This is the submitted version of the article: Badr, Eslam; Bars Cortina, Francesc. «Bielliptic smooth plane curves and quadratic points». International Journal of Number Theory, Vol. 17, Issue 4 (May 2021), p. 1047-1066. DOI 10.1142/S1793042121500238 This version is available at https://ddd.uab.cat/record/240649 under the terms of the license BIELLIPTIC SMOOTH PLANE CURVES AND QUADRATIC POINTS ESLAM BADR AND FRANCESC BARS Abstract. Let Ck be a smooth plane curve of degree d ≥ 4 defined over a global field k of characteristic p = 0 or p > (d − 1)(d − 2)/2 (up to an extra condition on Jac(Ck )). Unless the curve is bielliptic of degree 4, we observe that it always admits finitely many quadratic points. We further show that there are only finitely many √ quadratic extensions k( D) when k is a number field, in which we may have more points of Ck than these over k. In particular, we have this asymptotic phenomenon valid for the Fermat’s and the Klein’s equations. Second, we conjecture that there are two infinite sets E and D of isomorphism classes of smooth projective plane quartic curves over k with a prescribed automorphism group, such that all members of E (respectively, D) are bielliptic and have finitely (respectively, infinitely) many quadratic points over a number field k. We verify the conjecture over k = Q for G = Z/6Z and GAP(16, 13). The analogue of the conjecture over global fields with p > 0 is also considered. 1. Introduction Let Ck be a smooth projective curve defined over a global field k of characteristic p, which is neither rational nor elliptic. The set of K-points of Ck := Ck ⊗ k is denoted by C(K), where K/k is a field extension inside a fixed algebraic closure k of k. For a finite field extension L/k ⊆ k, the set of quadratic points of CL = Ck ⊗ L S is denoted by Γ2 (C, L). That is, Γ2 (C, L) = C(K) : K/L ⊆ k and [K : L] ≤ 2 . The work of Harris and Silverman over zero characteristic [19], and Schweizer over positive characteristic [37] (assuming an extra condition on the Jacobian variety Jac(Ck )) confirms that the set Γ2 (C, L) is infinite for some L if and only if Ck is hyperelliptic or bielliptic. Suppose now that Ck is a smooth plane curve of degree d ≥ 4, so it is never rational or hyperelliptic. Over characteristic p = 0, it is not hard to see that Ck is bielliptic if and only if it is a smooth quartic curve with an automorphism of order 2 (an involution). For instance, Bardelli and Del Centina [7] showed that a complex smooth plane quartic curve with an involution is the canonical model of a C-point of the locus M3,bi whose points are the C-isomorphism classes of smooth bielliptic curves of genus 3. In this direction, we introduce the notation M ^Pl (G) for the locus of k-isomorphism classes of smooth plane 3 quartic curves whose automorphism groups are exactly isomorphic to the group G. Next, motivated by the arithmetic geometry point of view, we find interest to study the set of its k-points M^Pl (G)(k) i.e. modulo k- 3 isomorphism of smooth plane non-hypererelliptic genus 3 curves defined over k (equivalently smooth plane non singular curves over k, by 3.1) whose automorphism group in k is isomorphic to G. in particular the bielliptic ones in M^ Pl (G) := M 3,bi ^Pl (G) ∩ M 3 . The research of Abramovich, Harris, and Silverman over number fields 3,bi allows us to say that a k-point C of M^ Pl (G)(k) has infinitely many quadratic points over k if and only if 3,bi Aut(Ck ) contains a (bielliptic) involution w defined over k such that the bielliptic quotient Ck /w is an elliptic curve of positive rank over k. In this case, we call C bielliptic over k. However, the arithmetic (geometric) progressions of those quadratic points must be finite [6]. Now, once a finite group G is fixed such that M^ Pl (G) 6= ∅, we partition the M^ 3,bi Pl (G)(k) into two disjoint 3,bi subsets E and D as follows: given C ∈ M^ Pl (G)(k), C ∈ E if and only if C has finitely many quadratic points 3,bi over k, and C ∈ D otherwise. In section §4, we conjectured that E and D are of infinite cardinality. We verified the claim using the twisting theory for curves if k = Q and G = Z/6Z or GAP(16, 13); see Theorems 4.2, 4.4, 4.7 and 4.9, also if k = Fp (T ) and G = Z/6Z; see Theorem 4.11. It is obvious to the reader that this has Key words and phrases. Plane curves; Bielliptic curves; Automorphism group; twist. F. Bars is partially supported by MTM2016-75980-P. 1 2 E. BADR AND F. BARS some connection with ranks distributions for families of elliptic curves. This topic finds a lot interest and it has been investigated by so many people around. Examples including, but not limited to Hulek-Kloosterman [21] or Salgado [35]. The first two sections are just a state of the art that can be interpreted as immediate consequences of well- known results. However, they are needed to introduce and motivate the problem. In our way to do this, we observed an asymptotic phenomenon for any smooth projective plane curve of degree d ≥ 5 over k, which is different from the asymptotic phenomenon for the Fermat’s equations [15]; see Corollary 3.7. 2. The interplay between hyperelliptic (respectively, bielliptic) curves and quadratic points Let k (respectively, k sep ) be a fixed algebraic (respectively, separable) closure of a field k of characteristic p 6= 2, once and for all. By Ck we mean a smooth projective curve defined over k such that the base extension Ck := C ⊗k k has genus g ≥ 2 and non-trivial automorphism group Aut(Ck ). The set of all k-points of Ck is denoted by C(k). An arithmetic geometer finds a lot of interest to investigate the cardinality of C(k) when k is a global field; (i) k is a finite field extension of either Q or (ii) k is a finite extension of the quotient field of the polynomial ring over Fp with the variable T . In the latter case, the finite field k ∩ Fp is denoted by Fq where q is some power of p. Over zero characteristic, we have the following result on Mordell’s Conjecture due to Faltings [13, 14]: Theorem 2.1 (Faltings). Given a smooth projective curve Ck as above defined over a number field k, the set C(k) is always finite. Over positive characteristic, the genus of Ck relative to k is the smallest integer gC,k such that, for any k-divisor D of C of sufficiently large degree, `(D) = deg(D) + 1 − gC,k where `(D) is the dimension of the associated k-(Riemann-Roch) vector space. For instance, the (geometric) genus g defined earlier is actually the genus of Ck relative to k. The relative genus may change under inseparable extensions of k inside k, see for example [41]. Also, the relative genus gC,k is an upper bound for the absolute genus g. In this sense, the curve Ck is conservative over k if g = gC,k , hence it is not genus-changing under inseparable extensions between k and k. For smooth plane curves, we have: Lemma 2.2. Let Ck be a smooth plane curve of degree d ≥ 4 over a global field k of characteristic p > (d − 1)(d − 2) + 1. Then, Ck and all its quotients are conservative over k. Proof. The result follows by [41, Corollary 2], since the relative genus gC,k = (d − 1)(d − 2)/2 < (p − 1)/2, and also all its quotients.  