Optimal Curves of Genus 3 over Finite Fields with Discriminant -19
aa r X i v : . [ m a t h . AG ] A ug OPTIMAL CURVES OVER FINITE FIELDS WITHDISCRIMINANT -19
E. ALEKSEENKO , S. ALESHNIKOV , N. MARKIN, A. ZAYTSEV
Abstract.
In this work we study the properties of maximal andminimal curves of genus 3 over finite fields with discriminant − −
19 of cardinality up to 997. We also show thatexistence of a maximal curve implies that there is no minimal curveand vice versa. Introduction
The number of rational points of an irreducible non-singular projec-tive curve C/ F q of genus g satisfies the Hasse-Weil-Serre bound: | C ( F q ) − q − | ≤ g [2 √ q ] . In case of equality, that is, C ( F q ) = q + 1 ± g [2 √ q ] the curve iscalled optimal over F q . When C ( F q ) = q + 1 − g [2 √ q ] it is called a maximal over F q curve and when C ( F q ) = q + 1 − g [2 √ q ] it is calleda minimal over F q curve.Let C be an optimal curve of genus g over F q . Then the Frobeniusendomorphism induces a homomorphismF : T l Jac( C ) ⊗ Z Q → T l Jac( C ) ⊗ Z Q , where T l Jac( C ) is the projective limit lim ←− Jac( C )[ l n ]. Moreover, if thecharacteristic polynomial of Jac( C ) splitsP Jac( C ) ( T ) = g Y i =1 ( T − α i ) , then the number of rational points on C equals to C ( F q ) = q + 1 − g X i =1 α i = q + 1 − g X i =1 ( α i + α i ) , with α i + g = α i . The eigenvalues of the endomorphism Frobenius Fhave following property: α i + ¯ α i = − [2 √ q ] when C is maximal and α i + ¯ α i = [2 √ q ] when C is minimal. Therefore, if a curve is optimal,then L -polynomial of this curve is L ( t ) = g Y i =1 (1 − α i t ) = g Y i =1 (1 ∓ [2 √ q ] t + qt ) , where the minus sign applies to the minimal case and the plus signto the maximal case. Then the theory of Honda-Tate shows that theJacobian Jac(C) of a maximal curve C is isogenous to a product ofcopies of a maximal elliptic curve, that is Jac( C ) ∼ E g , where E is amaximal elliptic curve over a finite field F q . The similar isogeny occursif the curve C and the elliptic curve E are minimal. The isogeny class of E over a finite field F q is characterized by the characteristic polynomialof the Frobenius endomorphism of E .The definition of the discriminant of a finite field is recalled. Definition 1.1.
For a finite field F q , the number [2 √ q ] − q is calledthe discriminant of a finite field F q .Now, we consider the equivalence between the category of ordinaryabelian varieties Jac( C ) over F q which are isogenous to E g (hence E is ordinary) and the category of R -modules, where R is the ring de-fine by the Frobenius endomorphism of E . In our case let C be asmooth irreducible projective algebraic curve over F q with discrimi-nant −
19. Therefore R = O K , where K = Q ( √− C ) bethe principal polarized Jacobian variety of C with theta-divisor θ . Bythe Torelli Theorem, the curve C is completely defined by (Jac( C ) , θ ),up to a unique isomorphism over an algebraic closure of F q . Con-sider the Hermitian module ( O gK ; h ), where O gK is a O K -module, and h : O gK × O gK → O K is a Hermitian form. The equivalence of categoriesis defined by the functor F : Jac( C ) −→ Hom( E, Jac( C )) and its in-verse V : O gK −→ O gK ⊗ O K E . Under this equivalence the principalpolarisation of Jacobian Jac( C ) corresponds to an irreducible Hermit-ian O K -form h . Therefore we can use the classification of unimodularirreducible Hermitian forms in order to study the isomorphism classesof Jac(C) . For detailed description of this equivalence of categories seethe Appendix by J.-P. Serre in [3].Deligne’s Theorem [1] yields that the number of isomorphism classesof abelian varieties isogenous to A equals the number of isomorphismclasses of R -modules, which may be embedded as lattices in the K -vector space K g , where K = Quot( R ). Since in our case there existsone isomorphism class of such R -modules, then there exists a uniqueisomorphism class of abelian varieties. Therefore Deligne’s Theoremtogether with 2.1 show that Jac( C ) is actually isomorphic to E g . PTIMAL CURVES OVER FINITE FIELDS WITH DISCRIMINANT -19 3
