +
8 4 2 3 4 5 0 7 7 5 2 6 5 4 0 1 5 * t ^8 + 1 3 7 7 4 1 1 6 4 5 9 8 0 * t ^6 +
9 2 5 2 0 2 7 0 1 5 * t ^4 + 2 4 5 9 4 * t ^2 + 1 ) / ( 4 0 9 9 * t ^2 - 4 1 0 0 * t + 1) Table 5.1 lists running times of our implementation when applied to the genus 3 nonhyperelliptic curve y3−(x4−37x3+ 2x2−102x+ 21) = 0. The absolute times should not be taken as a measure of the efficiency of the algorithm as we paid little attention to optimisation. There is, however, a separate column stating the increase in runtime for each quadrupling of the characteristic. Note that one expects a factor of roughly 2 by Theorem C. The table was calculated with version 1.0 of the package on a PC with Magma 2.15-8 and 2 gigabyte of RAM.
5.2. COMPUTING THE ABSOLUTE FROBENIUS ACTION IN LARGER CHARACTERISTIC71
p time time increase p time time increase
214+ 27 8.2 sec -.- 226+ 15 10.9 min 1.7 216+ 1 15.1 sec 1.8 228+ 3 21.9 min 2.0 218+ 3 30.3 sec 2.0 230+ 3 53.7 min 2.5
220+ 7 1.3 min 2.6 232+ 15 1.9 h 2.1
222+ 15 3.1 min 2.4 234+ 25 4.0 h 2.1
224+ 43 6.4 min 2.1 236+ 31 9.4 h 2.4
Table 5.1. Runtimes of our implementation of Algorithm 2 with varying primespand fixed inputa=3,h=x^4 -37x^3 +2x^2 -102x +21, andN=3.
APPENDIX A
CURVES OVER COMPLETE DISCRETE VALUATION RINGS
For this appendix, we redefine k and K. Let Λ be a complete discrete valuation ring with fraction field K and perfect residue field k. Fix a generatort of the maximal ideal ofΛ. SetS= Spec Λ.df
A.1. Importing properties from the special fibre
Lemma ([Gro65, Prop. 4.6.1 & Prop. 6.7.7]). — Letf:X →Speck be a locally of finite type morphism of schemes. IfX is reduced/normal/regular, then X is geomet-rically reduced/normal/regular.(1)
Lemma ([Gro66, Thm. 12.2.1 & Thm. 12.2.4]). — Let f: X → S be proper and flat morphism of schemes. If Xk is geometrically reduced/integral/normal/regular, then so isXK. IfXk is equidimensional of dimensionn, then so is XK.
Lemma. — Let X be a scheme andf:X →S be proper and flat. IfXk is a curve overk, thenX is a reduced curve overS andf is equidimensional. IfXk is geomet-rically integral/normal/regular, then so isX.
Proof. — By the above lemmas, it remains to show (1) thatX is reduced, (2) thatf is locally equidimensional [Gro66, Def. 13.3.2], [Gro67, Err. IV.35], and (3) thatX is geometrically integral/normal/regular, wheneverXk is.
Ad (2): LetU be an open neighbourhood ofx∈X. Sincef is flat, every irreducible component dominatesS. Moreover, every irreducible component of U ∩f−1(f(x0)) has the same dimension for allx0 ∈X since the fibres off are equidimensional of the same dimension. Hencef is locally equidimensional [Gro66, Prop. 13.3.1].
Ad (1), (3): We already know that the respective properties hold on the open sub-scheme XK of X. It remains to consider the points x ∈ Xk. Since f is flat, it is
(1)Geometric regularity means that the geometric fibres are regular, i.e. X is smooth overk.
universally open [Gro65, Thm. 2.4.6]. Hence, ifXk is geometrically reduced/integral at x, then so is X [Gro66, Prop. 15.3.1 & Cor. 15.3.3]. Finally, a flat, locally of finite type scheme over Λ with geometrically normal/regular fibres is geometrically normal/regular itself [Gro65, Cor. 6.5.2 & Cor 6.5.4].