Similar to Theorem 2.1, we have the next result over positive characteristic due to Grauert [17] and Samuel [36, Theorem 4 and 5b]: Theorem 2.3 (Grauert-Samuel). Let Ck be a smooth projective curve over a global field k of characteristic p > 0. If Ck is conservative, then C(k) is a finite set except possibly when Ck is an isotrivial curve, that is, when Ck ⊗k k sep isomorphic to a smooth projective curve over a finite field Fqn . Remark 2.4. Grauert-Samuel Theorem requires Ck to be geometrically non-singular instead of being conser- vative. But if Ck is conservative, then it is geometrically non-singular by [17]. This following is an example of Theorem 2.3 where an infinite number of points occurs. Example 2.5. Consider the Fermat curve Ck : X d + Y d = Z d over the global field k = Fp (w, v)/(wd + v d − 1) with d and p coprime. First, it is isomorphic to the Fermat curve over Fp . Second, it has infinitely many points over k namely, (f n (w), f n (v), 1) for each positive integer n. Those points are constructed from the point (w, v, 1) ∈ C(k) \ C(Fp ) via the Frobenious f : x 7→ xp . BIELLIPTIC SMOOTH PLANE CURVES AND QUADRATIC POINTS 3 Motivated by these results, it became natural to investigate the infinitude of the set Γ2 (C, L) of quadratic points of C over L. Definition 2.6. A smooth projective curve Ck is called hyperelliptic (respectively, bielliptic) over k if there exists a degree two k-morphism to a projective line P1k (respectively, to an elliptic curve Ek ) over k. We simply call it hyperelliptic (respectively, bielliptic) if Ck is hyperelliptic (respectively, bielliptic) over k. Clearly, if Ck is hyperelliptic over k and k is not a finite field, then Γ2 (C, k) is an infinite set. Also, if Ck is bielliptic over k and Ek has infinitely many k-points, then Γ2 (C, k) is again an infinite set. The following result is well-known in the literature (cf. [37] for (ii)). Proposition 2.7. Let Ck be a smooth projective curve over k with g ≥ 2, recall p 6= 2. Then, (i) Ck is hyperelliptic if and only if there exists a (hyperelliptic) involution w ∈ Aut(Ck ) having exactly 2g + 2 fixed points. In particular, if Ck is hyperelliptic, then w is unique, defined over a finite purely inseparable extension of k, and it is called the hyperelliptic involution of Ck . (ii) Ck is bielliptic if and only if there exists a (bielliptic) involution w̃ ∈ Aut(Ck ) having 2g − 2 fixed points. If Ck is bielliptic and g ≥ 6, then w̃ is unique, defined over a finite purely inseparable extension of k, and it belongs to the center of Aut(Ck ). 2.1. Number fields. Assume that k is a global field of characteristic zero, or simply a number field. Because inseparable extensions between k and k do not exist, Proposition 2.7 leads to the following well-known result (cf. [9, Lemma 2.5] and [19, Lemma 5]): Proposition 2.8. Let Ck be a smooth projective curve Ck defined over a number field k. It is hyperelliptic with hyperelliptic involution w such that C/hwi(k) 6= ∅ if and only if it is hyperelliptic over k. Also, for g ≥ 6, Ck is bielliptic if and only if it is bielliptic over k. The next result is known to the specialists and it follows from Abramovich-Harris in [1], or Harris-Silverman in [19]. One also can read a proof in [9, Theorem 2.14]. Theorem 2.9. Let Ck be a smooth projective curve over a number field k with g ≥ 2. The set Γ2 (C, k) is an infinite set if and only if Ck is hyperelliptic over k or Ck is bielliptic over k such that it exists a degree two k-morphism form Ck to an elliptic curve Ek of positive rank over k. 2.2. Global fields with p > 0. First, we prove: Proposition 2.10. Let Ck be a smooth projective curve defined over a global field k of characteristic p > 0, that is conservative over k. Assume also that Ck is hyperelliptic with hyperelliptic involution w defined over a finite purely inseparable extension `/k in k. Then, there is a (unique) degree two `-morphism ϕ to a conic Q over `. Moreover, if C` (or more generally if C` /hwi) has an `-point, then Q is `-isomorphic to P1` . Proof. By assumption C` /hwi is conservative genus 0 curve defined over ` (Lemma 2.2), so it corresponds to a conic over `. Next, the covering π : C` → C` /hwi is cyclic of order 2, hence it is Galois. Thus π is defined over a separable extension of ` inside k. Now, fix a separable closure `sep ⊆ k of ` and put Gal(`sep /`) for the absolute Galois group. By the uniqueness of the hyperelliptic involution w, π and σ π only differs by an automorphism ξσ of P1`sep for any σ ∈ Gal(`sep /`). In other words, for any σ ∈ Gal(`sep /`), we obtain ξσ ∈ PGL2 (`sep ); the projective general linear group of 2 × 2 matrices over `sep , where σ π = ξσ ◦ π. It can be easily checked that ξστ = σ ξτ ◦ ξσ for all σ, τ ∈ Gal(`sep /`). Therefore, ξ : Gal(`sep /`) → PGL2 (`sep ) : σ 7→ ξσ defines a 1-cocycle, in particular, an element of the first Galois cohomology set H1 (Gal(`sep /`), PGL2 (`sep )). Using the Twisting Theory for Varieties (cf. [38, III.1]), it exists a conic Q (a twist for P1` ) over ` and an isomorphism ϕ0 : Q → C` /hwi given by the rule ξσ = σ ϕ0 ◦ ϕ−1 0 , for all σ ∈ Gal(` sep /`). Consequently, ϕ := ϕ−1 0 ◦ π : C` → Q is an `-morphism from C` to Q. The rest is direct because a conic over ` with an `-point is the trivial twist for P1` . This obviously happens through the morphism ϕ if C` or C` /hwi have `-points.  