Lifting low-gonal curves for use in Tuitman's algorithm
aa r X i v : . [ m a t h . N T ] S e p LIFTING LOW-GONAL CURVES FOR USE IN TUITMAN’SALGORITHM
WOUTER CASTRYCK AND FLORIS VERMEULEN
Abstract.
Consider a smooth projective curve C over a finite field F q , equipped witha simply branched morphism C → P of degree d ≤
5. Assume char F q > d ≤
4, andchar F q > d = 5. In this paper we describe how to efficiently compute a lift of C tocharacteristic zero, such that it can be fed as input to Tuitman’s algorithm for computingthe Hasse–Weil zeta function of C/ F q . Our method relies on the parametrizations of lowrank rings due to Delone–Faddeev and Bhargava. Introduction
About 20 years ago, Kedlaya published an influential paper [21], showing how one canemploy Monsky–Washnitzer cohomology to efficiently compute Hasse–Weil zeta functionsof hyperelliptic curves over finite fields having small odd characteristic. Its many follow-upworks include several generalizations to geometrically larger classes of curves, first to su-perelliptic curves [17], then to C ab curves [12] and then further to non-degenerate curves [6],i.e., smooth curves in toric surfaces. A more significant step was taken in 2016, when Tuit-man [27, 28] published a Kedlaya-style algorithm that potentially covers arbitrary curves,and at the same time beats the methods from [6, 12] in terms of efficiency. Unfortunately,the user of Tuitman’s algorithm is expected to provide a lift of the input curve to character-istic zero that meets the technical requirements from [28, Ass. 1]. Beyond non-degeneratecurves, this is a non-trivial task. As a result, the exact range of applicability of Tuitman’smethod remains unclear.A partial approach to lifting curves having gonality at most four was sketched in [7],with concrete details being limited to curves of genus five. In the current paper we presenta different method, which is faster, works for curves of gonality at most five, and is mucheasier to implement. Concretely, we assume that we are given an absolutely irreduciblecurve over a finite field F q of characteristic p >
2, defined by a polynomial of the form(1) f d ( x ) y d + f d − ( x ) y d − + . . . + f ( x ) ∈ F q [ x, y ]for some d ≤
5. Moreover, the morphism ϕ from its non-singular projective model C to theprojective line, induced by ( x, y ) x , is assumed to be simply branched of degree d ; inother words, all fibers of ϕ should consist of either d − d geometric points. Finally, if d = 5 then it is assumed that p >
3. Then our method efficiently produces a lift satisfyingthe main requirement from [28, Ass. 1], which therefore can be fed as input to Tuitman’salgorithm, modulo Heuristic H discussed below.In terms of moduli, the locus of genus g curves admitting a simply branched morphism to P of degree at most 5 has dimension min { g + 5 , g − } by a result of Segre [26]. For g = 6and g ≥ g ≤ WOUTER CASTRYCK AND FLORIS VERMEULEN
Remark . Expecting our curve to be given in the form (1) is essentially equivalent toassuming knowledge of an F q -rational degree d morphism C → P that is simply branched,in contrast with the assumptions from [7]. If such a morphism to P exists but is not known,then one can try to resort to methods due to Schicho–Schreyer–Weimann [23] or Derickx [13, § Lifting strategy.
Write q = p n and fix a degree n number field K in which p is inert.Let O K denote its ring of integers and identify F q with O K / ( p ). To lift the curve C meansto produce a non-singular projective curve C/K whose reduction mod p is isomorphic to C/ F q ; necessarily, the genus of C should be equal to that of C . Our actual goal is to liftthe morphism ϕ , which means that we want to equip C with a morphism ϕ : C → P reducing to ϕ : C → P mod p , up to isomorphism. Our approach to solving this problemis based on the parametrization of low rank rings by Delone and Faddeev [16, Prop. 4.2],and Bhargava [2, 3], in combination with algorithms due to Hess for computing reducedbases [20]. In doing so, we will find concrete, typically non-planar equations for C over F q that have “free coefficients”, which can be lifted to O K naively, in order to obtain anon-singular projective curve C/K along with a morphism ϕ : C → P of the said kind. Werefer to Section 2 for a more elaborate discussion. Remark . In general, the polynomial (1), which defines a plane curve that is birationallyequivalent with C , is not liftable directly: there may be many singularities, which typicallydisappear when lifting the coefficients of (1) naively to O K , causing an increase of the genus. Remark . In Kedlaya’s original algorithm, corresponding to the case d = 2, an implicitfirst step is to rewrite (1) into Weierstrass form. Indeed, Weierstrass models have “freecoefficients” that can be lifted naively to O K , always resulting in a hyperelliptic curve over K having the same genus. From now on we assume d ≥ f d ( x ) y d + f d − ( x ) y d − + . . . + f ( x ) = 0, for polynomials f i ∈ O K [ x ] which, in general,do not reduce to f i mod p ; here, the lifted morphism ϕ again corresponds to ( x, y ) x .The change of variables y ← y/f d ( x ) yields a monic defining equation(2) Q ( x, y ) = y d + f d − ( x ) y d − + . . . + f ( x ) f d ( x ) d − , having the right shape to serve as input for Tuitman’s algorithm. All subsequent arithmeticin Tuitman’s algorithm is done in the p -adic completion Z q of O K (or rather its fractionfield Q q ), up to some finite p -adic precision. But for the lifting step it suffices to work over O K , and this has some implementation-technical advantages [7, Rmk. 2]. On Tuitman’s assumption.
