This is the submitted version of the article: Badr, Eslam; Bars Cortina, Francesc. «Plane model-fields of definition, fields of definition, and the field of moduli for smooth plane curves». Journal of Number Theory, Vol. 194 (January 2019), p. 278-283. DOI 10.1016/j.jnt.2018.07.010 This version is available at https://ddd.uab.cat/record/240664 under the terms of the license PLANE MODEL-FIELDS OF DEFINITION, FIELDS OF DEFINITION, THE FIELD OF MODULI OF SMOOTH PLANE CURVES ESLAM BADR AND FRANCESC BARS Abstract. Given a smooth plane curve C of genus g ≥ 3 over an algebraically closed field k, a field L ⊆ k is said to be a plane model-field of definition for C if L is a field of definition for C, i.e. ∃ a smooth curve C ′ defined over L where C ′ ×L k ∼ = C, and such that C ′ is L-isomorphic to a non-singular plane model F (X, Y, Z) = 0 in P2L . In this short note, we construct a smooth plane curve C over Q, such that the field of moduli of C is not a arXiv:1705.00901v1 [math.NT] 2 May 2017 field of definition for C, and also fields of definition do not coincide with plane model-fields of definition for C. As far as we know, this is the first example in the literature with the above property, since this phenomenon does not occur for hyperelliptic curves, replacing plane model-fields of definition with the so-called hyperelliptic model-fields of definition. 1. Introduction Consider F the base field for an algebraically closed field k. Let F ⊆ L ⊆ k be fields, given a smooth projective curve C over k, then C is defined over L if and only if there is a curve C ′ over L that is k-isomorphic to C, i.e. C ′ ×L k ∼ = C. In such case, L is called a field of definition of C. We say that C is definable over L if there is a curve C ′ /L such that C and C ′ ×L k are k-isomorphic. Definition 1.1. The field of moduli of a smooth projective curve C defined over k, denoted by KC , is the intersection of all fields of definition of C. It becomes very natural to ask when the field of moduli of a smooth projective curve C is also a field of definition. A necessary and sufficient condition (Weil’s cocycle criterion of descent) for the field of moduli to be a field of definition was provided by Weil [12]. If Aut(C) is trivial, then this condition becomes trivially true and so the field of moduli needs to be a field of definition. It is also quite well known that a smooth curve C of genus g = 0 or 1 can be defined over its field of moduli, where g is the geometric genus of C. However, if g > 1 and Aut(C) is non-trivial, then Weil’s conditions are difficult to be checked and so there is no guarantee that the field of moduli is a field of definition for C. This was first pointed out by Earle [4] and Shimura [11]. More precisely, in page 177 of [11], the first examples not definable over their field of moduli are introduced, which are hyperelliptic curves over C with two automorphisms. There are also examples of non-hyperelliptic curves not definable over their field of moduli given in [2, 5]. B. Huggins [6] studied this problem for hyperelliptic curves over a field k of characteristic p 6= 2, proving that a hyperelliptic curve C of genus g ≥ 2 with hyperelliptic involution ι can be defined over KC when Aut(C)/hιi is not cyclic or is cyclic of order divisible by p. On the other hand, one may define fields of definition of models of the same concrete type for a smooth projective curve C. For example, if C