4 E. BADR AND F. BARS Corollary 2.11. Let Ck be a smooth projective curve defined over a global field k of characteristic p > 0, that is conservative over k. Then, Ck is hyperelliptic if and only if there exists a finite extension L/k inside k such that Ck ⊗k L is hyperelliptic over L. In this situation, Γ2 (C, L) is an infinite set. Corollary 2.12. Let Ck be a smooth projective curve defined over a global field k of characteristic p > 2, that is conservative over k and such that Ck is not hyperelliptic. If Ck is bielliptic, then there exists a finite extension L/k inside k such that Ck ⊗k L is bielliptic over L. Consequently, Γ2 (C, L0 ) is an infinite set for some finite extension L0 /L inside k. Proof. Suppose that Ck is bielliptic, and consider a bielliptic involution w̃ ∈ Aut(Ck ) as in Proposition 2.7. Since g ≥ 2 and Aut(Ck ) is finite, we have only finitely many possibilities for the Galois group Gal(k/k)-action on w̃. So w̃ must be defined over a finite field extension L0 /k inside k. Because Ck is also conservative over k, then by making a finite extension L/L0 with L ⊆ k, we get a degree two L-morphism from Ck ⊗k L to a genus one curve (by Lemma 2.2) that has L-points, thus to an elliptic curve E over L, and hence Ck is bielliptic over L. Finally, it suffices to apply base extension to some L ⊆ L0 ⊆ k of finite index so that E ⊗L L0 has positive rank. Consequently, Γ2 (C, L0 ) is infinite and we conclude.  Furthermore, inspired by the case of number fields (Theorem 2.9 above), we have [37, Theorem 5.1]: Theorem 2.13 (Schweizer). Let Ck be a conservative smooth projective curve over a finite field extension k of Fq (T ); i.e. k is a global field of characteristic p > 2. Under the conditions: g ≥ 3, C(k) 6= ∅, Ck is not hyperelliptic over k, and that the Jacobian variety Jac(Ck ) over k has no non-zero homomorphic images defined over Fq , the set Γ2 (C, k) is infinite only if Jac(Ck ) over k contains an elliptic curve Ek of positive rank. Moreover, there is a degree two morphism from Ck to Ek . Remark 2.14. The condition that an abelian variety A over k has no non-zero homomorphic images defined over Fq means that A has no isogeny factor defined over Fq . The isogeny, however, can be defined over k. So what is excluded the existence of an abelian variety B over F q such that over k there exists an isogeny from a isogeny factor of the base change of A onto B. As an example, the conditions on Jac(Ck ) imply that Ek is not isotrivial (that is, its j-invariant does not belong to Fq ) and that any factor of the Jacobian is a Jacobian of an isotrivial curve. The curve Ck : Z 4 + X 4 + Y 4 − 3X 2 Y 2 = 0 is an example for which the theorem does not apply. It appears for its Jacobian an elliptic curve with j-invariant 1728, hence isotrivial. See the proof of Lemma 4.6 Remark 2.15. We observe that if g ≥ 6 and p > 2, then the degree two morphism from Ck to Ek described in Theorem 2.13 is defined over k, and Ck is a bielliptic curve over k. Effectively, if g ≥ 6, then the same argument of [19, Lemma 5] shows the result provided that Castellnuovo’s inequality holds over the function field extensions of k which are involved (all are of degree 2 in Theorem 2.13). Castellnouvo’s inequality is true when k is perfect, see [40, Theorem 3.11.3]. However, following the proof of [40, Theorem 3.11.3] under the assumption that Ck is conservative, Castellnouovo inequality remains true if the characteristic p is big enough to guarantee that inseparable extensions do not appear in the extensions of the fields involved in the inequality. Consequently, under the hypothesis of Theorem 2.13 together with the assumptions g ≥ 6 and p > 2, we get Ck is bielliptic over k. 3. Bielliptic smooth plane curves and their quadratic points Let C/k be a smooth projective curve over a field k. Suppose that L/k ⊆ k is a field extension such that CL := C ⊗k L is a non-singular plane curve F (X, Y, Z) = 0 over L. In such a case we call L a plane model field of definition for C and C is a smooth (k, L)-plane curve. We simply call C a smooth plane curve over k if L = k is plane model field of definition for C. We investigated (minimal) plane models fields of definition for smooth (k, k sep )-plane curves in [5, §2]. For instance, we showed: BIELLIPTIC SMOOTH PLANE CURVES AND QUADRATIC POINTS 5 Theorem 3.1 (Badr-Bars-Garcı́a). Given a smooth (k, k sep )-plane curve C of degree d ≥ 4, it does not neces- sarily have a non-singular plane model defined over the field k. However, it does in any of the following cases; if d is coprime with 3, if C(k) 6= ∅, or if the 3-torsion Br(k)[3] of the Brauer group Br(k) is trivial. In general, C is a smooth (k, L)-plane curve of degree d for some cubic Galois field extension L/k. When k is a plane model field of definition for C, then Ck has a unique gd2 -linear series modulo PGL3 (k)- conjugation, see [22, Lemma 11.28]. This allows us to embed Ck into Pk2 as a non-singular plane model over k of some degree d, in particular, the genus g := g = (d − 1)(d − 2)/2. Moreover, we have: Proposition 3.2. A smooth (k, k)-plane curve C of degree d ≥ 5 is neither hyperelliptic nor bielliptic. Also, C is never hyperelliptic when d = 4. Proof. It is a basic fact in algebraic geometry that a smooth (k, k)-plane curve of degree d ≥ 4 is non- hyperelliptic. On the other hand, the (geometric) gonality of a smooth projective curve C is defined to be the minimum degree of a k-morphism from Ck to the projective line P1k . For the special case of smooth (k, k)- plane curves C of degree d ≥ 4, the gonality equals to d − 1 (cf. Namba [33] for zero characteristic and Homma [23] for positive characteristic, also one can read the assertion in [45, page 341]). Consequently, the gonality of C is at least 5 when d ≥ 6 and C can not be bielliptic. For d = 5 and k = C, see [2, Chapter V, Exercises A-6 and A-7], but here we will introduce a general argument: By assumption, Ck admits a bielliptic involution, that is, an automorphism w̃ of order 2 fixing 2g − 2 = 10 points on Ck . By [22, Lemma 11.28], we can take w̃ : X 7→ X, Y 7→ Y, Z 7→ −Z up to PGL3 (k)-conjugation. In particular, Ck is defined by an equation of the form Z 4 L1,Z + Z 2 L2,Z + L5,Z = 0, where Li,Z denotes a homogenous degree i binary form in X, Y . This means that w̃ fixes exactly d + 1 = 6 < 10 points on Ck , which contradicts the fact that C is bielliptic.  