Let us discuss the specific requirements from [28, Ass. 1] inmore detail. A first assumption concerns the polynomial r ( x ) = ∆ / gcd(∆ , d ∆ /dx ) with ∆the discriminant of (2), when viewed as a polynomial in y over O K [ x ]:(a) the discriminant of r ( x ) is a unit in Z q .Next, consider the ring R = Z q [ x, /r, y ] / ( Q ) and write Q q ( x, y ) for the field of fractions of R ⊗ Q q and F q ( x, y ) for the field of fractions of R ⊗ F q . A second assumption is that weknow explicit matrices W ∈ GL d ( Z q [ x, /r ]) and W ∞ ∈ GL d ( Z q [ x ± , /r ])such that, if we write b j, = P d − i =0 ( W ) i +1 ,j +1 y i and b j, ∞ = P d − i =0 ( W ∞ ) i +1 ,j +1 y i , then: Lifting a ∈ F q \ { } naively to O K means: producing whatever element a ∈ O K such that a mod p = a . IFTING LOW-GONAL CURVES FOR USE IN TUITMAN’S ALGORITHM 3 (b) { b , , . . . , b d − , } is an integral basis for Q q ( x, y ) over Q q [ x ] and its reduction mod p is an integral basis for F q ( x, y ) over F q [ x ],(c) { b , ∞ , . . . , b d − , ∞ } is an integral basis for Q q ( x, y ) over Q q [ x − ] and its reductionmod p is an integral basis for F q ( x, y ) over F q [ x − ].Finally, writing R = Z q [ x ] b , + . . . + Z q [ x ] b d − , and R ∞ = Z q [ x − ] b , ∞ + . . . + Z q [ x − ] b d − , ∞ , it is assumed that(d) the discriminants of the finite Z q -algebras ( R / ( r )) red and ( R ∞ / (1 /x )) red are units.Here the subscript ‘red’ means that we consider the reduced ring obtained by quotientingout the nilradical. The geometric meaning of assumptions (a) and (d) is discussed in [28, Prop. 2.3]; seealso [27, Rmk. 2.3]. They express that all branch points of ϕ : C → P , as well as all pointslying over these branch points, should be distinct mod p . In our context, these propertiesare automatic. Indeed, since p > ϕ : C → P is simply branched, there is no wildramification, hence the ramification divisor of ϕ reduces mod p to that of ϕ . Thus, againbecause ϕ is simply branched, we see that the ramification points of ϕ must reduce to2 g + 2 d − ϕ , as wanted; here g denotes thegenus of C . We also see that ϕ is simply branched as well.Assumptions (b) and (c), on the other hand, ask for an explicit description of our lift ϕ : C → P in terms of two affine patches ϕ − ( P \{∞} ) and ϕ − ( P \{ } ), glued together using W = W − W ∞ , that is compatible with reduction mod p . In Tuitman’s own pcc p and pcc q code, the matrices W and W ∞ are found by computing integral bases for the function fieldextension K ( x ) ⊆ K ( C ) defined by (2), using the Magma intrinsic MaximalOrderFinite() ,and hoping that these have good reduction mod p . There is a non-zero probability thatthis approach fails, in which case Tuitman’s code outputs “bad model for curve”, but inpractice this probability become negligible very rapidly as q grows; see the tables in [7]. Wetherefore content ourselves with relying on the same bet, which we call Heuristic H: Definition 1.4 (informal) . The output (2) satisfies
Heuristic H if the associated integralbases of K ( C ) over K [ x ] and K [ x − ], computed using Magma as in Tuitman’s implemen-tation, meet the requirements from [28, Ass. 1].Of course, if through some other method one manages to find integral bases with goodreduction, then this would by-pass Heuristic H. In particular, if d = 3 then, as explained inRemark 3.4, such integral bases can be extracted as by-products of our lifting procedure. Combined runtime.
The running time of our lifting procedure is strongly dominated bythat of Tuitman’s algorithm, as should be clear from the discussions in Sections 3, 4 and 5below. We will therefore omit a detailed analysis, although it is crucial to note that liftingdoes not inflate the input size too badly. Concretely, if we let δ = max ≤ i ≤ d deg f i , then • the reader can check that all f i ’s are of degree O ( g ), which in turn is O ( δ ) thanksto Baker’s bound [1, Thm. 2.4], • when lifting coefficients from F q to O K naively, we can choose them to be of bit size O ( n log q ), and as a result the same asymptotic estimate applies to the size of thecoefficients of the f i ’s, This takes into account the erratum pointed out in https://jtuitman.github.io/erratum.pdf . https://github.com/jtuitman/pcc , see mat W0() and mat Winf() in coho p.m and coho q.m . WOUTER CASTRYCK AND FLORIS VERMEULEN • as discussed in [28, p. 313-314], the matrices W , W ∞ produced by the Magma in-trinsic, as well as their inverses, involve K ( x )-coefficients whose pole orders are in O ( δ ), as required by [28, Ass. 2]; for d = 3, the reader can check that the samebound applies to the integral bases from Remark 3.4.From [28, Thm. 4.10] it follows that e O ( pδ n ) bit operations suffice for computing the Hasse–Weil zeta function of any curve C/ F q of the form (1), where we recall our dependence onHeuristic H if d = 4 , Practical performance.
This paper comes with an implementation of our lifting proce-dure in Magma [4], which can be found at https://homes.esat.kuleuven.be/~wcastryc/ .Appendix A reports on how the code performs in combination with Tuitman’s implemen-tation for computing Hasse–Weil zeta functions. As discussed there, this gives satisfactoryresults for d = 3 and d = 4, leading to a substantial enlargement of the class of curvesadmitting fast computation of their zeta function (over finite fields with small odd charac-teristic). In degree d = 5 the combined code is considerably slower. This is almost entirelydue to the seemingly harmless “elimination of variables” step, which is needed to put thelifted curve C/K in the form (2) and which produces large hidden constants in the above O ( g ) and O ( n log q ) estimates. Nevertheless, here too, it is practically feasible to computezeta functions in a non-trivial range. Tracks for future work.
Besides mitigating the effect of variable elimination and gettingrid of Heuristic H, a challenging goal is to dispose of the conditions on p and of the conditionthat ϕ is simply branched. This seems to require changes to Tuitman’s algorithm that aresimilar to how Denef and Vercauteren managed to make Kedlaya’s algorithm work in evencharacteristic [11]. Also, as explained in Section 2, our naive lifting strategy using “freecoefficients” is closely related to Schreyer’s proof [24, Cor. 6.8] of the unirationality of H g,d ,the moduli space of simply branched degree d covers of P by curves of genus g , for d ≤ d ≥
7, where there is no hope for ourstrategy to work. This leaves d = 6 as an interesting open case, on which several partial(positive) results have been proved by Geiss [19], see [25, Fig. 1] for an overview. It seemsworth investigating how Geiss’ results combine with our approach. Acknowledgements.
We thank Jan Tuitman and Yongqiang Zhao for several inspiringconversations, and the anonymous reviewers for their many helpful comments. This workis supported by CyberSecurity Research Flanders with reference VR20192203 and by KULeuven with references C14/17/083 and C14/18/067.2.
Preliminaries
Reduced bases and Maroni invariants.
Let k be any field, which in the next sectionswill be specialized to k = F q and/or k = K . Consider a non-singular projective curve C/k of genus g , along with a k -rational degree d morphism ϕ : C → P . Consider the inclusion offunction fields k ( x ) ⊆ k ( C ) corresponding to ϕ . Let k [ C ] , resp. k [ C ] ∞ , denote the integralclosure of k [ x ], resp. k [1 /x ], inside k ( C ). Theorem 2.1.