is hyperelliptic, a field M is called a hyperelliptic model-field of definition for C if M , as a field of definition for C, satisfies that C is M -isomorphic to a hyperelliptic model of the form y 2 = f (x), for some polynomial f (x) of degree 2g + 1 or 2g + 2. By the work of Mestre [10], Huggins [5, 6], Lercier-Ritzenthaler [7], Lercier-Ritzenthaler-Sijsling [8] and Lombardo-Lorenzo in [9], one gets fair-enough characterizations for the interrelations between the three fields; the field of moduli, fields of definition and hyperelliptic model-fields of definition. For instance, if C is hyperel- liptic, then there are always two of these fields, which are equal. Summing up, one obtains the next table issued from Lercier-Ritzenthaler-Sijsling [8], where k = F is a perfect field of characteristic char(F ) 6= 2: E. Badr and F. Bars are supported by MTM2016-75980-P. 1 2 E. BADR AND F. BARS H = Aut(C)/hιi Conditions Fields of definition = The field of moduli= Hyperelliptic model-fields A field of definition Not tamely cyclic Yes Yes g odd,#Hodd No Yes Tamely cyclic with #H > 1 g even or #H even Yes No g odd No Yes Tamely cyclic with #H = 1 g even Yes No By tamely cyclic, we mean that the group is cyclic of order not divisible by the char(F ). Now, consider a smooth plane curve C, i.e. C viewed as a smooth curve over k admits a non-singular plane model defined by an equation of the form F (X, Y, Z) = 0 in Pk2 , where F (X, Y, Z) is a homogenous polynomial of degree d ≥ 4 over k with g = 21 (d − 1)(d − 2) ≥ 3. Similarly, we define a so-called plane model-fields of definition for C: Definition 1.2. Given a smooth plane curve C over k, a subfield M ⊂ k is said to be a plane model-field of definition for C if and only if the following conditions holds (i) M is a field of definition for C. (ii) ∃ a smooth curve C ′ defined over M , which is k-isomorphic to C, and M -isomorphic to a non-singular plane model F (X, Y, Z) = 0, for some homogenous polynomial F (X, Y, Z) ∈ M [X, Y, Z] of degree d ≥ 3. In this short note, we start with a smooth plane curve C over Q where the field of moduli is not a field of definition by the work of B. Huggins in [5]. Next, we go further, following the techniques developed in [1], to construct a twist of C, for which there is a field of definition for C, which is not a plane model-field of definition. Acknowledgments. We would like to thank Elisa Lorenzo and Christophe Ritzenthaler for bringing this problem to our attention, as a consequence of our discussion with them in BGSMath-Barcelona Graduate School in March 2017. 2. The example Consider the Hessian group of order 18, denoted by Hess18 , which is PGL3 (Q)-conjugate to the group generated by       1 0 0 0 10 10 0 S :=  0 ζ3 0  , T :=  0 0 1  , and R :=  0 0 1  .       0 0 ζ32 1 00 01 0 First, we reproduce an example, by B. Huggins in [5, Chp. 7, §2], of a smooth Q-plane curve of genus 10 not definable over its field of moduli, and with full automorphism groups Hess18 . Definition 2.1. A quaternion extension of a field K is a Galois extension K ′ /K such that Gal(K ′ /K) is isomorphic to the quaternion group of order 8. Definition 2.2. ( [5, Lemma 7.2.3]) A field K is of level 2 if −1 is not a square in K, but it is a sum of two squares in K. Lemma 2.3. ( [5, Lemma 7.2.3]) Let K be a field of level 2. Then, for u, v ∈ K ∗ \ (K ∗ )2 such that uv ∈ / (K ∗ )2 , √ √ √ K( u, v) is embeddable into a quaternion