3.1. Smooth plane quartic curves, which are bielliptic. Over the complex field C, Bardelli and Del Centina [7] showed that any non-singular plane quartic curve F (X, Y, Z) = 0 over C that admits an auto- morphism of order 2 is the canonical model of a bielliptic curve of genus 3. Moreover, they characterized a distinguished (canonical) equation for any (C, C)-plane quartic curve C, once a bielliptic structure on C is fixed. As a consequence, a nice configuration was obtained for the 84 proper second order Weierstrass point of the Klein quartic curve K : X 3 Y + Y 3 Z + Z 3 X. See [7, Section 2] for further details. We remark that the Klein quartic K is the unique C-point of the locus M3,bi ^ (PSL(2, 7)), which is 0- dimensional. However, this locus has several Q-points that are not isomorphic over Q, but isomorphic over C to the Klein quartic curve K above. In particular, those Q-points come up with different arithmetic properties. Since the paper is closer to arithmetics, we will be following the approach developed by Lercier-Ritzenthaler- Rovetta-Sijsling in [27]. Let Mg be the (coarse) moduli space, representing k-isomorphism classes of smooth curves of genus g. For a finite non-trivial group G, consider the locus M^Pl (G) ⊂ M consisting of all k-isomorphism classes of smooth g g plane curves [C] with automorphism group PGLg (k)-conjugated to %(G) for some injective representation % : G ,→ PGLg (k). Lercier-Ritzenthaler-Rovetta-Sijsling in [27, §2] introduced the notions of complete, and representative fam- ilies for loci of Mg when k is any field of characteristic p = 0 or p > 2g + 1. In the case of plane curves, a family C of smooth plane curves over k is said to be complete over k for M ^Pl (G) if, for any algebraic extension g 0 0 ^Pl (G)(k 0 ), there exists a k-isomorphic non-singular plane model for C k /k inside k and any k -point C in the M g defined over k 0 in the family C. If such a model in the family is always unique, then the family C is representative over k for M ^Pl (G). A family C is geometrically complete (respectively, geometrically representative) if C ⊗ k k g is complete (respectively, representative) over k. The following result is a reformulation of [7, Corollary 2.2 and Thoerem 2.5] in terms of the notions developed in [27]. Theorem 3.3 (Bielliptic quartic curves). Let k be a field of characteristic p = 0 or p > 7. A smooth (k, k)- plane quartic curve C is bielliptic if and only if C ⊗k k is isomorphic to a non-singular plane model of the form Z 4 + Z 2 L2,Z + L4,Z = 0, where Li,Z is a homogenous binary form in k[X, Y ] of degree i. 6 E. BADR AND F. BARS Table 1 below gives a geometrically complete family for each locus of M3 of smooth plane bielliptic quartic curves over k. Table 1. Bielliptic geometrically complete classification Aut(Ck ) Families Restrictions Z/2Z Z 4 + Z 2 L2,Z (X, Y ) + L4,Z (X, Y ) L2,Z (X, Y ) 6= 0, not below Z/2Z × Z/2Z Z + Z 2 (bY 2 + cX 2 ) + (X 4 + Y 4 + aX 2 Y 2 ) 4 a 6= ±b 6= c 6= ±a Z/6Z Z 4 + aZ 2 Y 2 + (X 3 Y + Y 4 ) a 6= 0 S3 (X 3 + Y 3 )Z + X 2 Y 2 + aXY Z 2 + bZ 4 a 6= b, ab 6= 0 D4 Z 4 + bXY Z 2 + (X 4 + Y 4 + aX 2 Y 2 ) b 6= 0, ± √2a1−a 4 4 4 2 2 √ GAP(16, 13) Z + (X + Y + aX Y ) ±a 6= 0, 2, 6, 2 −3 √ −1± −7 S4 Z 4 + aZ 2 (Y 2 + X 2 ) + (X 4 + Y 4 + aX 2 Y 2 ) a 6= 0, 2 Z 4 + X 4 + Y 4 + (4ζ3 + 2)X 2 Y 2  GAP(48, 33) − GAP(96, 64) Z 4 + (X 4 + Y 4 ) − PSL2 (F7 ) X3Y + Y 3Z + Z3X − The algebraic restrictions for the parameters over k, in the last column, are taken so that the defining equation is non-singular and has no bigger automorphism group. For example, the term “not below” means to assume more restrictions for no larger automorphism group to occur. Here ζ3 denotes a fixed primitive 3rd root of unity. The automorphism groups are given in GAP library [16] notations. Remark 3.4. The two highlighted cases in Table 1 are not in the prescribed form Z 4 + Z 2 L2,Z + L4,Z = 0, that is, diag(1, 1, −1) is not an explicit bielliptic involution. However, they do up to PGL3 (k)-conjugation. The way to see this is elementary; any 3 × 3 projective linear transformation of order two of the plane fixes (point-wise) a line L ⊂ P2k , called its axis. So, we may ask for a change of variables φ such that the transformed L becomes the projective line L0 : Z = 0. For example, for G = S3 , the change of variables   1 −1 1  1 −1 −1    0 2 0 does the job. It transforms the axis L : X − Y = 0 of involution X ↔ Y, Z 7→ Z to L0 : Z = 0. In particular, we obtain the k-equivalent family Z 4 − 2Z 2 X 2 − 8XY + (2a + 7)Y 2 + X 4 + 2(2a − 3)(XY )2 − 8(a − 1)XY 3 + (4a + 16b − 3)Y 4 .   Proof of Theorem 3.3. The assertion that non-singular plane models of the form Z 4 + Z 2 L2,Z + L4,Z = 0 characterizes smooth (k, k)-plane bielliptic quartic curves up to PGL3 (k)-equivalence can be found in [7]. The stratification by automorphism groups follows from the work of Henn [20] (see also [8]). The geometrically complete families are issued from [27, 28].  Remark 3.5. Any locus of the shape M ^Pl (G) corresponds to a unique representation % : G ,→ PGL (k), 3 3 provided that it is non-empty. The uniqueness of % up to conjugation in PGL3 (k) is not guaranteed for d > 4, since it might happen that two smooth (k, k)-plane curves C, C 0 of degree d > 4 have isomorphic non-conjugated automorphism groups. In particular, C and C 0 are not k-isomorphic. See [3, 4] for more details and examples. Theorem 3.6. Let C be a smooth plane curve of degree d ≥ 4 over a number field k. Unless C is a smooth plane bielliptic quartic curve, the set Γ2 (C, L) has finite cardinality for any finite field extension L/k ⊆ k. Proof. We know by Proposition 3.2 that C is never hyperelliptic. Moreover, by Proposition 3.2 and Theorem 3.3, it is not bielliptic if (i) d ≥ 5 or (ii) d = 4 such that Aut(Ck ) ∼ = 1, Z/3Z, or Z/9Z. In those situations, we BIELLIPTIC SMOOTH PLANE CURVES AND QUADRATIC POINTS 7 guarantee that Γ2 (C, L) is finite due to Theorems 2.13, and 2.9. Otherwise, Γ2 (C, L) would be an infinite set for some finite extension L/k ⊆ k by Theorem 2.9.  The next asymptotic phenomenon is probably known to specialists, but we present a quite simple proof: Corollary 3.7. Let C be a smooth plane curve of degree d ≥ 5 over a number field k. Then, there are only finitely many quadratic field extensions k 0 /k ⊆ k such that C(k 0 ) 6= C(k). In particular, for the Fermat curve C : X d + Y d − Z d = 0 or the Klein curve C : X d−1 Y + Y d−1 Z + Z d−1 X = 0 of degree d ≥ 5 over Q, we √ have C(Q) = C(Q( D)) for all but finitely many square-free integers D. This means that only finitely many √ √ quadratic extensions Q( D) satisfy #C(Q( D)) 6= #C(Q). Proof. By definition, C(k) ⊆ C(k 0 ) ⊆ Γ(C, k) for any quadratic field extension k ⊂ k 0 ⊆ k. By Theorem 3.6, Γ2 (C, k) is a finite set. So C(k) ( C(k 0 ) occurs for for finitely many k 0 .  