There exist unique negative integers r ≥ r ≥ . . . ≥ r d − for which thereis a basis , α , . . . , α d − of k [ C ] over k [ x ] such that , x r α , . . . , x r d − α d − is a basis of k [ C ] ∞ over k [1 /x ] . See [20] for a proof; it is standard to call e i = − r i − Maroni invariants of C withrespect to ϕ (e.g., if ϕ is a degree 2 cover, then there is just one Maroni invariant, namely IFTING LOW-GONAL CURVES FOR USE IN TUITMAN’S ALGORITHM 5 g − , α , . . . , α d − is called a reduced basis . In our cases of interest,the integers r i and an accompanying reduced basis can be computed efficiently: if k is afinite field or a number field, then the Magma command ShortBasis() takes care of this.
Remark . In more geometric language, the integers r i are characterized by the sheafdecomposition ϕ ∗ O C ∼ = O P ⊕ O P ( r ) ⊕ O P ( r ) ⊕ . . . ⊕ O P ( r d − ) which, according to atheorem due to Grothendieck, is indeed unique. As a consequence to the Riemann–Rochtheorem, the Maroni invariants satisfy the following basic properties: (i) − ≤ e ≤ e ≤ . . . ≤ e d − , (ii) e + e + . . . + e d − = g − d + 1, and (iii) e d − ≤ (2 g − /d . Models with “free coefficients”.
As mentioned in the introduction, every cover ϕ : C → P of degree 3 ≤ d ≤ F q to O K . This follows from Schreyer’s proof [24, Cor. 6.8] ofthe unirationality of H g,d for d ≤
5. The natural ambient space for this model is a rationalnormal scroll , which can be obtained by gluing together( P \ {∞} ) × P d − and ( P \ { } ) × P d − in a non-standard way; the gluing depends on the Maroni invariants e , . . . , e d − of C withrespect to ϕ . We refer to [14, 24] for more details on this construction, as well as on theclaims below. For the sake of conciseness we only describe what the model looks like on theleft copy A × P d − , which we equip with coordinates x, Y , . . . , Y d − .First assume that d = 3. Then C admits a defining equation of the form(3) X l + l =3 f l ,l ( x ) Y l Y l = 0with deg f l ,l ≤ l e + l e +4 − g , such that ϕ corresponds to projection on the x -coordinate.Conversely, every irreducible polynomial of the form (3) defines a curve having genus at most g ; this can also be seen using Baker’s bound [1, Thm. 2.4], because the dehomogenizationwith respect to Y is supported on the polygon from Figure 2.1. If equality holds then this (0 ,
0) (2 e − e + 2 , e − e + 2 , , Figure 2.1.
Polygon describing covers of degree 3. polynomial defines a non-singular projective curve (on the entire rational normal scroll) andprojection on the x -coordinate yields a degree 3 morphism to P whose associated Maroniinvariants are e , e .Next, assume that d = 4. Then C arises as the intersection of two surfaces defined by(4) X l + l + l =2 f i,l ,l ,l ( x ) Y l Y l Y l = 0for i = 1 ,
2, where deg f i,l ,l ,l ≤ l e + l e + l e − b i for unique integers − ≤ b ≤ b with b + b = g −
5, called the Schreyer invariants of C with respect to ϕ . Conversely,every irreducible such intersection defines a curve of genus at most g ; this too can beseen using (a three-dimensional version of) Baker’s bound [22, Thm. 1], by noting that the WOUTER CASTRYCK AND FLORIS VERMEULEN (0 , ,
2) (0 , , , ,
0) (2 e − b i , ,
0) (2 e − b i , , e − b i , , Figure 2.2.
Polytope describing covers of degree 4. dehomogenizations with respect to Y are supported on the polytopes from Figure 2.2. Ifequality holds then it concerns a non-singular projective curve, and projection on the x -coordinate defines a degree 4 morphism to P with associated Maroni invariants e , e , e and Schreyer invariants b , b .Finally, assume d = 5, which comes with five Schreyer invariants b ≤ . . . ≤ b summingup to 2 g −
12. In this case C can be viewed as the intersection of five hypersurfaces, whichare all obtained from a single 5 × M over k [ x ][ Y , Y , Y , Y ] whose( i, j )-th entry is of the form(5) M ,i,j ( x ) Y + M ,i,j ( x ) Y + M ,i,j ( x ) Y + M ,i,j ( x ) Y with M r,i,j ( x ) ∈ k [ x ] of degree at most e r + b i + b j + 6 − g . More precisely, our hypersurfacesare cut out by the five 4 × of M . Conversely, whenever the 4 × g . If equality holds then itconcerns a non-singular projective curve, and projection on the x -coordinate defines a degree5 morphism to P with Maroni invariants e , e , e , e and Schreyer invariants b , b , b , b , b . Lifting strategy revisited.
In the next sections we show how results on ring parametriza-tions due to Delone–Faddeev [16, Prop. 2.4] and Bhargava [2, 3] can be used to efficientlyproduce such a “free coefficient” model for our input curve C/ F q . Then, by the abovediscussion, and using that the genus cannot increase under reduction mod p , any naivecoefficient-wise lift of this model to O K will define a non-singular projective curve C/K along with a morphism ϕ : C → P lifting C and ϕ . Remark . From a non-algorithmic viewpoint, the fact that the Delone–Faddeev andBhargava correspondences produce non-singular curves in rational normal scrolls might havebeen known to some specialists (e.g., for d = 3 this can be read in Zhao’s Ph.D. thesis [30]).3. Lifting curves in degree d = 3For R a PID, we recall that a ring of rank d over R is a commutative R -algebra whichis free of rank d as a module over R . Every ring S of rank d over R admits an R -basisof the form 1 , α , ..., α d − . This can be seen by applying the structure theorem for finitelygenerated free modules over PIDs to the submodule R · S . Parametrizing cubic rings.