For the remainder of this appendix, letX be a curve overΛ.
A.2. Picard group
Let s ∈ S. The morphism ρk(s):Xk(s) = X ×Speck(s) → X induces a group homomorphism
ρ∗k(s): Pic(X)→Pic(Xk(s)).
The image of an invertible sheafL onX is denoted byLk(s)and called therestriction ofL toXk(s). Recall that the degree ofLk(s)isχ(Lk(s))−χ(OXk(s)). By invariance of the Euler-Poincar´e characteristic [Liu06, Prop. 5.3.28],
degLK = degLk.
The open immersionρK induces a group morphism Div(X)→Div(XK). On the level of divisor classes, this maps sits in a commutative diagram
CaCl(X) _ ////
CaCl(X _ K)
Pic(X) ρ∗K
////Pic(XK)
Let us now consider the closed immersion ρk. A divisor Z on X is relative (to S) if every irreducible component of Z (as subscheme of X) is flat over S.(2) Equiva-lently, a relative divisor does not contain any generic point of Xk. Let G(Xk/X) be the group of relative divisors. The map ρk then induces a group morphism ρ∗k: G(Xk/X) → Div(Xk). It is related to ρ∗k: Pic(X) → Pic(Xk) by the canon-ical isomorphism OXk(ρ∗k(Z)) ∼= ρ∗kOX(Z) [Liu06, Lem. 7.1.29]. So there is the commutative diagram
G(Xk/X) //
Div(Xk)
Pic(X) ρ∗k //Pic(Xk)
Lemma. — IfX is smooth over S, thenρ∗k: Pic(X)→Pic(Xk)is onto.
Proof. — By the above it is enough to show thatρ∗k:G(Xk/X)→Div(Xk) is onto.
Since X is smooth, we may also identify divisors with Weil divisors, i.e. cycles of codimension 1. So let P be a closed point of Xk and consider it as divisor on Xk.
(2)Liu calls these divisorshorizontal[Liu06, Def. 8.3.5].
A.4. ACYCLICITY 75
There exists a closed pointPK ofXK of the same degree asP [Liu06, Prop. 10.1.40].
Its topological closure in X is{PK, P} and defines a relative divisorP onX. Since Suppρ∗k(P) = SuppP∩Xk, the image ofP isP.
A.3. Cohomological flatness
Lemma. — Lets∈S. For any quasi-coherent sheafF onX that is flat overS and for all i∈N, there is a natural embeddingHi(F)⊗k(s),→Hi(Fk(s))of k(s)-spaces.
In case sis the generic point, the map is an isomorphism.
Proof. — The map is defined in [Liu06, Lem. 5.2.26]. Injectivity follows from [Liu06, Lem. 5.3.19]. Surjectivity in casek(s) =K is [Liu06, Cor. 5.2.27].
An invertible sheafL onXiscohomologically flatif the inclusion Hi(L)⊗k ,→Hi(Lk) is an isomorphism for alli∈N.
Theorem ([Liu06, Thm. 5.3.20 & Rem. 5.3.30]). — Let L be an invertible sheaf onX. For all i∈N the following holds.
(1) hi(LK)≤hi(Lk).
(2) hi(LK) = hi(Lk)if and only if Hi(L)is free andHi(L)⊗k ,→Hi(Lk)is an isomorphism.
(3) Hi(L)⊗k ,→Hi(Lk)is an isomorphism if and only if Hi+1(L)is free.
Since the fibres ofX are 1-dimensional, Hi(F) vanishes for any quasi-coherent sheaf on X whenever i is at least 2 [Liu06, Rem. 5.2.25]. Also, by duality H0(L) is free [Liu06, Rem. 6.4.21]. Hence the above says thatL is cohomologically flat if and only if H1(L) is free, or equivalently if hi(LK) = hi(Lk) fori= 0 ori= 1.