The main result of this paper is putting this theory to practical use.We give a characterization of isomorphism classes of optimal curves ofgenus 3 over finite fields with discriminant −
19 in such a way that weare able to give an explicit description of all such curves. In particular,we produce maximal and minimal curves of genus 3 over prime finitefields with discriminant −
19 of cardinality up to 997.We also would like to mentioned recent article of Christophe Ritzen-thaler ”Explicit computations of Serre’s obstruction for genus 3 curvesand application to optimal curves ” (see [9]). Beside of all other in-teresting results of this article we would like to point out the methodof detecting maximal and minimal curves and the model of a modu-lar curve which can be reduced to optimal curve over finite field withdiscriminant − Optimal Curves of Genus and Optimal Elliptic Curves.
In this subsection we explore optimalelliptic curves over F q and produce concrete calculations for the finitefields F q of the discriminant −
19 and q ≤ E ) of an elliptic curve E is the setof all isogenies φ : E ( F q ) → E ( F q ), with multiplication correspond-ing to composition. If a curve E has complex multiplication, then byDeuring’s theory [7] the endomorphism ring End( E ) is an order in theimaginary quadratic field K = Q ( √ d ). The theory of complex mul-tiplication and Deuring’s lifting theory give us the following: given aquadratic field K , the number of isomorphism classes of elliptic curvesover F q whose endomorphism rings are isomorphic to the maximal order O K is equal the number of ideal classes h K of K . Proposition 2.1.
Let F q be a finite field with discriminant − . Thereexist exactly two F q -isomorphism classes of optimal elliptic curves E over F q , namely the class of maximal and the class of minimal ellipticcurves over F q .Proof. Deuring’s Theorem provides the existence of maximal and min-imal elliptic curves over a finite field with discriminant −
19. Let E besuch a curve. Then End F q (E) contains the ring Z [ x ] / ( x + mx + q ),where m = ± [2 √ q ] is the trace of the Frobenius endomorphism of E . Therefore we have End F q (E) ∼ = O K ∼ = Z [ x ] / ( x + mx + q ), where K is the imaginary quadratic field Q ( √−
19) with discriminant − O K = Z [ − √− ] and the Minkovski bound is B K ≈ , ≤ ,
77. We verify that 2 O K is a principal ideal to conclude E. ALEKSEENKO , S. ALESHNIKOV , N. MARKIN, A. ZAYTSEV that h K = 1. From the class number and the mass formula (see ?? )it follows that there exists a unique class of isomorphic elliptic curvesover F q . (cid:3) Remark . Alternatively, we can find the number of F q -isomorphismclasses of elliptic curves over F q within a given isogeny class by usingthe following properties. In case when two elliptic curves are givenby E : y = x + ax + b and E ´ : y = x + a ´ x + b ´ , then E ∼ = E ´ over F q if and only if the the following relations on the coefficients hold: a ´= ac , b ´= bc for a some c ∈ F q . Example 2.3.
We give examples of maximal and minimal ellipticcurves over finite fields over F q with discriminant −
19 for all q < .q Maximal Minimal47 y = x + x + 38 y = x + 32 x + 2761 y = x + 6 x + 29 y = x + 32 x + 57137 y = x + x + 36 y = x + 61 x + 47277 y = x + 2 x + 61 y = x + 61 x + 47311 y = x + x + 50 y = x + 18 x + 308347 y = x + 2 x + 96 y = x + 174 x + 12467 y = x + 2 x + 361 y = x + 234 x + 337557 y = x + 3 x + 132 y = x + 140 x + 295761 y = x + x + 82 y = x + 592 x + 454997 y = x + 6 x + 493 y = x + 500 x + 9342.2. Optimal Curves of Genus . We start with a proposition whichwas proven in [8].