4. Quadratic points on smooth plane curves fixing the base field Let C be a smooth (k, k)-plane quartic curve as in Theorem 3.3. Because the degree d = 4 is coprime with 3, we deduce by Theorem 3.1 that k is not only a field of definition for C, but also a plane-model field of definition. We find some interest to conjecture the following over a number field k. Conjecture 4.1. Fix a non-empty loci M3,biell ^(G), and denote its k-points by M3,biell^(G)(k). Then, there ^(G)(k) such that each element C of E (respectively, D) exists an infinite subset E (respectively, D) of ⊆ M3,biell corresponds to a k-isomorphism class of smooth plane quartic curves over k with Γ2 (C, k) finite (respectively, infinite). We refer to section §4.3 for a version of this conjecture over positive characteristic. By the work of Lercier-Ritzenthaler-Rovetta-Sijsling in [27, §2], we have a parametrization of the loci M ^Pl (G) 3 in terms of complete and representative families over Q, except when G = Z/2Z (a representative family over R does not exist). On the other hand, the work of Garcı́a in [28] and [29] detailed the study of the twists of smooth plane quartic curves over Q. These results help us to support the previous conjecture in two different situations over k = Q. The main idea is to start with a family of smooth plane quartic curves whose defining equation over Q involves many parameters. This in turns allows us to construct subfamilies of infinite cardinalities of non Q-isomorphic smooth plane quartic curves with the same automorphism group G, up to group isomorphism. The members of these families are mapped to concrete elliptic curves of rank zero (respectively, positive rank) over Q. Notations. We use ζn for a fixed primitive n-th root of unity inside k provided that the characteristic of k is coprime with n. A projective linear transformation A = (ai,j )1≤i,j≤3 of the projective plane P2k is often written as [a1,1 X + a1,2 Y + a1,3 Z : a2,1 X + a2,2 Y + a2,3 Z : a3,1 X + a3,2 Y + a3,3 Z], where {X, Y, Z} are the homogenous coordinates of P2k . ^ 4.1. The conjecture 4.1 is true for M3,biell ^ (Z/6Z) = MPl 3 (Z/6Z). Let k be a field of characteristic p = 0 or p > 7. The one parameter family Ca defined by Ca : aZ 4 + Y 2 (Y 2 + aZ 2 ) + X 3 Y = 0, ^ where a 6= 0, 4, is a representative family over k for MPl 3 (Z/6Z), see [27, Theorem 3.3]. Thus, any smooth plane quartic curve C over k with automorphism group isomorphic to Z/6Z (up to k-isomorphism) has a non-singular plane model in Ca for an unique a ∈ k. That is, there exists a finite extension L/k ⊆ k such that C ⊗k L is L-isomorphic to an unique non-singular polynomial equation of the form aZ 4 + Y 2 (Y 2 + aZ 2 ) + X 3 Y = 0 for some a ∈ k. In particular, Γ2 (C, L) = Γ2 (aZ 4 + Y 2 (Y 2 + aZ 2 ) + X 3 Y = 0, L). We also note that the above family is k-isomorphic to the geometrically complete family described in Theorem 3.3 through a diagonal change 8 E. BADR AND F. BARS of variables [27]. So we can assume that w̃ = diag(1, 1, −1) is the unique (bielliptic) involution for any smooth curve in the family Ca . Theorem 4.2. Consider a smooth quartic plane bielliptic curve Ca of the form aZ 4 + Y 2 (Y 2 + aZ 2 ) + X 3 Y = 0 for some a ∈ Q \ {0, 4}. The quotient Ca /hw̃i is an elliptic curve of positive rank over Q. In particular, the family Ca for a ∈ Q \ {0, 4} contains infinitely many smooth plane quartic bielliptic curves C over Q with automorphism group Z/6Z, and such that Γ2 (C, Q) is infinite. Proof. We work affine with az 4 + az 2 + x3 + 1 = 0 by taking Y = 1, in particular, Ca /hw̃i : az 2 + az + x3 + 1 = 0. It is straightforward to see that Ca /hw̃i is Q-isomorphic to the elliptic curve E/Q : z 2 = x3 − a3 (1 − a/4) whose j-invariant equals to zero. One can use MAGMA to check that the point Pa := (x, z) = (a, a2 /2) on E/Q has order m > 10 and m 6= 12. So it is a non-torsion point on E/Q (cf. [31, Theroem 7’]). Thus rankQ (Ca /hw̃i) ≥ 1 for any a ∈ Q \ {0, 4}, and Γ2 (Ca , Q) is an infinite set by Theorem 2.9. Finally, the family C is representative for MPl^ (Z/6Z) over Q. This means that two curves C and C 0 in the a 3 a a 6 a0 ∈ Q can not be Q-isomorphic. Thus, we have infinitely many smooth plane quartic curves family with a = each of which admits infinitely many quadratic points over Q.  Now, if we aim to parameterize the set of k-isomorphism classes of smooth plane quartic curves over k with automorphism group isomorphic to Z/6Z, then we may use [28, Proposition 3.2.8] (or [29, Proposition 5.3]) to deduce that the family CA,n,m : Am2 Z 4 + mY 2 Z 2 + nX 3 Y + Y 4 = 0, 3 2 where (A, n, m) ∈ k ∗ × k ∗ /k ∗ × k ∗ /k ∗ is such a parametrization1. More precisely, any smooth plane quartic curve over k with automorphism group Z/6Z is k-isomorphic to a non-singular plane model in the family CA,n,m 3 2 for some triple (A, n, m) ∈ k ∗ × k ∗ /k ∗ × k ∗ /k ∗ . In particular, it is bielliptic with unique bielliptic involution w̃ = diag(1, 1, −1). Moreover, two such curves are k-isomorphic if and only if they have the same parameter 3 2 A ∈ k ∗ . Therefore, it suffices to assume Γ2 (CA,n,m , k) for (A, n, m) ∈ k ∗ ×k ∗ /k ∗ ×k ∗ /k ∗ in order to investigate the infinitude of quadratic points over k of smooth plane quartic curves inside MPl ^ (Z/6Z) over k. 3 ∗ ∗ ∗3 Lemma 4.3. For an arbitrary but a fixed (A, n) ∈ k × k /k , the set Γ2 (CA,n,m , k) being finite or infinite is independent of the choice of m. 2 2 Proof. One finds that CA,n,m /hw̃i is k-isomorphic to the elliptic curve DA,n /k : z 2 = x3 + n A (1−4A) 4 over k. Indeed, one works affine Y = 1 to easily obtain that CA,n,m /hw̃i is k-isomorphic to the elliptic curve Az 2 + z + nx3 + 1 = 0 over k, in particular, its defining equation and its rank is independent from m. To reach 1 the Weierstrass form DA,n , we follow the usual way (cf. [39]); first, by the change of variables z 7→ z − 2A and 2 3 1−4A 2 x 7→ −x, we get z = (n/A)x + 4A2 , After by z 7→ (1/nA )z and x 7→ (1/An)x.  Theorem 4.4. There are infinitely many smooth plane quartic curves C over Q in the family CA,n,m , with (A, n, m) ∈ Q∗ × Q∗ /Q∗ 3 × Q∗ /Q∗ 2 , such that Γ2 (C, Q) is a finite set. Proof. For example, consider the subfamily CA(t),n(t),m : A(t)m2 Z 4 + mY 2 Z 2 + n(t)X 3 Y + Y 4 = 0 where A(t) := (108t2 + 1)/4 and n(t) := 4/t(108t2 + 1), for t = a/b ∈ Q∗ with a, b are odd coprime integers. In this situation, CA(t),n(t),m /hw̃i is always Q-isomorphic to DA(t),n(t) : z 2 = x3 − 27 of rank 0 over Q. Second, two curves in the family associated to the triples (A(t), n(t), m) and (A(t0 ), n(t0 ), m0 ) with A(t) 6= A(t0 ) are not Q-isomorphic. Accordingly, we have infinitely many smooth plane quartic curves with infinitely many quadratic points over Q, which was to be shown.  