Let R be a PID. Cubic rings over R admit a parametrizationusing binary cubic forms over R , considered modulo a natural action by GL ( R ): for anelement A = (cid:18) a bc d (cid:19) ∈ GL ( R ) , The square roots of the determinants of the five 4 × IFTING LOW-GONAL CURVES FOR USE IN TUITMAN’S ALGORITHM 7 and f = f Y + f Y Y + f Y Y + f Y a cubic form over R , we let A ∗ f ( Y , Y ) = 1det A f ( aY + cY , bY + dY ) . Theorem 3.1 (Delone–Faddeev) . There is a canonical bijection between the set of cubic R -rings up to isomorphism and binary cubic forms over R , modulo the action of GL ( R ) . For a proof, see e.g. [16, Prop. 4.2]. For use below we briefly describe how this bijectionis constructed. Let S be a cubic R -ring with basis 1 , α , α . By adding elements of 1 · R to α and α we can assume that α α is in R . We call such bases normal . Now write out themultiplication table of S :(6) α α = − g ,α = − g + f α − f α ,α = − g + f α − f α . By associativity of S we have α · α = α · ( α α ) and α · α = ( α α ) · α . This gives(7) g = f f ,g = f f ,g = f f , so the g i are determined by the f i . One then associates to S the cubic form f = f Y + f Y Y + f Y Y + f Y . Conversely, given such a form f , associate to this the cubicring, formally equipped with basis 1 , α , α and multiplication defined by (6) and (7). TheGL ( R )-action on cubic forms corresponds precisely to changing one normal basis to anotheron the level of cubic rings. Remark . A cubic form f = f Y + f Y Y + f Y Y + f Y is irreducible if and onlyif its associated cubic R -ring is a domain. In this case, we may describe it as the subring ofFrac (cid:18) R [ y ]( f y + f y + f y + f ) (cid:19) generated by 1 , α = f y, α = − f y − = f y + f y + f . This point of view is especiallynice when R = k [ x ] for some field k . Indeed, then f ( y,
1) = 0 defines a curve in A over k and the cubic ring associated to f has as its field of fractions the function field of this curve. Lifting degree covers. Consider the function field F q ( C ) = Frac (cid:18) F q [ x, y ]( f y + f y + f y + f ) (cid:19) defined by our input polynomial, and consider the integral closure F q [ C ] of F q [ x ] insideit; this is a cubic F q [ x ]-ring. Let e , e be the Maroni invariants of C with respect to ϕ and let 1 , α , α be a corresponding reduced basis. After adding to α and α elements of F q [ x ] we may assume that this basis is normal. In more detail, if α α = aα + bα + c ,for a, b, c ∈ F q [ x ], then we replace α by α − b and α by α − a . This operation will notchange the fact that the basis is reduced. Applying the Delone–Faddeev correspondence tothis basis produces a new cubic form f ( Y , Y ) = f Y + f Y Y + f Y Y + f Y whose coefficients we, abusingly, again denote by f i . WOUTER CASTRYCK AND FLORIS VERMEULEN
Lemma 3.3.
Let f be obtained through the Delone–Faddeev correspondence as above. Thenthis is a model for C of the form (3) .Proof. Note that the curve f ( y,
1) = 0 is indeed birationally equivalent with C , in view ofRemark 3.2. Denote by e , e the Maroni invariants of C . Since 1 , α , α is a reduced basis,the elements 1 , x − e − α , x − e − α form a basis for F q [ C ] ∞ , the integral closure of F q [ x − ]inside F q ( C ). Writing out the multiplication for this ring gives x − e − e − α α = − x − e − e − f f ,x − e − α = − x − e − f f + x − e − f x − e − α − x − e + e − f x − e − α ,x − e − α = − x − e − f f + x − e + e − f x − e − α − x − e − f x − e − α . Since the coefficients of this table must be elements of F q [ x − ] we see that deg f i ≤ ( i − e +(2 − i ) e +2 for i = 1 ,
2, hence f ( y,
1) is supported on the polygon from Figure 2.1. (cid:3)
Thus we can proceed as follows. We compute a reduced basis for the function field F q ( C )over F q [ x ], make it normal if needed, and apply the Delone–Faddeev correspondence to it toobtain a model f = 0 of the form (3). As discussed in Section 2, any naive coefficient-wiselift of the polynomial f ( y,
1) to a polynomial f = f y + f y + f y + f ∈ O K [ x ] definesa good lift. After making the polynomial f monic as in (2), it can be fed to Tuitman’salgorithm to compute the zeta function of C over F q . Remark . Our discussion also shows that 1 , f y, f y − = f y + f y + f is an integralbasis of K ( C ) over K [ x ] that reduces to an integral basis of F q [ C ] over F q [ x ]. Using thevariable change x = x − and y = y/x e − e we find the patch f recipr.3 ( x ) y + f recipr.2 ( x ) y + f recipr.1 ( x ) y + f recipr.0 ( x )above infinity, which admits an analogous integral basis. Here f recipr. i denotes the degree( i − e + (2 − i ) e + 2 reciprocal of f i . We can supply these bases as additional input toTuitman’s algorithm, thereby by-passing Heuristic H.4. Lifting curves in degree d = 4 Parametrizing quartic rings.
The parametrization of quartic R -rings S is due to Bhar-gava [2]. This time, the objects involved are pairs of ternary quadratic forms, up to anaction of GL ( R ) × GL ( R ). For an element( A, B ) ∈ GL ( R ) × GL ( R ) , and a pair of ternary quadratic forms ( Q , Q ) over R represented as 3 × A, B ) ∗ ( Q , Q ) = B · (cid:18) AQ A T AQ A T (cid:19) . Concretely, the quadratic forms associated with a quartic ring are obtained by specifying a cubic resolvent (the next paragraph provides more details):
Theorem 4.1 (Bhargava) . There is a canonical bijection between pairs ( S, S ′ ) where S isa quartic ring over R and S ′ is a cubic resolvent for S , considered up to isomorphism, andpairs of ternary quadratic forms over R , up to the action of GL ( R ) × GL ( R ) . IFTING LOW-GONAL CURVES FOR USE IN TUITMAN’S ALGORITHM 9
See [2, Thm. 1], although we will not explicitly rely on this theorem. But we will recycleits central map φ , whose construction we briefly recall, while zooming in on our main caseof interest, namely where S is a domain, say with field of fractions F . We assume moreoverthat F is a separable S -extension of K = Frac R , i.e., its Galois closure E/K has as Galoisgroup the full symmetric group S . Then a cubic resolvent for S is a certain full-ranksubring S ′ ⊆ E D =: F res , where D = h (12) , (1324) i , see [2, Def. 8] for a precise definition.In general, there might be more than one cubic resolvent ring, but for maximal rings it isunique [2, Cor. 5]. Note that if F = K [ y ] / ( f ) with f = ( y − r )( y − r )( y − r )( y − r ) = y + ay + by + cy + d then F res = K [ y ] / (res f ) withres f = ( y − r r − r r )( y − r r − r r )( y − r r − r r )= y − by + ( ac − d ) y − ( a d + c − bd ) . This polynomial is famously known as
Lagrange’s cubic resolvent . The most importantfeature of the Bhargava correspondence is the natural quadratic map˜ φ : F → F res : α α (1) α (2) + α (3) α (4) , where the α ( i ) denote the conjugates of α inside E (numbered compatibly with the roots r i ). This map turns out to descend to a quadratic map of R -modules φ : SR → S ′ R .