Proposition 2.4.
Up to an isomorphism over the field F q there existsexactly one maximal (resp. minimal) optimal curve C of genus over F q , viz., the fibered product over P of the two maximal (resp. minimal)optimal elliptic curves E : y = f ( x ) and E : y = f ( x )( α x + β ) . Example 2.5.
Here we produce examples of elliptic curves E fromthe proposition above and maximal curve of genus 2 over the finite field PTIMAL CURVES OVER FINITE FIELDS WITH DISCRIMINANT -19 5 F q of the discriminant −
19 and q < q Maximal elliptic curve Maximal curve of genus two47 y = ( x + x + 38)( x + 30) z = x + 4 x + 22 x + 3361 y = ( x + 6 x + 29)( x + 2) z = x + 55 x + 18 x + 9137 y = ( x + x + 36)( x + 18) z = x + 83 x + 14 x + 77277 y = ( x + 2 x + 61)(2 x + 80) z = 104 x + 247 x + 185 x + 245311 y = ( x + x + 50)( x + 134) z = x + 220 x + 66 x + 19347 y = ( x + 2 x + 96)( x + 166) z = x + 196 x + 84 x + 316467 y = ( x + 2 x + 361)( x + 47) z = x + 326 x + 91 x + 118557 y = ( x + 3 x + 132)(2 x + 266) z = 209 x + 318 x + 356 x + 421761 y = ( x + 3 x + 132)( x + 257) z = x + 751 x + 288 x + 98997 y = ( x + 3 x + 132)( x + 760) z = x + 711 x + 20 x + 30Note that the corresponding minimal curves of genus 2 can be ob-tained by twisting of maximal curves.3. A degree of a projection
We can calculate the degree of the maps C → E , obtained via theembedding of C into Jac( C ) ∼ = E g and projections onto E .The following result can be found in [8], we include it here with theproof for the sake of completeness. Note that proof relies on the factthat the hermitian lattice corresponding to Jac(C) is a free O K -module,which holds in the case when F q has discriminant − Proposition 3.1.
Let C be an optimal curve over F q . Fix an iso-morphism Jac( C ) ∼ = E g such that the theta divisor corresponds to thehermitian form ( h ij ) on O gK on the canonical lift of Jac( C ) . Thendegree of the k -th projection f k : C ֒ → Jac( C ) ∼ = E g pr k −→ E equals det( h ij ) i,j = k .Proof. We denote the abelian variety E g by E × . . . × E g , where E i = E ,and consider the first projection. The degree of the map f equals theintersection number [ C ] · [ E × . . . × E g ]. The cohomology class [ C ] of C in an appropriate cohomology theory is [Θ g − / ( g − L is a line bundle on an abelian variety A of dimension g then by theRiemann-Roch theorem one has ( L g /g !) = deg( ϕ L ), and deg( ϕ L ) =det( r ij ) , where the matrix ( r ij ) gives the hermitian form correspondingto the first Chern class of the line bundle L . Since the hermitianform ( h ij ) i,j =1 corresponds to the line bundle Θ | E × ... × E g on the abelian E. ALEKSEENKO , S. ALESHNIKOV , N. MARKIN, A. ZAYTSEV variety E × . . . × E g the degree of f is given by[ C ] · [ E × . . . × E g ] = 1( g − | E × ... × E g ) g − = det(( h ij ) i,j =1 ) . (cid:3) Properties the Automorphism Group
In this section we prove that an optimal curve of genus 3 over a finitefield with the discriminant is −
19 is not hyperelliptic. Furthermore, weprove that there exists either a maximal or a minimal curve.From the table of classification of hermitian modules with discrimi-nant −
19 along with the lemma 4.2 proved that an order of an auto-morphism group of an optimal curve of genus 3 over a finite field withthe discriminant −
19 is 6.