Remark 4.5. We can prove Theorems 4.2 and 4.4 without using the twisting theory of curves. For instance, following Henn’s table (cf. [8, Theorem 29]), smooth plane curves in the family Ca,b : y 4 + ay 2 = x3 + b have automorphism group equals to Z/6Z and their Q-isomorphism classes are uniquely determined by the ratio b/a2 . Imposing a2 +4b = 8 yields infinitely many non-isomorhic smooth plane curves in the family Ca,b corresponding to 1It remains to determine the algebraic restrictions on the parameters to ensure non-singularity and no larger automorphism p group. For example, A 6= 1/4, since we get the singular points (0 : ± m/2 : 1) on C1/4,n,m , otherwise. BIELLIPTIC SMOOTH PLANE CURVES AND QUADRATIC POINTS 9 the elliptic curve y 2 = x3 + 2 of rank one. Similarly, imposing a2 + 4b = 4 yields infinitely many non-isomorphic smooth plane curves corresponding to the elliptic curve y 2 = x3 + 1 of rank zero. ^ 4.2. The conjecture 4.1 is true for M3,biell (GAP(16, 13)). The family defined by Ca : Z 4 + (X 4 + Y 4 + aX 2 Y 2 ) = 0, √ ^ where ±a 6= 0, 2, 6, 2 −3, is a geometrically complete family for the locus MPl 3 (GAP(16, 13)) over Q. In this case, there are exactly seven (bielliptic) involutions, namely ι1 := diag(1, 1, −1), ι2 := diag(−1, 1, 1), ι3 := diag(1, −1, 1), ι4 := [Y : X : Z], ι5 := [Y : X : −Z], ι6 := [Y : −X : ζ4 Z] and ι7 := [Y : −X : −ζ4 Z]. Lemma 4.6. For a ∈ Q \ {0, ±2, ±6}, suppose that Ca /hιi i has a Q-point. Then, Ca /hιi i is Q-isomorphic to an elliptic curve E/Q of one of the forms: Involution E j-invariant ι1 z 2 = x3 − ax2 − 4x + 4a (16a6 + 576a4 + 6912a2 + 27648)/(a4 − 8a2 + 16) ι2 , ι3 z 2 = x3 + (a2 − 4)x 1728 ι4 , ι5 , ι6 , ι7 z 2 = x3 + (1 − a2 /4)x 1728 Proof. First, one can use MAGMA to verify that Ca /hιi i is a smooth curve of genus 1 over Q. Second, by assumption, Ca /hιi i has rational points. So it is Q-isomorphic to its Jacobian variety, which is an elliptic curve over Q of one the prescribed forms. Third, the condition that Ca /hιi i has rational points is verified in many 3 cases. For example, we can see that Ca /hι1 i is defined inside PQ by the two quadrics −X2 X3 + X42 = 0 and 2 p X12 + X22 + X32 + aX2 X3 = 0 over Q. Hence, if we impose −(a + 2) ∈ Q∗ \ {4}, then ( −(a + 2) : 1 : 1 : 1) is an obvious Q-point.  0 Consider the family CA : AX 4 + Y 4 + Z 4 + X 2 Y 2 = 0, where ±A 6= {1/4, 1/36, 1/12}. This is a representative family over Q (cf. [28, pages 36-37]), in particular, any smooth plane quartic curve over Q with automorphism 0 group isomorphic to GAP(16, 13) is isomorphic (not necessarily over Q) to a smooth plane curve in the family CA ∗ for a unique A ∈ Q \ {±1/4, ±1/36, ±1/12}. On the other hand, the transformed seven bielliptic involutions 0 of any smooth plane quartic curve in the family CA over Q are ι0i := P −1 ιi P , for i = 1, . . . , 7, where P := 4 diag( √4 1 A ∗ ∗ , 1, 1). If we restrict A ∈ Q \ Q , then ι1 = ι1 , ι02 = ι2 , ι03 = ι3 are the only bielliptic involutions 0 defined over Q. By the work of Garcı́a in [28, Chp. 3], we know that any diagonal twist of CA , for a fixed A, in the family 0 CA is Q-isomorphic to CA,m,q : mAX 4 + q 2 mY 4 + Z 4 + qmX 2 Y 2 = 0, 4 for some A, m, q ∈ Q∗ . Two of the twists {A, m, q} and {A0 , m0 , q 0 } are Q-isomorphic if A = A0 , m ≡ m0 mod Q∗ 2 4 and q ≡ q 0 mod Q∗ . First, we consider smooth curves of the form CA,m,q with (A, m, q) ∈ Q∗ × Q∗ /Q∗ × 2 Q∗ /Q∗ . Next, the quotient curve CA,m,q /hι03 i is a genus one curve that is Q-isomorphic to y 2 + (1/m)z 4 − 2 (1/4 − A) = 0 independently from the parameter q ∈ Q∗ /Q∗ . Assuming that 1/4 − A is a square implies that CA,m,q /hι03 i has Q-points, and its Weierstrass form in the variables u, ν reduces to ν 2 = u3 + ((1 − 4A)/m) u. Theorem 4.7. Consider a smooth bielliptic quartic plane curve in the family CA,m,q : mAX 4 + q 2 mY 4 + 2 4 Z 4 + qmX 2 Y 2 = 0, for some q ∈ Q∗ \ Q∗ and A, m ∈ Q∗ \ Q∗ , such that 1/4 − A is a square. Then, the quotient CA,m,q /hw̃i, where w̃ = diag(1, −1, 1), is an elliptic curve of positive rank over Q. In particular, CA,m,q gives an infinite family of smooth plane quartic curves C over Q whose automorphism group is isomorphic to GAP(16, 13), and Γ2 (C, Q) is an infinite set. Proof. We have seen above that CA,m,q /hw̃i is Q-isomorphic to DA,m : ν 2 = u3 + Du with D := (1 − 4A)/m 2 independently from the parameter q ∈ Q∗ /Q∗ . It suffices now to specialize A, m so that rankQ (DA,m ) is positive, so Γ2 (C, Q) is infinite. For example, if we take D := (1 − 4A)/m = p, where p is a prime integer < 1000 and congruent to 5 modulo 8, then the rank is always 1 in accordance with the conjecture of Selmer and Mordell (see [10]). If we take D := (1 − 4A)/m = −p, where p is a Fermat or a Mersenne prime, then the ranks 10 E. BADR AND F. BARS 0, 1 and 2 were found (see [26]). If we take D := (1 − 4A)/m = −n, where n is related to the positive integer solutions of the diophantine equation n = α4 + β 4 = γ 4 + µ4 , then the rank is at least 3 (see [24]). If we take D := (1 − 4A)/m = −pq, where p and q are two different odd primes, then, up to an extra condition, a family of rank 4 was found (see [30]), etc... Finally, by [28, Proposition 3.2.9], two twists CA,q,m and CA,q0 ,m0 , with 2 4 4 (A, m, q), (A, m0 , q 0 ) ∈ Q∗ × Q∗ /Q∗ × Q∗ /Q∗ are Q-isomorphic only if m = m0 in Q∗ /Q∗ . Consequently, for any fixed A, we can run m and q as before to obtain infinitely many non-Q-isomorphic smooth plane quartic curves with infinitely many quadratic points over Q.  