On the density of rational points on rational elliptic surfaces
aa r X i v : . [ m a t h . AG ] J u l On the density of rational points on rational elliptic surfaces
Julie DESJARDINS
Abstract
Let E → P Q be a non-trivial rational elliptic surface over Q with base P Q (with a section).We conjecture that any non-trivial elliptic surface has a Zariski-dense set of Q -rational points.In this paper we work on solving the conjecture in case E is rational by means of geometricand analytic methods. First, we show that for E rational, the set E ( Q ) is Zariski-dense when E is isotrivial with non-zero j -invariant and when E is non-isotrivial with a fiber of type II ∗ , III ∗ , IV ∗ or I ∗ m ( m ≥ ). We also use the parity conjecture to prove analytically the densityon a certain family of isotrivial rational elliptic surfaces with j = 0 , and specify cases for whichneither of our methods leads to the proof of our conjecture. Let E be an elliptic surface over P Q , i.e. a projective algebraic surface E defined over Q endowed with a morphism π : E → P Q such that, for all t ∈ P Q except a finite number, the fiber E t := π − ( t ) is a smooth projective curve of genus 1. Moreover, we suppose that there exists asection σ : P Q → E for π .Such an elliptic surface can be written as the set of solutions in P Q × P Q of a Weierstrassequation E : y z = x + A ( T ) xz + B ( T ) z , (1)with A ( T ) , B ( T ) ∈ Z [ T ] . We call generic fiber of E the elliptic curve over Q ( T ) , denoted by E T ,whose model is given by the equation (1). We classify the elliptic surfaces according to their j -invariant function:1. E is non-isotrivial if j ( T ) is non-constant,2. E is isotrivial otherwise, and admits a Weierstrass equation of the form(a) y = x + af ( T ) x + bf ( t ) where a, b non-zero integers and f ∈ Q [ T ] , (if j ( T ) ∈ Q \{ , } )(b) y = x + A ( T ) where A ∈ Q [ T ] (if j ( T ) = 0 ),(c) y = x + B ( T ) x where B ∈ Q [ T ] (if j ( T ) = 1728 ).For almost every t ∈ P Q , the fiber E t = π − ( t ) is an elliptic curve over Q and the set E t ( Q ) admits a group structure. By Mordell-Weil’s theorem, this group decomposes as the sum of afinitely generated free group (isomorphic to Z r ) and a finite group (the torsion points). Theinteger r = rk( E t ) is called the Mordell-Weil rank (or simply the rank ) of E t over Q .We put forward the following conjecture, a variant of a conjecture of Mazur [Maz92, Conjec-ture 4], where "real density" is replaced by "Zariski density". This conjecture is already implicitin the literature, particularly in [CCH05]. Conjecture 1.1.
Let E → P Q be a non-trivial elliptic surface over Q . Then E ( Q ) is Zariski-dense in E . emark . In case E is a trivial surface, i.e. there exists a curve E such that E ≃ E × P Q over Q . One has E ( Q ) = E ( Q ) × P Q ( Q ) , and this can be dense or not dense depending on thenumber of points of E ( Q ) .There is two approaches to solve this conjecture. We can either prove that the rank is non-zero by means of geometric arguments, or we can use the parity conjecture (a weaker version ofthe Birch and Swinnerton-Dyer conjecture) which links the root number of an elliptic curve E to the parity of its rank : W ( E ) = ( − rk E . We already have evidence (see [Man95, Hel03, Des16b]) that the rational points should bedense when the elliptic surface is non-isotrivial, because of the variation of the root number ofthe fibers. The articles mentioned above additionally use two conjectures of analytic numbertheory, the squarefree conjecture and Chowla’s conjecture, which are known only for polynomialsof low degree.In this article, we work on proving the conjecture in case E is a rational elliptic surface, i.e. Q -birational to P . This is the class of elliptic surfaces with the simplest geometry, and thus agood starting point. It is also interesting to observe that it leads to results on Del Pezzo surfaces,due to the relation between those classes of surfaces. In certain cases, we can even prove thestronger property of unirationality : a surface S over a field k is unirational if there is a dominantrational map P E over k . Note that if a projective surface S is Q -unirational, one has inparticular that the set of its rational points S ( Q ) is Zariski-dense in S .Finally, note that in section 7.1, we state on which rational elliptic surfaces the results of[Hel03] unconditionnally imply the variation of the root number.We prove the following theorem : Theorem 1.2.