Proposition 4.1.
There exists an optimal curve C of genus over F q , namely the double covering of a maximal or minimal elliptic curverespectively.Proof. The equivalence of categories as described in the Introductiontells us that a polarization of the Jacobian corresponds to a class ofirreducble unimodular hermitian forms. According to the classification[6] of unimodular hermitian modules, there is a unique class of irre-ducible unimodular hermitian forms. This class can be represented bythe unimodular hermitian matrix below. −
11 3 − √− − − −√− . Therefore by the Theorem of Oort and Ueno [4], there exists a unique F q -isomorphism class of optimal curves over F q . By Proposition 3.1 thedegree of f : C → E is equal to the determinantdet − √− − −√− ! which is 2. Hence C is a double covering of an optimal elliptic curve,as desired. (cid:3) Now we show an optimal curve of genus 3 is not hyperelliptic.
Lemma 4.2.
Let C be an optimal curve of genus over a finite field F q with discriminant − . Then C is non-hyperelliptic. PTIMAL CURVES OVER FINITE FIELDS WITH DISCRIMINANT -19 7
Proof.
For the sake of contradiction suppose that C is a hyperellipticcurve. Then there are two involutions, the first involution τ is thehyperelliptic involution and the second involution σ corresponds to thedouble cover f : C → E from the previous proposition. So C/ h σ i isan optimal elliptic curve and C/ h τ i is a projective line. The subgroup h σ, τ i is isomorphic to Z / Z × Z / Z and we have the following diagramof coverings C w w pppppppppppp (cid:15) (cid:15) GGGGGGGGGG C/ h σ i ∼ = E & & NNNNNNNNNNNN C/ h στ i (cid:15) (cid:15) P { { P Furthermore the formal relation of groups2 · { id, τ, σ, στ } + { id } = 12 { id, σ } + 12 { id, τ } + 12 { id, στ } implies the relation between idempotents in End(Jac( C )) (see [2]) andtherefore we have an isogeny(4.1) Jac( C ) ∼ Jac( C/ h σ i ) × Jac( C h σ ◦ τ i ) . From the isogeny above and Hurwitz’ formula, it follows that C → C/ h σ i is an unramified double covering. Therefore the number of ra-tional points C ( F q ) is even. On other hand C ( F q ) = q + 1 ± m isodd since m is odd. (cid:3) Next lemma shows that a given finite field F q with discriminant − Lemma 4.3.
Let F q be a finite field with discriminant − . Then F q cannot admit minimal and maximal curves simultaneously.Proof. Suppose there exist a maximal curve C M and a minimal curve C m over F q . Then Jac( C M × F q F q ) ∼ = Jac( C m × F q F q ) and hence wehave an F q -isomorphism ( C M × F q F q ) ∼ = ( C m × F q F q ).We denote C M × F q F q by C . Then there are automorphisms F M and F m on C which are induced by corresponding Frobenius endomor-phisms. In other words if F q ( C M ) ∼ = F q ( x, y ) and F q ( C m ) ∼ = F q ( u, w ) ⊂ F q ( C ) then F q ( C ) = F q ( x, y ), F M : ( F q ( C ) −→ F q ( C ) P α ij x i y j P β lm x l y m P α qij x i y j P β qlm x l y m , and E. ALEKSEENKO , S. ALESHNIKOV , N. MARKIN, A. ZAYTSEV F m : ( F q ( C ) = F q ( u, w ) −→ F q ( u, w ) P α ij u i w j P β lm u l w m P α qij u i w j P β qlm u l w m , From the construction of the automorphisms F M , F m it follows thatthe quotient curves C/ h F M i and C/ h F m i are defined over F q and F q ( C/ h F M i ) = F q ( C ) h F M i = F q ( C M ) , F q ( C/ h F m i ) = F q ( C ) h F m i = F q ( C m ) . The automorphisms F m and F M induce automorphisms on Jac( C )which we, by abuse of notation, denote by F m and F M , respectively. InEnd F q2 (Jac(C)) we have the relation F m = F M and hence F m = − F M ,since the two are distinct. On the other hand the automorphism F m and F M induce two different automorphisms of C . Therefore Torelli’sTheorem C must be a hyperelliptic curve. But we showed that this isimpossible in Lemma 4.2. (cid:3) Equations of Optimal Curves of Genus − Theorem 5.1.
