Addition in the Jacobian of non-hyperelliptic genus 3 curves and rationality of the intersection points of a line with a plane quartic

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1 Addition in the Jacobian of non-hyperelliptic genus 3 curves and rationality of the intersection points of a line with a plane quartic Stéphane Flon, Roger Oyono, Christophe Ritzenthaler C. Ritzenthaler, C.N.R.S. Institut de Mathématiques de Luminy Luminy Case 930, F13288 Marseille CEDEX 9 ritzenth@iml.univ-mrs.fr web : http ://iml.univ-mrs.fr/ ritzenth/ Stéphane Flon, Roger Oyono, ChristopheAddition Ritzenthaler in the () Jacobian of non-hyperelliptic genus 3 curves and rationality 1 / of11the

2 General setting Non-hyperelliptic genus 3 curves = smooth plane quartics. Choice of a good divisor at innity condition ( ) : There is a rational line l which crosses the quartic C in four k-points P 1, P 2, P 3, P 4. Let D = P 1 + P 2 + P 3. Because Sym 3 C Jac(C), D + D + D is surjective, a element D Jac(C) is a sum of three points called D +. Question : how do we compute (eciently) D 1 + D 2 in terms of D + 1, D+ 2? Stéphane Flon, Roger Oyono, ChristopheAddition Ritzenthaler in the () Jacobian of non-hyperelliptic genus 3 curves and rationality 2 / of11the

3 A geometric addition algorithm Let D 1, D 2 Jac(C)(k). Then D 1 + D 2 is equivalent to a divisor D = D + D, where the points in the support of D + are given by the following algorithm : 1 Take a cubic E which goes (with multiplicity) through the support of D +, 1 D+ 2 and P 1, P 2, P 4. This cubic also crosses C in the residual eective divisor D 3. 2 Take a conic Q which goes through the support of D 3 and P 1, P 2. This conic also crosses C in the residual eective divisor D +. Why? Because (l C) κ, (Q C) 2κ and (E C) 3κ. Stéphane Flon, Roger Oyono, ChristopheAddition Ritzenthaler in the () Jacobian of non-hyperelliptic genus 3 curves and rationality 3 / of11the

4 A chord construction Stéphane Flon, Roger Oyono, ChristopheAddition Ritzenthaler in the () Jacobian of non-hyperelliptic genus 3 curves and rationality 4 / of11the

5 Special forms of the curve C admits an equation of the form with deg(f 4 ) 4 and C : y 3 + h 1 (x)y 2 + h 2 (x)y = f 4 (x) Stéphane Flon, Roger Oyono, ChristopheAddition Ritzenthaler in the () Jacobian of non-hyperelliptic genus 3 curves and rationality 5 / of11the

6 Special forms of the curve C admits an equation of the form C : y 3 + h 1 (x)y 2 + h 2 (x)y = f 4 (x) with deg(f 4 ) 4 and 1 deg(h 1 ) 2 and deg(h 2 ) 3 if P 1 = P 2 (tangent case) ; Stéphane Flon, Roger Oyono, ChristopheAddition Ritzenthaler in the () Jacobian of non-hyperelliptic genus 3 curves and rationality 5 / of11the

7 Special forms of the curve C admits an equation of the form with deg(f 4 ) 4 and C : y 3 + h 1 (x)y 2 + h 2 (x)y = f 4 (x) 1 deg(h 1 ) 2 and deg(h 2 ) 3 if P 1 = P 2 (tangent case) ; 2 deg(h 1 ) 1 and deg(h 2 ) 3 if P 1 = P 2 = P 4 (ex case). If char(k) 3 we can write C : y 3 + h 2 (x)y = f 4 (x). Stéphane Flon, Roger Oyono, ChristopheAddition Ritzenthaler in the () Jacobian of non-hyperelliptic genus 3 curves and rationality 5 / of11the

8 Special forms of the curve C admits an equation of the form with deg(f 4 ) 4 and C : y 3 + h 1 (x)y 2 + h 2 (x)y = f 4 (x) 1 deg(h 1 ) 2 and deg(h 2 ) 3 if P 1 = P 2 (tangent case) ; 2 deg(h 1 ) 1 and deg(h 2 ) 3 if P 1 = P 2 = P 4 (ex case). If char(k) 3 we can write C : y 3 + h 2 (x)y = f 4 (x). 3 deg(h 1 ) 1 and deg(h 2 ) 2 if P 1 = P 2 = P 3 = P 4 (hyperex case). These curves are the C 3,4 -curves. Stéphane Flon, Roger Oyono, ChristopheAddition Ritzenthaler in the () Jacobian of non-hyperelliptic genus 3 curves and rationality 5 / of11the