Now, we aim to construct an infinite family of smooth quartic curves over Q inside MPl ^ 3 (GAP(16, 13)), such that the quotient by any bielliptic involution may only produces elliptic curves of rank 0 over Q. In particular, the set quadratic points over Q is finite. To do this, we turn our attention to non-diagonal twists of CA , for a fixed A ∈ Q∗ . Those twists are parameterized by Ca,b,m,q,A : 2aX 4 + 8bmX 3 Y + 12amX 2 Y 2 + 8bm2 XY 3 + 2am2 Y 4 + q(X 2 − mY 2 )2 + Z 4 = 0 where m ∈ Q∗ , a, b, q ∈ Q satisfy a2 −b2 m = q 4 A (see [28, Proposition 3.2.9], or alternatively see [29, Proposition 5.5]). Two such twists {a, b, m} and {a0 , b0 , m0 } for CA are equivalent if and only if m ≡ m0 mod Q∗ 2 and that √ √ √ there exist c, d ∈ Q such that a + b m = (c + d m)4 (a0 + b0 m). The transformed seven bielliptic involutions of any smooth plane quartic curve Ca,b,m,q,A in the family Ca,b,m,q,A over Q are ι00i := P 0−1 ι0i P 0 , for i = 1, . . . , 7, where p 4 √ √ p 4 √  a + b m m a + b m 0 √ √ p √ P 0 :=  4 a − b m − m 4 a − b m 0  . p  0 0 1 4 / Q∗ One can check that diag(1, 1, −1) is the unique bielliptic involution defined over Q for Ca,b,m,q,A , when A ∈ ∗2 and m ∈ / Q . Thus, the only way to obtain degree two Q-morphisms to elliptic curves over Q is to quotient by w̃ := diag(1, 1, −1). Suppose that the quotient family Ca,b,m,q,A /hw̃i has Q-points, which is verified when q = −2a through the point (0 : 1 : 0), for example. Thus Ca,b,m,q=−2a,A /hw̃i is Q-isomorphic to its Jacobian variety given by: Em,q : y 2 = x3 + 8qmx2 + 16q 2 m3 x (1−m) 4 when a = −b. In this case, A = 4q 2 / Q∗ . and we always impose A ∈ Lemma 4.8. The two parameter family Em,q : y 2 = x3 + 8qmx2 + 16q 2 m3 x, for q ∈ Q and m ∈ Q∗ \ Q∗ 2 such 4 that 1−m / Q∗ , contains infinitely many elliptic curves Em,q of rank 0 over Q. More precisely, for any fixed 4q 2 ∈ 2 m ∈ Z \ Z2 with m ∈ / {1 − h2 | h ∈ Z}, there exist infinitely many q ∈ Q∗ mod Q∗ with m q square-free integer, where rankQ (Em,q ) = 0. Proof. Set m, m q ∈ Z such that m ∈ / Z2 and mq is square-free. Then, the elliptic curve Em,q is Q-isomorphic to the quadratic twist ED : Dy 2 = x3 + 8x2 + 16mx for E : y 2 = x3 + 8x2 + 16mx over Q, where D = mq. Indeed, to obtain the equation of Em,q we multiply the defining equation for ED by D3 and next we apply the change of variables y 7→ (1/D2 )y and x 7→ (1/D)x. We know that E/Q is modular by the work of so many people; mainly C. Breuil, B. Conrad, F. Diamond, R. Taylor and A. Wiles (cf. [11, 12, 42, 44]). Therefore, we deduce from M. R. Murty and V. K. Murty [32, Chapter 6, Theorem 1.1] (cf. the first line of the abstract in [34]) that, for a fixed m, there are infinitely many square-free integers D = mq such that ED has rank 0 over Q, where each D is congruent to 1 mod 4n, where n is the conductor of the elliptic curve E/Q. In particular, we 2 get infinitely many elliptic curves Em,q of rank 0 over Q, corresponding to infinitely many such q’s mod Q∗ . The condition that m ∈ / Z2 with m ∈ / {1 − h2 | h ∈ Z} is to ensure that A is not a fourth power.  Theorem 4.9. The two-parameters family Cm,q : −qX 4 + 4qmX 3 Y − 6qmX 2 Y 2 + 4qm2 XY 3 − qm2 Y 4 + q(X 2 − mY 2 )2 + Z 4 = 0, 2 2 4 for q ∈ Q∗ \ Q∗ and m ∈ Q∗ /Q∗ such that A = 1−m / Q∗ , contains infinitely many, non Q-isomorphic, 4q 2 ∈ smooth plane curves C over Q that have only finitely many quadratic points over Q. BIELLIPTIC SMOOTH PLANE CURVES AND QUADRATIC POINTS 11 Proof. If we specialize q = −2a = 2b in the family Ca,b,m,q,A mentioned above, then we get the subfamily Cm,q . In particular, any smooth plane curve in Cm,q is bielliptic and has only one bielliptic involution w̃ over Q. Furthermore, by Lemma 4.8, for any m ∈ Z \ Z2 with m ∈ / {1 − h2 | h ∈ Z}, there exists an infinite number ∗ ∗2 of q ∈ Q mod Q with mq a square-free integer, such that Cm,q /hw̃i is an elliptic curve of rank zero over Q. Therefore, the number of quadratic points over Q is finite by Theorem 2.9. Moreover, if any two curves Cm,q and Cm,q0 , which give rank 0, are Q-isomorphic, then C(1−m)/4q2 : 1−m 4 4 4 2 2 4q 2 X + Y + Z + X Y = 0 and C(1−m)/4q : 02 1−m 4 4 4 2 2 4q 02 X + Y + Z + X Y = 0 are Q-isomorphic (recall that C m,q and C m,q 0 are twists for C(1−m)/4q 2 and 0 C(1−m)/4q02 , respectively). But this contradicts the fact that the family CA : AX 4 + Y 4 + Z 4 + X 2 Y 2 = 0 is a ^ 13)). representative family over Q for the locus MPl (GAP(16,  3 4.3. A version of Conjecture 4.1 over the global field Fp (T ). It is to be noted that formulating the con- jecture 4.1 over global fields of positive characteristic has more complications than this over zero characteristic. For instance, a similar statement to Theorem 2.9 is not valid for the setting, also Theorem 2.13 does not apply when isotrivial elliptic curves appear in the picture. In this setting, we reformulate the conjecture in terms of the behavior of bielliptic quotients instead of quadratic points as follows: Conjecture 4.10. Fix a prime p > 7 and consider a non-empty loci M ^ Pl (G), where all of its F (T )-points are 3 p ^ Pl bielliptic curves. Then, there is an infinite set D (respectively, E) in M3 (G)(Fq (T )), of Fp (T )-isomorphism classes of smooth plane quartic curves over Fp (T ), such that for each d ∈ D one of their bielliptic quotients is an elliptic curve of positive Fp (T )-rank (respectively, for each e ∈ E all bielliptic quotients have zero rank over Fp (T )). ^ Theorem 4.11. Conjecture 4.10 is true for M3,biell (Z/6Z). Proof. Since G = Z/6Z contains only one bielliptic involution, then any Fp (T )-point of MPl ^ 3 (Z/6Z) has a unique 2 bielliptic quotient. Let Ea,b be the elliptic curve y = x + b, which is Fp (T )-isomorphic to y 2 = x4 + (b + a4 ). 2 3 Let Ca,b be the double cover y 4 +ay 2 = x3 +b. For generic a and b in Fp (T ), Ca,b has automorphism group Z/6Z, and its Fp (T )-isomorphism class is uniquely determined by b/a2 . In particular, for (a, b), (a0 , b0 ) ∈ Fp (T )2 , we say that (a, b) ∼ (a0 , b0 ) if and only if b/a2 = b0 /(a0 )2 . First, taking (a, b) ∈ Fp (T )2 / ∼ such that 4b+a2 = 4(t2 −t3 ) gives infinitely many Ca,b covering the isotrivial elliptic curve E : y 2 = x3 + t2 − t3 with (t, t) ∈ E(Fp (T )). By [18, Proposition 2.4], E(Fp (t))tors = {0}, so rankFp (T ) (E) ≥ 1 and we get an inifinte set D as described in Conjecture 4.10. Second, taking (a, b) ∈ Fp (T )2 / ∼ such that 4b + a2 = 4t reduces the problem to checking the Fp (t)-rank of the elliptic curve E : y 2 = x3 + t that has a bad additive reduction at t and 1/t. By the degree formula (cf. [43, Theorem 9.3]), the L-function L(E, s) has degree zero as polynomial in p−s . Because the Birch-Swinnerton-Dyer conjecture over elliptic curves over Fp (T ) is known for isotrivial elliptic curve by Tate [43, Theorem 12.2], we can conclude that E has rank zero. This constructs an infinite set E as required.  