Let C be an optimal curve over F q . Then C can bewritten in one of the following forms: (cid:26) z = α + α x + α x + β y,y = x + ax + b, (cid:26) z = α + α x + α x + ( β + β x ) y,y = x + ax + b, (cid:26) z = α + α x + α x + α x + ( β + β x ) y,y = x + ax + b, with coefficients in F q and the equation y = x + ax + b correspondingto an optimal elliptic curve.Proof. Let C be an optimal curve of genus 3 over a finite field F q and let f : C → E be a double covering of C with the equation y = x + ax + b .Set D = f − ( ∞ ′ ) = P P |∞ ′ e ( P |∞ ′ ) · P ∈ Div( C ), where ∞ ′ ∈ E liesover ∞ ∈ P by the action E → P , deg D = 2.By Riemann-Roch Theoremdim D = deg D + 1 − g + dim ( W − D ) = dim ( W − D ) , PTIMAL CURVES OVER FINITE FIELDS WITH DISCRIMINANT -19 9 where W is a canonical divisor of the curve C . Consequently, D isequivalent to the positive divisor W − D , where deg D = 2. Concludedim D = dim ( W − D ) < dim W = 3. Taking into account that C is anon-hyperelliptic curve and deg D = 2, we conclude dim D = 1.Consider the divisor 2 D . By Clifford’s Theoremdim 2 D ≤ D. Therefore, dim 2 D ≤ D = 3.Then there exist linearly independent elements 1 , x, z ′ ∈ L (2 D ).Seven elements 1 , x, x , y, z ′ , ( z ′ ) , zx lie in the vector space L (4 D ).Since dim 4 D = 6, then there exists relation a z ′ + a z ′ + a z ′ x = a + a x + a x + a y, where a , a , a , a , a , a , a ∈ F q . Recall that a = 0, otherwisethe equation for z ′ over k ( x, y ) will be of degree 1, which is acontradiction, since [ k ( C ) : k ( x, y )] = 2. Dividing both parts ofthe equation by a and making the substitution z = z ′ + ( a a + a a x ) /
2, we obtain the equation z = α + α x + α x + β y. (2) Suppose dim (2 D ) = 2 and D = Q + Q , where Q = Q , Q , Q ∈ C ( F q ).Then we have dim (2 D + Q ) = 3, by Riemann-Roch Theo-rem. The elements 1 , x, x , y, z, z , xz, yz, xz ∈ L (4 D + 2 Q )are linearly dependent since dim (4 D + 2 Q ) = 8 and x ∈ L (2 D ) , z ∈ L (2 D + Q ) , y ∈ L (3 D ). Therefore, z ( α + α x ) + z ( β + β x + β y ) + ( γ + γ x + γ x + δy ) = 0 . Denoting the expressions in brackets by ϕ , ϕ , ϕ respectively,we rewrite the expression above as z ϕ + zϕ + ϕ = 0 . Knowing that ϕ = α + α x = 0 (otherwise v P ( x ) = 0) andthe equation above can be rewritten as( z + ϕ ϕ ) + ϕ ϕ − ϕ ϕ = 0 . After appropriate substitutions we get the desired equation z = α + α x + α x + α x + ( β + β x ) y. (3) Suppose dim (2 D ) = 2 and D = Q + Q = 2 Q , where Q = Q = Q ∈ C ( F q ).In order to manage this case we prove that the elements1 , x, z, y, x , z , xy, xz are linearly dependent. As a corollaryof this fact we obtain the equation of the second type.In this case the functions x ∈ L (2 D ) , y ∈ L (3 D ) have poledivisors ( x ) ∞ = 4 Q, ( y ) ∞ = 6 Q , and there is a function z ∈ L (2 