Upon taking bases for
S/R and S ′ /R we obtain our two ternary quadratic forms over R .Changing bases of these modules then corresponds to an element of GL ( R ) × GL ( R ). Lifting degree covers. We can assume that f = 1, i.e., our input polynomial (1) ismonic. Let F q ( C ) denote the function field it defines, which is a separable S -extension of F q ( x ) because ϕ is simply branched [15, Lem. 6.10]. Similarly, consider the cubic resolvent(8) y − f y + ( f f − f ) y − ( f f + f − f f )defining F q ( C res ) := F q ( C ) res . We let F q [ C ] and F q [ C res ] be the respective integral closuresof R = F q [ x ] inside these fields. It can be argued that F q [ C res ] is the unique cubic resolventring S ′ for S = F q [ C ] , but for our needs it suffices to know that S ′ ⊆ F q [ C res ] , which isimmediate since F q [ C res ] is maximal.Let e , e , e be the Maroni invariants of C with respect to ϕ , and let b , b be its Schreyerinvariants. Take reduced F q [ x ]-bases 1 , α , α , α ∈ F q [ C ] and 1 , β , β ∈ F q [ C res ] . Withrespect to these bases, the map φ above gives us two ternary quadratic forms Q , Q ∈ F q [ x ][ Y , Y , Y ]. To properly bound the degrees of their coefficients, we have to understandhow the Maroni invariants of the resolvent curve C res relate to data associated with C .Surprisingly, up to a small shift, these turn out to be the Schreyer invariants of C withrespect to ϕ : Theorem 4.2.
Let k be a field of characteristic = 2 and consider a smooth projectivecurve over k equipped with a simply branched degree morphism to P , say with Schreyerinvariants b , b . Then the Maroni invariants of its cubic resolvent are b + 2 , b + 2 .Proof. This result is due to Casnati [5, Def. 6.4], although he formulated it in terms of Recil-las’ trigonal construction, which is the geometric counterpart of Lagrange’s cubic resolvent,as pointed out in [18, § (cid:3) Lemma 4.3.
The quadratic forms Q , Q obtained through Bhargava’s correspondence asabove are a model of C of the form (4) .Proof. Note that the polynomials indeed cut out a curve that is birationally equivalent with C , in view of [3, § Since 1 , α , α , α and 1 , β , β are reduced bases, by Theorem 4.2 wehave that 1 , x − e − α , x − e − α , x − e − α and1 , x − b − β , x − b − β are bases of F q [ C ] ∞ , resp. F q [ C res ] ∞ , the integral closures of F q [ x − ] in F q ( C ), resp. F q ( C res ).Now the quadratic map ˜ φ : F q ( C ) → F q ( C res )from above also descends to a quadratic map of F q [ x − ]-modules φ ′ : F q [ C ] ∞ F q [ x − ] → F q [ C res ] ∞ F q [ x − ] . With respect to the above bases, φ ′ is defined by two quadratic forms over F q [ x − ], whichare necessarily obtained from Q and Q by applying the corresponding (diagonal) changeof basis matrices. In other words, φ ′ is represented by the quadratic forms x b +4 Q ( x − e − Y , x − e − Y , x − e − Y ) ,x b +4 Q ( x − e − Y , x − e − Y , x − e − Y ) . But these have coefficients in F q [ x − ]. Hence the degree of the Y i Y j -coefficient in Q can beat most e i + e j − b , and similarly for Q . In other words, the dehomogenized polynomials Q ( y , y ,
1) and Q ( y , y ,
1) are supported on the polytopes from Figure 2.2. (cid:3)
To compute these liftable quadrics Q , Q in practice we will not directly compute theresolvent map φ with respect to reduced bases for F q ( C ) and F q ( C res ). Instead, we computethe map φ with respect to certain naive bases for F q ( C ) and F q ( C res ) and then applychange of basis to a reduced basis. In more detail, denoting by f ′ i the coefficients of thecubic resolvent polynomial of f as in (8), we consider the bases1 , − f y − , y, y for F q ( C ) and(9) 1 , y, − f ′ y − for F q ( C res ) . Computing the representation of the resolvent map φ with respect to these bases can bedone symbolically by means of Vieta’s formulas, yielding the quadrics(10) Q ′ = f f − f f − f f , Q ′ = − f − f . Now let 1 , α , α , α and 1 , β , β be reduced bases for F q [ C ] , resp. F q [ C res ] , as above. Tocompute the cubic resolvent map with respect to these bases, we simply apply the change ofbasis action from the naive bases in (9) to these reduced bases. We note that this involveselements of GL ( F q ( x )) × GL ( F q ( x )) rather than GL ( F q [ x ]) × GL ( F q [ x ]). The resulting Alternatively, the reader can check that res y ( Q ′ ( y , y , , Q ′ ( y , y , y + f y + f y + f y + f ,where Q ′ and Q ′ are the quadratic forms from below. IFTING LOW-GONAL CURVES FOR USE IN TUITMAN’S ALGORITHM 11 quadrics Q , Q will be our model of the form (4). Then, as explained in Section 2, we cantake any Q , Q ∈ O K [ x ][ y , y ] lifting the Q i ( y , y , y ( Q , Q ), which is indeed ofdegree 4 in y = y . After making it monic, it can be fed as input to Tuitman’s algorithm.5. Lifting curves in degree d = 5 Parametrizing quintic rings.