Remark 4.12. Rewriting the argument of Lemma 4.6 for MPl ^ 3 (GAP(16, 13)) over positive characteristic yields E : y = x + Dx with D ∈ Fp (T ) of positive rank (by taking D = −t2 /(t + 1)3 , for example). This allows us 2 3 ^ 13))(F (T )). to get a D inside MPl (GAP(16, 3 p Acknowledgements The authors are grateful to Andreas Schweizer and Douglas Ulmer for their e-mails that improved the part over global fields of positive characteristic. We also would like to thank Luis Dieulefait, Joaquim Roé, and Xavier Xarles for their interesting comments. We also thank the referee for his/her recommendations that improved the paper in its present form, especially for pointing out to us the work of Bardelli and Del Centina in the area. 12 E. BADR AND F. BARS References [1] D. Abramovich and J. Harris, Abelian varieties and curves in Wd (C). Compo. Math. 78 (1991), 227-238. [2] E. Arbarello, M. Cornalba and P. Griffiths, Geometry of algebraic curves, volume II with a contribution by J. D. Harris. Vol. 268. Springer Science and Business Media, 2011. [3] E. Badr, On the stratification of smooth plane curves by automorphism groups. PhD thesis, September 2017, Universitat Autònoma de Barcelona. [4] E. Badr and F. Bars, On the locus of smooth plane curves with a fixed automorphism group, Mediterr. J. Math. 13 (2016), 3605-3627. [5] E. Badr, F. Bars and E. Lorenzo Garcı́a, On twists of smooth plane curves. Math. Comp. 88 (2019), 421-438. [6] E. Badr and M. Sadek, On quadratic progression sequences on smooth plane curves. J. Number Theory 213 (2020), 445-452. [7] F. Bardelli and A. Del Centina, Bielliptic curves of genus three:canonical models and moduli space. Indag.Mathem., N.S., 10(2), (1999), 183–190. [8] F. Bars, On the automorphisms groups of genus 3 curves. Surv. Math. Sc. 2 (2012), no. 2, 83-124. [9] F. Bars, On quadratic points of classical modular curves Momose Memorial Volume. Number theory related to modular curves Momose memorial volume, 17-34, Contemp. Math. 701, Amer. Math. Soc., Providence, RI, 2018. [10] A. Bremner, J. W. S. Cassels, On the equation Y 2 = X(X 2 + p). Math. Comp. 42 (1984), 257-264. [11] C. Breuil, B. Conrad, F. Diamond and R. Taylor, On the Modularity of Elliptic Curves Over Q: Wild 3-Adic Exercises. J. Amer. Math. Soc. 14 (2001), 843-939. [12] B. Conrad, F. Diamond and R. Taylor, Modularity of Certain Potentially Barsotti-Tate Galois Representations. J. Amer. Math. Soc. 12 (1999), 521-567. [13] G. Faltings, Endlichkeitsstze fr abelsche Varietten ber Zahlkrpern [Finiteness theorems for abelian varieties over number fields]. Inventiones Mathematicae (in German) 73 (3) (1983): 349-366, doi:10.1007/BF01388432. [14] G. Faltings, Erratum: Endlichkeitsstze fr abelsche Varietten ber Zahlkrpern. Inventiones Mathematicae (in German) 75 (2) (1984): 381. [15] N. Freitas and S. Siksek, The asymptotic Fermat’s last theorem for five-sixths of real quadratic fields. Compos. Math. 151 (2015), no. 8, 1395-1415. [16] GAP Group, Gapgroups, algorithms, and programming system for computational discrete algebra (2008), available at http://www.gap-system.org. version 4.4.11. [17] H. Grauert, Mordells Vermutung ber rationale Punkte auf algebraischen Kurven und Funktionenkrper. Publ. Math., I.H.E.S. 25 (1965), 131-149. [18] R. Griffon and D. Ulmer On the arithmetic of a family of twisted constant elliptic curves arXiv:1903.03901, 2019. [19] J. Harris and J. H. Silverman, Bielliptic curves and symmetric products. Proc. Amer. Math. Soc. 112 (1991), 347-356 . [20] P. Henn, Die Automorphismengruppen dar algebraischen Functionenkorper vom Geschlecht 3, Inagural-dissertation, Heidel- berg, 1976. [21] K.Hulek and R.Kloosterman Calculating the Mordel-Weil rank of elliptic threefolds and the cohomology of singular hypersur- faces. Ann. de L’Institute Fourier, 61, n3 (2011), 1122-1179. [22] J. W. P. Hirschfeld, G. Korchmáros, and F. Torres, Algebraic Curves over Finite Fields, Princeton Series in Applied Mathe- matics, 2008. [23] M. Homma, Funny plane curves in characteristic p > 0. Commun. Algebra, 15 (1987), 1469-1501. [24] F. A. Izadi, F. Khoshnam, and K. Nabardi, Sums of two biquadrates and elliptic curves of rank ≥ 4. Math. J. Okayama Univ. 56 (2014), 51-63. √ [25] a, The Fermat equation over Q( 2). J. Number Theory 109 (2004), no. 1, 182-196 [26] T. Kudo, K. Motose, On group structure of some special elliptic curve. Math. J. Okayama Univ. 47 (2005), 81-84. [27] R. Lercier, C. Ritzenthaler, F. Rovetta, and J. Sijsling, Parametrizing the moduli space of curves and applications to smooth plane quartics over finite fields, LMS J. Comput. Math. 17 (2014), no. suppl. A, 128-147. [28] E. Lorenzo Garcı́a, Arithmetic properties of non-hyperelliptic genus 3 curves. PhD dissertation, Universitat Politècnica de Catalunya (2015), Barcelona. [29] E. Lorenzo Garcı́a Twists of non-hyperelliptic curves of genus 3. Int. J. Number Theory 14 (2018), no. 6, 17851812. [30] M. Maenishi, On the rank of elliptic curves y 2 = x3 − pqx. Kumamoto J. Math. 15 (2002), 1-5. [31] B. Mazur, Modular curves and the Eisenstein ideal. Publ. Math., I.H.E.S. 47 (1977), 33-186. [32] M. R. Murty and V. K. Murty, Non-vanishing of L-functions and applications. Birkhauser Verlag, Basel, 1997. [33] M. Namba, Families of meromorphic functions on compact Riemann surfaces. Lect. Notes Math. 767 , Springer-Verlag, Berlin, 1979. [34] K. Ono, Rank zero quadratic twists of modular elliptic curves. Compos. Math. 104 (1996), 293–304. [35] C.Salgado, On the rank of the fibres of rational elliptic surfaces. Algebra and number Theory 6:7(2012), 1289–1318. [36] P. Samuel Compléments à un article de Hans Grauert sur le conjecture de Mordell, Publ. I.H.E.S., tome 29 (1966), 55–62. [37] A. Schweizer, Bielliptic Drinfeld modular curves. Asian J. Math. 5 (2001), 705-720. [38] J. P. Serre, Cohomologie galoisienne. fifth ed., Lecture Notes in Mathematics, vol. 5, Springer-Verlag, Berlin, 1994.. BIELLIPTIC SMOOTH PLANE CURVES AND QUADRATIC POINTS 13 [39] J. H. Silverman, The Arithmetic of elliptic curves, GTM 106 Springer, Ed.(1992). [40] H. Stichtenoth, Algebraic Function Fields and Codes, Springer Verlar, GTM254, 2nd Edition, (2008). [41] J. T. Tate, Genus change in inseparable extensions of function fields, Proc. Amer. Math. Soc. 3 (1952), 400-406. [42] R. Taylor and A. Wiles, Ring-Theoretic Properties of Certain Hecke Algebras. Ann. Math. 141 (1995), 553-572. [43] D. Ulmer, Elliptic curves over function fields. Arithmetic of L-functions, 211280, IAS/Park City Math. Ser., 18, Amer. Math. Soc., Providence, RI, 2011. [44] A. Wiles, Modular Elliptic-Curves and Fermat’s Last Theorem. Ann. Math. 141 (1995), 443-551. [45] H. Yoshihara, Function field theory of plane curves by dual curves, J. Algebra 239 (2001), 340-355. • Eslam Badr Department of Mathematics, Faculty of Science, Cairo University, 12613 Giza-Egypt E-mail address: eslam@sci.cu.edu.eg • Francesc Bars Departament Matemàtiques, Edif. C, Universitat Autònoma de Barcelona, 08193 Bellaterra, Catalonia E-mail address: francesc@mat.uab.cat