D + Q ) such that ( z ) ∞ = 5 Q .The element z is an integral element over F q [ x, y ]. Indeed,either 1 , x, z, y, x , z , xy, xz ∈ L (10 D )or 1 , x, y, z, x , zx, xy, z , zy, x , zx , xyz, z ∈ L (15 Q )are linearly dependent and in both cases we have relations withnonzero leading coefficients at z . This yields that z is integralover F q [ x, y ].It is clear that z F q ( x, y ) (otherwise 2 divides v Q ( z ) = 5).The minimal polynomial of z has degree 2 and coefficientsin F q [ x, y ], since the degree of extension [ F q ( C ) : F q ( x, y )] is 2.Therefore we have that z + X i ≥ a i zyx i + X j ≥ b j zx j + X l ≥ c l x l + X s ≥ d s yx s = 0 , and hence(5.1) z + c + c x + c x + d y + b z + b zx + d xy == − z ( b x + . . . ) + zy ( a + a x + . . . ) + ( c x + . . . ) + y ( d x + . . . ) . Furthermore, we have • v Q ( zx i ) = − − i ≡ • v Q ( zyx j ) = − − − i ≡ • v Q ( x l ) = − l ≡ • v Q ( yx i ) = − − i ≡ v Q ( z + c + c x + c x + d y + b z + b zx + d xy ) ≤ − v Q ( z + c + c x + c x + d y + b z + b zx + d xy ) ≥ − . PTIMAL CURVES OVER FINITE FIELDS WITH DISCRIMINANT -19 11
Therefore the right part of the equation above is zero, i. e. theelements 1 , x, z, y, x , z , xy, xz are linearly dependent. (cid:3) Example 5.2.
We produce examples of optimal curves over finite fieldswith discriminant −
19. It suffices to find either a maximal or a minimalcurve as their existence is mutually exclusive. q Maximal optimal curve Minimal optimal curve47 y = x + x + 38, z = 5 + 45 x + 30 x + 10 y -61 y = x + 6 x + 29, z = 2 + 35 x + 48 x + 6 y -137 y = x + x + 36, z = 3 + 85 x + 82 x + 45 y -277 y = x + 2 x + 61, z = x + 33 x + 212 + 5 y -311 - y = x + x + 261, z = 140 + 46 x + 11 x + 78 y
347 - y = x + 2 x + 251, z = 182 + 74 x + 2 x + 5 y y = x + 2 x + 361, z = 259 + 209 x + 6 x + 10 y -557 - y = x + 2 x + 151, z = 439 + 322 x + 5 x + 122 y y = x + 4 x + 105, z = 406 + 131 x + 3 x + 247 y -997 - y = x + 500 x + 934, z = x + 336 x + 564 + 196 y References [1] Pierre Deligne. Vari´et´es ab´eliennes ordinaires sur un corps fini.
Invent. Math. ,8:238–243, 1969.[2] E. Kani and M. Rosen. Idempotent relations and factors of Jacobians.
Math.Ann. , 284(2):307–327, 1989.[3] Kristin Lauter. The maximum or minimum number of rational points on genusthree curves over finite fields.
Compositio Math. , 134(1):87–111, 2002. With anappendix by Jean-Pierre Serre.[4] Frans Oort and Kenji Ueno. Principally polarized abelian varieties of dimensiontwo or three are Jacobian varieties.
J. Fac. Sci. Univ. Tokyo Sect. IA Math. ,20:377–381, 1973.[5] Gerard van der Geer and Marcel van der Vlugt. Supersingular curves of genus2 over finite fields of characteristic 2.
Math. Nachr.
J. Symbolic Comput. , 26(4):487–508, 1998. [7] William C. Waterhouse. Abelian varieties over finite fields.