Let E be a rational elliptic surface.1. Suppose E is isotrivial with non-zero j -invariant.(a) Then the set of rational points E ( Q ) is Zariski-dense in E .(b) Moreover, if the j -invariant is not , the surface E is Q -unirational.2. Suppose E has a fiber of type II ∗ , III ∗ , IV ∗ or I ∗ m ( m ≥ ). Then E is Q -unirational. When E is isotrivial with j ( T ) = 0 , this theorem says nothing on the Zariski-density of therational points. Such surfaces are given by a Weierstrass equation of the form y = x + g ( T ) ,where g ( T ) ∈ Z [ T ] is a polynomial of degree at most 6. According to Várilly-Alvarado [VA11],the root number of the fibers of one of these surfaces always takes infinitely many negativevalues, except possibly when all P i irreducible factors of g are such that µ ⊆ Q [ t ] /P i ( t ) . However, it may happen that the polynomial g does not respect this condition, in particular g ( T ) = AT + B , where A, B ∈ Z are such that AB is a rational square. Theorem 6.1, givesprecise conditions on the integers A and B for the surface E A,B,C : y = x + AT + B to have a constant root number of the fibers always +1 . This gives a description of surfaces forwhich our methods does not allow to prove Conjecture 1.1.We also study in Section 6.2 the variation of the root number on a specific family of rationalelliptic surfaces with j ( T ) = 1728 . It is interesting to note that in the case we study the proofof Theorem 1.2 gives a section of finite order. In Theorem 6.2, give the conditions under whichthere can be a constant root number on the fibers of an elliptic surface F A,B,C given by thefollowing Weierstrass equation: F A,B,C : y = x + C ( A T + B ) x , A, B, C ∈ Z and A, B coprime. .1 Previous results Various geometric arguments allow one to prove unconditionnally the density of rationalpoints on rational elliptic surfaces (or its associated del Pezzo surface).Rohrlich [Roh93, Theorem 3] shows the Zariski-density on f ( t ) y = x + ax + b where f is a quadratic polynomial and a, b are non-zero integer, under the additional assumptionthat there exists a fiber of positive rank.In [Sal12], Salgado studies the problem of comparing the rank of the special fibers over anumber field k with that of the generic fiber over k ( P ) . She proves for a large class of rationalelliptic surfaces the existence of infinitely many fibers whose rank exceeds the generic rank of atleast 2.In [Ula08] and [Ula07], Ulas proves the density of rational points on certain families ofisotrivial rational elliptic surfaces with j -invariant 0 and 1728 by constructing a multisectionwith infinitely many points on those families. Jabara generalizes one of Ulas’ work in [Jab12,Theorems C and D] and proves the density when B ( t, is monic and the pair of coefficients ( A, B ) is sufficiently general. An article of Salgado and Van Luijk [SvL14] improves Ulas con-struction, and proves the Zariski-density of set of rational points of a del Pezzo surface of degree1 satisfying certain conditions. For instance it suffices to suppose that the elliptic surface ob-tained by the blowup of the anticanonical point has a fiber of type II over a certain k -rationalpoint of P .The approach of Bettin, David and Delaunay [BDD16] is another way to find whether or nota rational elliptic surface has a section over Q . They study specifically the elliptic surfaces givenby a Weierstrass equation y = x + a ( T ) x + a ( T ) x + a ( T ) where a , a , a ∈ Z [ X ] wih noplace of multiplicative reduction except possibly at infinity. They find different classes of suchfamilies such that deg a i ( t ) ≤ for i = 2 , , and on each of them compute the generic rank. Inparticular, this proves the density of rational points on those of them with a non-zero genericrank.Rohrlich pioneered the study of variations of root numbers on algebraic families of ellipticcurves in [Roh93]. Many authors followed: see, for example, [Man95, GM91, Riz03, CCH05,Hel03, VA11]. Some authors (notably [CS82],[VA11]) remarked that it can happen that theroot number of the fibers might all be of the same value, when the elliptic surface considered isisotrivial, i.e. its modular invariant j ( E ) has no T -dependence. In Section 2, we give a few reminders on rational elliptic surfaces and del Pezzo surfaces. InSection 3, we recall the definition and the properties of the root number.In Section 4, we prove the unirationality of rational elliptic surfaces with j -invariant differentfrom or (the