The parametrization of quintic R -rings S is also due toBhargava [3]. We assume that char R = 2 ,
3. The objects involved in the parametrizationare now quadruples of 5 × R . There is a natural action ofGL ( R ) × GL ( R ) on such objects, given by( A, B ) ∗ M = B · AM A T AM A T AM A T AM A T , with M = ( M , M , M , M ) a quadruple of 5 × A, B ) ∈ GL ( R ) × GL ( R ). Here too, the parametrization requires us to specify a sextic resolvent (see the next paragraph for details): Theorem 5.1 (Bhargava) . There is a canonical bijection between pairs ( S, S ′ ) where S is aquintic ring and S ′ is a sextic resolvent for S , considered up to isomorphism, and quadruplesof × skew-symmetric matrices over R , up to the action of GL ( R ) × GL ( R ) . See [3], although as in the previous sections, we will not explicitly rely on this theorem.But we will need the fundamental resolvent map (11) below. Let us again focus on thesetting where S is a domain with field of fractions F , and let K = Frac R . We assume that F is a separable S -extension of K , i.e., its Galois closure E/K has as Galois group the wholeof S . Consider the order 20 subgroup H = H (1) = AGL ( F ) = h (12345) , (1243) i ⊆ S .Then a sextic resolvent for S is a certain full-rank subring S ′ ⊆ E H =: F res ; for a precisedefinition we refer to [3, Def. 5]. In general, such a sextic resolvent ring is not unique, butfor maximal quintic rings it is [3, Cor. 19]. If F = K [ y ] / ( f ) with f = ( y − r )( y − r )( y − r )( y − r )( y − r ) = y + ay + by + cy + dy + e, then F res = K [ y ] / (res f ) with res f = ( y − ρ )( y − ρ )( y − ρ )( y − ρ )( y − ρ )( y − ρ ), where ρ = ( r r + r r + r r + r r + r r − r r − r r − r r − r r − r r ) and { ρ , ρ , . . . , ρ } is the orbit of ρ under the natural S -action permuting the r i ’s. Notethat ρ is stabilized by H (1) . We choose ρ i to be stabilized by the conjugate subgroup H (2+ i ) = (12345) − i h (13254) , (3245) i (12345) i , for 0 ≤ i ≤ . The polynomial res f is known as Cayley’s sextic resolvent ; concrete expressions for itscoefficients in terms of a, b, c, d, e can be found in [10, Proof of Prop. 13.2.5]. For an element α ∈ F res we denote by α ( i ) the conjugates of α inside E , labeled so that α ( i ) is fixed by H ( i ) . Consider bases α = 1 , α , . . . , α for S/R and β = 1 , β , . . . , β for Or it can be found hard-coded in our accompanying Magma file precomputed 5.m . S ′ /R , and define √ disc S = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . . . α (1)1 α (2)1 . . . α (5)1 ... ... . . . ... α (1)4 α (2)4 . . . α (5)4 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . The central tool in Bhargava’s correspondence is the fundamental resolvent map , which isthe bilinear alternating form(11) g : F res × F res → F : ( α, β )
7→ √ disc S · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α (1) + α (2) α (3) + α (6) α (4) + α (5) β (1) + β (2) β (3) + β (6) β (4) + β (5) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . This turns out to descend to a well-defined map ˜ S ′ × ˜ S ′ → ˜ S , where˜ S = Rα ∗ + Rα ∗ + Rα ∗ + Rα ∗ ⊆ F, ˜ S ′ = Rβ ∗ + Rβ ∗ + Rβ ∗ + Rβ ∗ + Rβ ∗ ⊆ F res are defined in terms of the dual bases α ∗ , . . . , α ∗ and β ∗ , . . . , β ∗ with respect to the tracepairing, i.e., Tr F/K ( α i α ∗ j ) = δ ij (with δ ij the Kronecker delta), and similarly for β ∗ j . Notethat the extensions F/K and F res /K are both separable and so their trace pairings arenon-degenerate. With respect to the bases { β ∗ i } i and { α ∗ i } i , the map g is represented by aquadruple M = ( M , M , M , M ) of 5 × S ′ and ˜ S then corresponds to an element of GL ( R ) × GL ( R ). Remark . Our fundamental resolvent map differs from Bhargava’s original map by afactor 4 /
3, which is not an issue in view of our restrictions on the field characteristic.
Lifting degree covers. As in the d = 4 case, we assume that our input polynomial f from (1) is monic (i.e., f = 1). Let F q ( C ) be the corresponding function field; this isa separable S -extension of F q ( x ) because ϕ is simply branched [15, Lem. 6.10]. We alsoconsider Cayley’s sextic resolvent associated with our input polynomial, defining F q ( C res ) := F q ( C ) res . Let F q [ C ] and F q [ C res ] be the respective integral closures of R = F q [ x ] insidethese two function fields; it can be argued that F q [ C res ] is the unique sextic resolvent ring S ′ for S = F q [ C ] , but as in the d = 4 case it suffices to observe that S ′ ⊆ F q [ C res ] .Let e , e , e , e be the Maroni invariants of C with respect to ϕ , and let b , b , b , b , b beits Schreyer invariants. Take reduced F q [ x ]-bases 1 , α , . . . , α ∈ F q [ C ] and 1 , β , . . . , β ∈ F q [ C res ] and consider the quadruple ( M , M , M , M ) of 5 × F q [ x ] arising along the above construction. We represent this by the single matrix M = M Y + M Y + M Y + M Y ∈ k [ x ][ Y , Y , Y , Y ]whose entries are now linear and homogeneous in the Y i . To get a handle on the degrees oftheir coefficients, we should again express the Maroni invariants of the resolvent curve C res in terms of data associated with C . As in the case of the cubic resolvent, this can be donein a surprisingly explicit way: Theorem 5.3.
Let k be a field of characteristic = 2 and consider a smooth projective curveover k equipped with a simply branched degree morphism to P , say with Schreyer invariants b , . . . , b . Then the Maroni invariants of its sextic resolvent are g − − b , . . . , g − − b .Proof. This theorem seems new and is part of a generalization of Theorem 4.2, which iscurrently being elaborated in collaboration with Yongqiang Zhao [8]. In the meantime, aproof of Theorem 5.3 can be found in the master thesis of the second listed author [29]. (cid:3)
IFTING LOW-GONAL CURVES FOR USE IN TUITMAN’S ALGORITHM 13
Lemma 5.4.