9 Special forms of the curve C admits an equation of the form with deg(f 4 ) 4 and C : y 3 + h 1 (x)y 2 + h 2 (x)y = f 4 (x) 1 deg(h 1 ) 2 and deg(h 2 ) 3 if P 1 = P 2 (tangent case) ; 2 deg(h 1 ) 1 and deg(h 2 ) 3 if P 1 = P 2 = P 4 (ex case). If char(k) 3 we can write C : y 3 + h 2 (x)y = f 4 (x). 3 deg(h 1 ) 1 and deg(h 2 ) 2 if P 1 = P 2 = P 3 = P 4 (hyperex case). These curves are the C 3,4 -curves. 4 If char(k) 3, C : y 3 = f 4 (x) (Picard curves) iif P 1 is a rational Galois point. : the more special, the better. Stéphane Flon, Roger Oyono, ChristopheAddition Ritzenthaler in the () Jacobian of non-hyperelliptic genus 3 curves and rationality 5 / of11the

10 Some complexities In the case char(k) > 5 and C has a model y 3 + h 2 (x)y = f 4 (x) with deg(h 2 ) 3. Remarks : Salem and Makdisi work with a good choice of Riemann-Roch spaces. Basiri, Enge, Faugère and Gürel work with ideals in function elds. Others (Blache, Cherdieu, Sarlabous,... ) work on the more general problem of reduction and give only asymptotics. Stéphane Flon, Roger Oyono, ChristopheAddition Ritzenthaler in the () Jacobian of non-hyperelliptic genus 3 curves and rationality 6 / of11the

11 Study of the condition ( ) over F q 1 A rational hyperex. Quartics with a hyperex form a sub-variety of codimension 1. But generically, if there is a hyperex, it is rational. Stéphane Flon, Roger Oyono, ChristopheAddition Ritzenthaler in the () Jacobian of non-hyperelliptic genus 3 curves and rationality 7 / of11the

12 Study of the condition ( ) over F q 1 A rational hyperex. Quartics with a hyperex form a sub-variety of codimension 1. But generically, if there is a hyperex, it is rational. 2 A rational ex. When char(k) > 3, smooth plane quartics have 24 exes counted with multiplicities. Heuristics and computations seem to show that the probability that C has at least one rational ex is asymptotically (q ) about 0.63 (probability that a degree 24 polynomial has at least one root). This seems true even in characteristic 2 and 3. Stéphane Flon, Roger Oyono, ChristopheAddition Ritzenthaler in the () Jacobian of non-hyperelliptic genus 3 curves and rationality 7 / of11the

13 Study of the condition ( ) over F q 1 A rational hyperex. Quartics with a hyperex form a sub-variety of codimension 1. But generically, if there is a hyperex, it is rational. 2 A rational ex. When char(k) > 3, smooth plane quartics have 24 exes counted with multiplicities. Heuristics and computations seem to show that the probability that C has at least one rational ex is asymptotically (q ) about 0.63 (probability that a degree 24 polynomial has at least one root). This seems true even in characteristic 2 and 3. Caution : in char. 2, 3 the computations of the exes cannot be done with the ordinary Hessian (which is zero) nd a good substitute. Stéphane Flon, Roger Oyono, ChristopheAddition Ritzenthaler in the () Jacobian of non-hyperelliptic genus 3 curves and rationality 7 / of11the

14 The generic case Follow an idea of Diem-Thomé. 1 Let P C(k). Consider the separable geometric cover φ : C κ P = P 1 of degree 3 induced by the linear system κ P. Stéphane Flon, Roger Oyono, ChristopheAddition Ritzenthaler in the () Jacobian of non-hyperelliptic genus 3 curves and rationality 8 / of11the

15 The generic case Follow an idea of Diem-Thomé. 1 Let P C(k). Consider the separable geometric cover φ : C κ P = P 1 of degree 3 induced by the linear system κ P. 2 Using eective Chebotarev's density theorem for function elds, one gets estimation on the number of completely split divisors in κ P. Stéphane Flon, Roger Oyono, ChristopheAddition Ritzenthaler in the () Jacobian of non-hyperelliptic genus 3 curves and rationality 8 / of11the