extension of K if and only if −u is a norm from K( −v) to K (i.e. −u = x2 + vy 2 for some x, y ∈ K). √ For instance, the field K := Q(ζ3 ) is of level 2, since (ζ32 )2 + ζ32 = −1 and −1 ∈ / K. It is easily shown √ √ √ √ √ that ±2 are not norms from K( −13) to K. So neither K( 2, 13) nor K( −2, 13) are embeddable into a quaternion extension of K. Now fix K to be the field Q(ζ3 ), and define the following: φ := XY Z, ψ := X 3 + Y 3 + Z 3 , χ := (XY )3 + (Y Z)3 + (XZ)3 . PLANE MODEL-FIELDS OF DEFINITION, FIELDS OF DEFINITION, THE FIELD OF MODULI OF SMOOTH PLANE CURVES 3 √ √ Suppose that u, v ∈ Q∗ , such that L := K( u, v) is a Z/2Z × Z/2Z extension of K that can not be embedded into a quaternion extension of K. Let √ √ √ cφ2 := ζ3 u + v + ζ32 uv, √ √ √ cφψ := ζ32 u + v + ζ3 uv, √ √ √ 1 cψ2 := u + v + uv − . 12 Fix an algebraic closure Q of Q containing L as above. Theorem 2.4. (B. Huggins, [5, Lemma 7.2.5 and Proposition 7.2.6]) Following the above notations, let F√u,√v (X, Y, Z) := cφ2 φ2 − 6cφψ φψ − 18cψ2 ψ 2 + χ. Then the equation F√u,√v (X, Y, Z) = 0 such that F√u,√v (X, 1, 1) is square free, defines a smooth Q-plane curve C over Q, with automorphism group Hess18 . The field of moduli KC is K = Q(ζ3 ), but it is not a field of definition. Remark 2.5. The condition that F√u,√v (X, 1, 1) is square free is possible. For example, with u = 2 and v = 13, ∂F the resultant of F√2,√13 (X, 1, 1) and ∂X (X, 1, 1) is not zero. Lemma 2.6. Let C be a smooth curve defined over an algebraically closed field k, with F = k and k perfect. An k-isomorphism φ : C ′ → C does not change the field of moduli or fields of definition, that is both C and C ′ have the same fields of moduli and fields of definitions. Proof. A field L ⊆ k is a field of definition for C if and only if there exists a smooth curve C ′′ over L, such that C ′′ ×L k is k-isomorphic to C through some ψ : C ′′ ×L k → C. Hence φ−1 ◦ ψ : C ′′ ×L k is a k-isomorphism, and L is a field of definition for C ′ . The converse is true by a similar discussion. Consequently, the field of moduli for C and C ′ coincides, being the intersection of all fields of definition.  Corollary 2.7. Consider a smooth Q-plane curve C defined by an equation of the form cφ2 6cφψ 1 1 1 1 1 1 1 (XY Z)2 − (XY Z)(X 3 + Y 3 + 2 Z 3 )−18cψ2 (X 3 + Y 3 + 2 Z 3 )2 + X 3 Y 3 + 3 (Y Z)3 + 2 X 3 Z 3 = 0, p2 p p p p p p p p √ √ where p ∈ Q, in particular C admits Q( u, v, ζ3 ) as a plane model-field of definition for C. Then Aut(C) is isomorphic to Hess18 . Moreover, the field of moduli KC is K = Q(ζ3 ), but it is not a field of definition. √ Proof. Since C is Q( 3 p)-isomorphic to F√u,√v (X, Y, Z) = 0 through a change of variables of the shape φ = √ p diag(1, 1/ 3 p, 1/ 3 p2 ), therefore they have conjugate automorphism groups. Moreover, fields of definition and the field of moduli of both curves are the same by Lemma 2.6. Consequently, the field of moduli KC is K = Q(ζ3 ), but it is not a field of definition, using Theorem 2.4.  