Proposition. — The sheafOX is cohomologically flat. IfX is smooth overS, then ΩX is cohomologically flat. In particular, the genera of the fibres ofX coincide.
Proof. — Since X is proper and flat over S, OX is cohomologically flat [Gro63, Prop. 7.8.6]. IfX is smooth, then ΩX is the dualising sheaf [Liu06, Thm. 6.4.32].
Cohomological flatness thus follows from the one ofOX[Liu06, Rem. 6.4.21]. There-fore, the genera h0(ΩXK) and h0(ΩXk) coincide.
A.4. Acyclicity
A quasi-coherent sheaf F (of abelian groups) on X (or one of its fibres) is called acyclic ornonspecial if Hi(F) vanishes wheneveriis greater than 0. Since Hi(F) is zero wheneveriis greater than 1,F is acyclic if and only if H1(F) is trivial.
IfF is flat and coherent overS, thenF is acyclic wheneverFk is. This is a conse-quence of the existence of the inclusion Hi(L)⊗k ,→Hi(Lk) and Nakayama’s lemma.
In particular, ifXk is of genusg, then Riemann-Roch says that an invertible sheaf is acyclic whenever its degree is at least 2g−1.
A.5. Strict relative normal crossing divisors
Definition. — Assume that X is smooth over S. A reduced divisor Z on X has strict relative normal crossings and is called astrict relative normal crossing divisor ifXcan be covered by open affine subschemesUsuch that eachU is ´etale overA1W(k) andZ|U is either empty or the fibre product ofU and the coordinate hyperplane. In other words, there exists an ´etale mapW(k)[x]→OX(U) that mapsxtotsuch that OX(−Z)(U) is eitherOX(U) ortOX(U).
Lemma. — Assume thatX is smooth overS. A reduced divisor Z on X has strict relative normal crossings if and only if it is a relative divisor and SuppZK and SuppZk have the same cardinality.
Proof. — Since X is smooth, we can cover the curve with finitely many open affines Ui = SpecAi such that each Ui is ´etale over A1Λ, i.e. there are ´etale maps ϕi: Λ[x] →Ai. The prime divisors of X are the irreducible components ofXk and closures inX of closed point onXK [Liu06, Prop. 8.3.4]. We writeZi forZ|Ui. Assume thatZ is a relative divisor and that the supports of its fibres have the same cardinality. ThenZ does not contain a generic point ofXk. We may therefore shrink the Ui so that each contains at most one point ofZk. If Zi is empty, then there is nothing to show. If Zi is not empty, then there is a unique irreducible component of Z that intersects Ui. So OX(−Z)(Ui) is a prime ideal of height 1 in Ai. Since ϕi is ´etale, the preimage of OX(−Z)(Ui) is a prime ideal also of height 1. Since Λ[x] is normal, the preimage is a principal ideal, generated by, say, fi. The map Λ[y] →Λ[x] that sends y tofi is ´etale atfiΛ[x]. Therefore, shrinking theUi again, we find Ui → Spec Λ[x] → Spec Λ[y] ´etale and Zi is the pullback of the coordinate hyperplaney= 0.
Assume thatZ is a strict relative normal crossing divisor. SinceUi→A1S is ´etale, it is quasi-finite [Liu06, Prop. 4.3.23]. In particular, its restriction toZi is quasi-finite and soZ is a relative divisor. Shrinking theUi, we may assume that each contains at most one point of Zk. Fix aUi whereZi is nonempty. LetP be the unique point in the support ofZi. AtP, the divisorZkcorresponds to the idealOXk(−Zk)P ofOXk,P. By assumption, this ideal is generated by the image ofxunder the mapk[x]→Ai⊗k.
Moreover, sincek[x]→Ai⊗k is ´etale,OXk(−Zk)P is maximal. If there existed two irreducible components of Z both with closed point P, then OXk(−Zk)P would not be radical and in particular not maximal. Hence the supports of the fibres ofZ have the same cardinality.
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