16 The generic case Follow an idea of Diem-Thomé. 1 Let P C(k). Consider the separable geometric cover φ : C κ P = P 1 of degree 3 induced by the linear system κ P. 2 Using eective Chebotarev's density theorem for function elds, one gets estimation on the number of completely split divisors in κ P. Theorem If q 127, there is always a line satisfying ( ). Remark : The number of Galois points (i.e. P such that φ is Galois) is at most 4 if char(k) 3 and 28 if char(k) = 3. Stéphane Flon, Roger Oyono, ChristopheAddition Ritzenthaler in the () Jacobian of non-hyperelliptic genus 3 curves and rationality 8 / of11the

17 The tangent case Let T : C Sym 2 (C), p T p (C) C 2p be the tangential correspondence. We associate to it its correspondence curve X = {(p, q) C C : q T (p)} which is dened over k. Let φ : X C be the rst projection. We want to prove that there is a rational point on X when q is big enough. Stéphane Flon, Roger Oyono, ChristopheAddition Ritzenthaler in the () Jacobian of non-hyperelliptic genus 3 curves and rationality 9 / of11the

18 The tangent case Let T : C Sym 2 (C), p T p (C) C 2p be the tangential correspondence. We associate to it its correspondence curve X = {(p, q) C C : q T (p)} which is dened over k. Let φ : X C be the rst projection. We want to prove that there is a rational point on X when q is big enough. Proposition (Aubry, Perret) Let X dened over F q be a geometrically irreducible curve of arithmetic genus π X. Then #X (F q ) (q + 1) 2π X q. In particular if q (2π X ) 2 then X has a rational point. Question : How to show that X is absolutely irreducible (+ estimate π X )? Stéphane Flon, Roger Oyono, ChristopheAddition Ritzenthaler in the () Jacobian of non-hyperelliptic genus 3 curves and rationality 9 / of11the

19 The tangent case (continued) Lemma Let φ : X Y be a separable morphism of degree 2 between two projective curves over k such that Y is smooth and abs. irreducible ; There exists a point P 0 Y such that φ is ramied at P 0 and φ 1 (P 0 ) is not singular. Then X is abs. irreducible. Stéphane Flon, Roger Oyono, ChristopheAddition Ritzenthaler in the () Jacobian of non-hyperelliptic genus 3 curves and rationality 10 / of11the

20 The tangent case (continued) Lemma Let φ : X Y be a separable morphism of degree 2 between two projective curves over k such that Y is smooth and abs. irreducible ; There exists a point P 0 Y such that φ is ramied at P 0 and φ 1 (P 0 ) is not singular. Then X is abs. irreducible. Proposition we have the following properties : 1 the ramication points of φ are the bitangence points. 2 If char(k) 2, the only possible singular points of X are the points (P, P) where P is a hyperex of C. Stéphane Flon, Roger Oyono, ChristopheAddition Ritzenthaler in the () Jacobian of non-hyperelliptic genus 3 curves and rationality 10 / of11the

21 The tangent case (end) Lemma Suppose char(k) 2. There is always a bitangence point which is not a hyperex except if char(k) = 3 and C is geometrically isomorphic to the Fermat quartic x 4 + y 4 + z 4 = 0. Stéphane Flon, Roger Oyono, ChristopheAddition Ritzenthaler in the () Jacobian of non-hyperelliptic genus 3 curves and rationality 11 / of11the

22 The tangent case (end) Lemma Suppose char(k) 2. There is always a bitangence point which is not a hyperex except if char(k) = 3 and C is geometrically isomorphic to the Fermat quartic x 4 + y 4 + z 4 = 0. Theorem Suppose char(k) 2. If q there exists a tangent to C which cuts C at rational points only. Remark : estimation of π X = 33 by Hurwitz formula or thanks to the general theory of correspondences. Question : The characteristic 2 case? (P, Q) with P a bitangence point is a singular point on X. There seem to exist cases where X is not abs. irreducible (for instance the Klein quartic x 3 y + y 3 z + z 3 x = 0). Stéphane Flon, Roger Oyono, ChristopheAddition Ritzenthaler in the () Jacobian of non-hyperelliptic genus 3 curves and rationality 11 / of11the