Theorem 2.8. Consider the family Cp of smooth plane curves over the plane model-field of definition L = √ √ Q( u, v, ζ3 ) given by an equation of the form cφ2 6cφψ 1 1 1 1 1 1 1 2 (XY Z)2 − (XY Z)(X 3 + Y 3 + 2 Z 3 )−18cψ2 (X 3 + Y 3 + 2 Z 3 )2 + X 3 Y 3 + 3 (Y Z)3 + 2 X 3 Z 3 = 0, p p p p p p p p p where p is a prime integer such that p ≡ 3 or 5 mod 7. Given a smooth plane curve C over L in Cp , then there exists a twist C ′ of C over L which does not have L as a plane model-field of definition. Moreover, the field of moduli of C ′ is Q(ζ3 ), and is not a field of definition for C ′ . √ Proof. Consider the Galois extension M ′ /L with M ′ = L(cos(2π/7), 3 p), where all the automorphisms of C := C ×L Q are defined. Let σ be a generator of the cyclic Galois group Gal(L(cos(2π/7))/L). We define a √ 1-cocycle on Gal(M ′ /L) ∼ = Gal(L(cos(2π/7))/L) × Gal(L( 3 p)/L) to Aut(C) by mapping (σ, id) 7→ [Y : Z : pX] and (id, τ ) 7→ id. This defines an element of H1 (Gal(M ′ /L), Aut(C)), coming from the inflation of an element in H1 (Gal(L(cos(2π/7))/L), Aut(C))Gal(M /L(cos(2π/7))) ). ′ This 1-cocycle is trivial if and only if p is a norm of an element of L(cos(2π/7) over L. However, this is not the case, since Q(cos(2π/7)) and L are disjoint with [L : Q] and [Q(cos(2π/7)) : Q] coprime, and moreover p is 4 E. BADR AND F. BARS not a norm of an element of Q(cos(2π/7)) over Q being inert by our assumption. Consequently, the twist C ′ is not L-isomorphic to a non-singular plane model in P2L by [1, Theorem 4.1]. That is, L is not a plane model-field of definition for C ′ . The last sentence in the theorem follows by Lemma 2.6 and Corollary 2.7.  Remark 2.9. By our work in [1], we know that a non-singular plane model of C ′ exists over at least a degree degree 3 extension of L. References [1] E. Badr, F. Bars, E. Lorenzo Garcı́a, On twists of smooth plane curves, arXiv:1603.08711v1. [2] R. Hidalgo, Non-hyperelliptic Riemann surfaces with real field of moduli but not definable over the reals, Arch. Math. 93 (2009), 219-224. [3] B. Huggins; Fields of moduli and fields of definition of curves. PhD thesis, Berkeley (2005), see http://arxiv.org/abs/math/0610247v1. [4] C. J. Earle, On the moduli of closed Riemann surfaces with symmetries, Advances in the Theory of Riemann Surfaces. Ann. Math. Studies 66 (1971), 119-130. [5] B. Huggins, Fields of moduli and fields of definition of curves. PhD thesis, Berkeley (2005), arxiv.org/abs/math/0610247v1. [6] B. Huggins; Fields of moduli of hyperelliptic curves. Math. Res. Lett. 14 (2007), 249-262. [7] R. Lercier and C. Ritzenthaler. Hyperelliptic curves and their invariants: geometric, arithmetic and algorithmic aspects. J. Algebra, 372:595636, 2012. [8] R. Lercier, C. Ritzenthaler, and J. Sijsling. Explicit galois obstruction and descent for hyperelliptic curves with tamely cyclic reduced automorphism group. Math. Comp, To appear. [9] D. Lombardo, E. Lorenzo Garcı́a;Computing twists of hyperelliptic curves, arXiv:1611.04856, November 2016. [10] J.-F. Mestre. Construction de courbes de genre 2 a partir de leurs modules. In Effective methods in algebraic geometry (Castiglioncello, 1990) , volume 94 of Progr. Math. , pages 313334. Birkhäuser Boston, Boston, MA, 1991. [11] G. Shimura, On the field of rationality for an abelian variety, Nagoya Math. J. 45 (1971), 167-178. [12] A. Weil, The field of definition of a variety, American J. of Math. vol. 78, n17 (1956), 509-524. • Eslam Essam Ebrahim Farag Badr Departament Matemàtiques, Edif. C, Universitat Autònoma de Barcelona, 08193 Bellaterra, Catalonia, Spain E-mail address: eslam@mat.uab.cat Department of Mathematics, Faculty of Science, Cairo University, Giza-Egypt E-mail address: eslam@sci.cu.edu.eg • Francesc Bars Cortina Departament Matemàtiques, Edif. C, Universitat Autònoma de Barcelona, 08193 Bellaterra, Catalonia E-mail address: francesc@mat.uab.cat