Denote by M r,i,j the ( i, j ) -th entry of the matrix M r constructed throughBhargava’s correspondence as above. Then deg M r,i,j ≤ e r + b i + b j + 6 − g . In particular,this defines a model for C of the form (5) .Proof. The fact that the sub-Pfaffians of M cut out a curve birational to C follows againfrom [3, § F q [ C ] ∞ the integral closure of F q [ x − ] in F q ( C ). Let g be thefundamental resolvent form attached to the basis 1 , α , . . . , α of F q [ C ] over F q [ x ], and let g ∞ be the fundamental resolvent form attached to the basis 1 , x − e − α , . . . , x − e − α of F q [ C ] ∞ over F q [ x − ]. We have that, for all u, v ∈ F q ( C res ), g ( u, v ) = q disc F q [ C ] q disc F q [ C ] ∞ g ∞ ( u, v ) = x g +4 g ∞ ( u, v ) . Let α ∗ , . . . , α ∗ , resp. β ∗ , . . . , β ∗ , be dual bases for 1 , α , . . . , α , resp. 1 , β , . . . , β . Then thecorresponding dual bases for the rings F q [ C ] ∞ and F q [ C res ] ∞ are α ∗ , x e +2 α ∗ , . . . , x e +2 α ∗ for F q [ C ] ∞ ,β ∗ , x e ′ +2 β ∗ , . . . , x e ′ +2 β ∗ for F q [ C res ] ∞ , where the e ′ i are the Maroni invariants of the resolvent. We now compute, for i, j > g ∞ ( x e ′ i +2 β ∗ i , x e j +2 β ∗ j ) = x e ′ i + e ′ j +4 x − g − g ( β ∗ i , β ∗ j )(12) = X l =1 x − e l − g − e ′ i + e ′ j ( M l ) ij ( x e l +2 α ∗ l ) . (13)It follows that g ∞ is represented by the matrix whose entries have coefficients x − e l − g − e ′ i + e ′ j ( M l ) ij , i, j = 1 , . . . , , l = 1 , . . . , . But these coefficients belong to F q [ x − ]. Hence we find that deg( M l ) ij ≤ e l + b i + b j + 6 − g by Theorem 5.3, as wanted. (cid:3) To compute such a liftable matrix in practice, we follow a similar approach as in the caseof degree 4 covers. Namely, we will not be computing the fundamental resolvent map withrespect to our reduced bases directly, but rather compute this for certain naive bases andapply change of basis. Concretely, consider the naive bases1 , y, y , y , y for F q ( C ) , and1 , y, y , y , y , y for F q ( C res ) , along with the slightly altered fundamental resolvent map g ′ : F q ( C res ) × F q ( C res ) → F q ( C ) : ( α, β ) q disc f · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α (1) + α (2) α (3) + α (6) α (4) + α (5) β (1) + β (2) β (3) + β (6) β (4) + β (5) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) where p disc f = det(( y i ) ( j ) ) ≤ i ≤ , ≤ j ≤ . We compute the M ′ ( r ) ij ∈ F q [ x ] for which g ′ ( y i , y j ) = X r =0 M ′ ( r ) ij y r , giving five 5 × M ′ (0) , . . . , M ′ (4) ; here we used that M ′ ( r ) ij = 0 assoon as i or j is zero, allowing us to disregard these terms. We call this the naive model . Remark . It is important to note that these expressions can be computed symbolically interms of the coefficients f i of f , by means of Vieta’s formulas. Therefore this computationonly has to be done once for all curves. This is in complete analogy with the degree 4 case,see (10). However, there the naive model was very simple, whereas this time the expressionsinvolved are rather long. However, a computer has no trouble with these computations.Now compute reduced bases 1 , α , . . . , α for F q [ C ] and 1 , β , . . . , β for F q [ C res ] alongwith their corresponding dual bases. Acting on the naive model with a change of basisfrom the naive bases to the duals of these reduced bases, yields the altered resolvent map g ′ with respect to these dual reduced bases. Note that this action will be by an elementof GL ( F q ( x )) × GL ( F q ( x )) rather than GL ( F q [ x ]) × GL ( F q [ x ]). To obtain instead theresolvent map g we have to multiply by q disc F q [ C ] p disc f . Since we already have the reduced bases at hand, this factor is easiest to compute as thedeterminant of the change of basis matrix from the naive basis for F q ( C ) to the reducedbasis 1 , α , . . . , α .At this point, we have a representation of the fundamental resolvent map g with respectto the duals of the reduced bases for F q [ C ] and F q [ C res ] as a 5 × M with entries in k [ x ][ Y , Y , Y , Y ], linear and homogeneous in the Y i . This is the desiredmodel, which we can lift naively, in a skew-symmetry preserving way, to a matrix havingentries in O K [ x ][ Y , Y , Y , Y ]. Computing its five 4 × References [1] P. Beelen,
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Cosic, research group at imec and KU LeuvenKasteelpark Arenberg 10/2452, 3001 Leuven (Heverlee), BelgiumDepartment of Mathematics: Algebra and Geometry, Ghent UniversityKrijgslaan 281 – S25, 9000 Gent, BelgiumSection of Algebra, Department of Mathematics, KU LeuvenCelestijnenlaan 200B, 3001 Leuven (Heverlee), Belgium [email protected]
Appendix A. Magma implementation: discussion and examples
The approximate timings mentioned below were obtained using Magma V2.25-2 on kraitchik , acomputer with 12 Intel Xeon E5-2630 v2 processors and 128GB of memory, running Ubuntu 16.04.
Degree d = 3 . In the accompanying Magma file lifting lowgonal 3.m , the user can choose afinite field F q of characteristic p >
2, along with a suitable pair of integers e and e . Running thecode • first generates a random degree 3 cover C → P over F q whose Maroni invariants are e , e ,of which it chooses a somewhat scrambled defining polynomial having the form (1); thisserves as test input for our lifting procedure, • next applies the Delone–Faddeev correspondence to this input, thereby procuding a naivelyliftable defining polynomial, as discussed in Section 3, • finally carries out the naive lift and, after making the result monic, prints it to a file inputcurve 3.m , which can be loaded as input to Tuitman’s pcc p.m or pcc q.m imple-mentation.E.g., over F , a run of our code generated the random trigonal curve (6 x +7) y +(7 x +10 x +2 x +3 x +8 x +7 x +1) y +(7 x +10 x +4 x +6 x +5 x +3 x +2 x +4 x + x + x +8 x +8 x +3) y +6 x +7 x +2 x +3 x +8 x +7 x + x +4 x + x + x +8 x +8 x +3 x +10 x +4 x +7 x +2 x +6 x + x +9 x +3 x +5=0 of genus 8, having prescribed Maroni invariants { , } . Under the Delone–Faddeev correspondencethis was transformed into (10 x +8) y +(8 x + x +10 x +7 x +1) y +(9 x +5 x +3 x +2 x +4 x +7 x +9) y + x +4 x +5 x + x +4 x +9 x +6 x +9=0 . After taking a naive lift having coefficients in {− , . . . , } ⊆ Z and making the result monic in y ,this was fed to Tuitman’s code, which determined the numerator of the Hasse–Weil zeta functionas T − T +30116537 T − T +2459688 T − T +151855 T − T +8366 T − T +1255 T − T +168 T − T +17 T − T +1 . On a larger scale, for a random trigonal genus 9 curve over F having Maroni invariants { , } , thesame procedure computed its Hasse–Weil zeta function in about 20 minutes. For a random trigonalgenus 8 curve over F having Maroni invariants { , } we obtained its Hasse–Weil zeta functionusing roughly 2 hours of computation. In both cases, the lifting step took less than 0 . Degree d = 4 . In the accompanying Magma file lifting lowgonal 4.m , the user chooses a finitefield F q of characteristic p >
2, along with a suitable quintuple of integers e , e , e , b , b . Runningthe code • first generates a random degree 4 cover C → P over F q with Maroni invariants e , e , e and Schreyer invariants b , b , of which it chooses a somewhat scrambled monic definingpolynomial; this serves as test input for our lifting procedure, • next applies the Bhargava correspondence to this input, thereby procuding a naively liftablepair of quadratic forms (i.e., symmetric matrices in F q [ x ] × ), as discussed in Section 4, • finally carries out the naive lift and, after taking a resultant and making the outcome monic,prints it to a file inputcurve 4.m , which can be loaded as input to Tuitman’s pcc p.m or pcc q.m implementation. IFTING LOW-GONAL CURVES FOR USE IN TUITMAN’S ALGORITHM 17
E.g., over F a run of our code generated the random tetragonal curve y +(4 x +6 x +2 x +3 x +5 x +6) y +(6 x +4 x +6 x +2 x + x +4 x +4 x +2 x + x + x +6 x ) y +(4 x +4 x +6 x +2 x + x +4 x + x +4 x +2 x +2 x +5 x +6 x +4 x +6 x + x +5 x +6 x +6 x +2 x +5 x +5) y + x +6 x +2 x +3 x +5 x +6 x +4 x +2 x + x + x +6 x +6 x +4 x +6 x + x +5 x +6 x +3 x +5 x +4 x + x +2 x + x +5 x + x +3 x +2 x + x +3 x + x +2 of genus 10, having Maroni invariants { , , } and Schreyer invariants { , } . Bhargava’s corre-spondence then produced the pair of matrices (cid:18) x +4 2 x +5 x +4 x +54 x +4 2 x +6 x +2 6 x +6 x +52 x +5 x +4 x +5 6 x +6 x +5 5 x +3 x +6 x +2 x +2 x +1 (cid:19) , x +3 x +20 1 2 x + x +6 x +45 x +3 x +2 2 x + x +6 x +4 3 x +6 x +4 x +6 x +4 . These matrices were then lifted naively to characteristic zero, i.e., to matrices over Z [ x ] whose entrieshave coefficients in {− , . . . , } . After taking a resultant of the corresponding (dehomogenized)quadratic forms and making the result monic, we obtained a polynomial of the form (4) whichwas fed as input to Tuitman’s code. The numerator of the Hasse–Weil zeta function was thendetermined as T +161414428 T +80707214 T +24706290 T +5764801 T − T − T − T − T − T − T − T − T − T − T − T +49 T +30 T +14 T +4 T +1 . On a larger scale, for a random tetragonal genus 8 curve over F having Maroni invariants { , , } and Schreyer invariants { , } we obtained its Hasse–Weil zeta function using about 1 hour ofcomputation. For a random tetragonal genus 7 curve over F with Maroni invariants { , , } and Schreyer invariants { , } we computed its zeta function in roughly 9 hours. In both cases thelifting step took less than five seconds, of which the lion’s share was accounted for by the resultantcomputation. Degree d = 5 . The accompanying Magma file precomputed 5.m , which can be reproduced by run-ning precomputation 5.m , contains hard-coded expressions for Cayley’s sextic resolvent and for thealtered fundamental resolvent map g ′ from Section 5. It is invoked by the file lifting lowgonal 5.m ,in which the user chooses a finite field F q of characteristic p >
3, along with a suitable sequence ofnine integers e , e , e , e , b , b , b , b , b . Running the code • first generates a random degree 5 cover C → P over F q with Maroni invariants e , e , e , e and Schreyer invariants b , b , b , b , b , of which it chooses a somewhat scrambled monicdefining polynomial; this serves as test input for our lifting procedure, • next applies the Bhargava correspondence to this input, thereby procuding a quadruple ofskew-symmetric matrices in F q [ x ] × , as discussed in Section 5, • finally naively lifts these matrices to characteristic zero, after which it considers theirlinear combination with coefficients 1 , y , y , y ; then it takes the five 4 × y , y and making theresult monic in y = y , gives rise to a lift of the form (2); this polynomial is then printedto a file inputcurve 5.m , which can be loaded as input to Tuitman’s pcc p.m or pcc q.m implementation.E.g., over F , a run of our code generated the random pentagonal curve y +(5 x + x +7 x +8 x +6 x +12 x +10 x +6 x +11) y +(10 x +4 x +11 x +15 x +14 x +8 x +16 x +5 x +3 x +3 x +7 x +11 x +15 x +16 x +4 x +12 x +8 x +9) y +(10 x +6 x +8 x +14 x +6 x +3 x +5 x +13 x +7 x +3 x +6 x +13 x +14 x +11 x +6 x +16 x +4 x +13 x +5 x +2 x +16 x +11 x +15 x +16 x +6 x +3 x +3 x +16) y +(5 x +4 x +11 x +15 x +11 x +13 x +2 x + x +15 x +14 x +2 x +6 x +4 x +4 x +9 x +5 x +13 x +2 x +9 x +8 x +15 x +11 x +14 x +4 x +4 x +7 x + x +9 x +8 x +11 x +12 x +6 x +14 x +2 x +6) y + x + x +7 x +8 x +13 x +6 x +3 x +4 x +14 x +3 x +4 x +7 x +12 x +12 x +9 x +11 x + x +11 x +2 x +9 x +11 x +8 x +7 x +16 x +11 x + x +8 x +10 x +15 x +14 x +16 x +4 x +6 x +3 x +5 x +4 x +9 x +15 x +7 x +10 x +5 x +5 x +12 x +15 x +7 of genus 9, having Maroni invariants { , , , } and Schreyer invariants { , , , , } . The Bhargavacorrespondence then produced the quadruple of skew-symmetric matrices x +15 x +9 2 x +10 14 x +3 13 x +2 x +8 0 3 6 x +15 015 x +7 14 0 5 03 x +14 11 x +2 12 0 04 0 0 0 0 , x +14 6 x +10 3 x +6 10 x +3 0 7 x +16 10 x x +7 10 x +1 0 1 014 x +11 7 x
16 0 07 8 0 0 0 ! , x +3 14 x +11 16 x +11 1016 x +14 0 10 x +14 13 x +1 163 x +6 7 x +3 0 3 0 x +6 4 x +16 14 0 07 1 0 0 0 , x + x +12 x +6 12 x +8 11 x +16 x +15 11 x +169 x +16 x +5 x +11 0 4 x +1 7 x +8 x +11 5 x +55 x +9 13 x +16 0 11 x +6 16 x + x +2 10 x +9 x +6 6 x +11 0 106 x +1 12 x +12 16 7 0 . These matrices were then lifted to characteristic zero, i.e., to matrices over Z [ x ] whose entries havecoefficients in {− , . . . , } ; note that this coefficient range forces the lifted matrices to be skew-symmetric. After taking the linear combination with coefficients 1 , y = y , y , y , computing thefive 4 × y , y andmaking the outcome monic in y , we ended up with a polynomial of the form (2) which was fed asinput to Tuitman’s code. The numerator of its Hasse–Weil zeta function was then determined tobe T − T +4513725403 T − T +271192687 T − T +12616584 T − T +924732 T − T +54396 T − T +2568 T − T +191 T − T +11 T − T +1 . This basic example took 7 . F having Maroni invariants { , , , } and Schreyer invariants { , , , , }}