second point of Theorem 1.2). In Section 5, we exhibit a section on a rationalelliptic surface with j -invariant equal to and from this deduce the density of its rationalpoints. This section is not always of infinite order, but its existence completes the proof of thestatement on isotrivial rational elliptic surfaces of Theorem 1.2.In Section 6.1, we find conditions on the coefficients of an rational elliptic surface with zero j -invariant give by the equation y = x + AT + B (with A, B ∈ Z ) so that the root number ofthe fibers always takes the value +1 . In Section 6.2, we find conditions on the coefficients of somerational elliptic surfaces with j -invariant given by the equation y = x + xC ( A T + B ) (where A, B, C ∈ Z ) so that the root number of the fibers always takes the value +1 .We end the article in Section 7 with the completion of the proof of Theorem 1.2. We alsogive various small results on non-isotrivial elliptic surfaces. .3 Acknowledgements I thank my supervisor, M. Hindry, for numerous helpful conversations and suggestions andfor his encouragement. I thank D. Rohrlich and J.-M. Couveignes for their careful reading ofearlier versions of this work. I also thank the anonymous referee for good suggestions.Most of the mathematics of this paper were done at Institut de Mathématiques de Jussieu -PRG. I thank the Institut Fourier of Université Grenoble Alpes and the Max Planck Institutein Bonn for providing good working environment.
Let E → P Q be an elliptic surface over Q given by a minimal Weierstrass equation E : y = x + A ( T ) x + B ( T ) , where A, B ∈ Z [ T ] . The discriminant is the homogeneous polynomial defined as ∆ E ( X, Y ) = Y k − deg ∆ ∆( X/Y ) where ∆( T ) = 4 A ( T ) + 27 B ( T ) , and k is the smallest integer such that k ≥ deg ∆( T ) . Notethat one has thus deg ∆ E = 12 k . Proposition 2.1. (Criteria of rationality [Mir89])An elliptic surface is rational, if and only if < max { A, B } ≤ We observe thus that the discriminant ∆ E actually gives the following classification of ellipticsurfaces: deg ∆ E = E is trivial E is rational E is a K -surface ... ...Rational elliptic surfaces are the (non-trivial) elliptic surfaces with discriminant of lowestdegree, and studying the density on them is a first step towards the resolution of Conjecture 1.1. The following theorem due to Iskovskikh links rational elliptic surfaces to Del Pezzo surfaces.
Theorem 2.2. [Isk79, Thm. 1]Let E be a rational elliptic surface defined over Q .Then, it has a minimal model X/ Q that is :1. either a conic bundle of degree ≥ ,2. or a Del Pezzo surface. A del Pezzo surface X is a non-singular projective algebraic surface whose anticanonicaldivisor is ample. Its degree is the integer d ∈ { , . . . , } corresponding to the self-intersectionnumber ( K X , K X ) of the canonical divisor of X .When X is a conic bundle, the work of Kollar and Mella [KM14] guarantees that the surfaceis Q -unirational, i.e. it is dominated by the projective plane P X . In particular, the set ofrational points is dense.Suppose that X is a del Pezzo surface of degree d . When d ≥ , one knows by the work ofSegre and Manin [Man74] that the existence of one rational point on X implies that the surface s Q -unirational. When d = 2 , Salgado, Testa and Várilly-Alvarado [STVA14], based on a workof Manin [Man74, Thm 29.4], showed that if X contains a rational point that does not lie on anexceptional curve nor a certain quartic, then X ( Q ) is Zariski-dense. If d = 1 , the surface X hasautomatically a rational point: the base point of the anticanonical system. However, the resultsconcerning density of rational points are still partial (for instance [SvL14] and [VA11]). If we blow up the anticanonical point on X , a del Pezzo surface of degree 1, we obtain arational elliptic surface E such that the image of the neutral section is the exceptional divisor.Thus, the rational points of X are dense if and only if the rational points of E are dense.By studying the singular points on rational elliptic surfaces, we obtain the following lemma: Lemma 2.3.
Let E be a minimal rational elliptic surface. We denote by X the surface obtainedfrom E by contracting its section at infinity. Then X is a del Pezzo surface of degree 1 if andonly if the only singular fibers of E have type II or I .Proof. A del Pezzo surface is smooth by definition. Therefore, the blow-up of its base point alsogives a smooth elliptic surface, meaning that the only singular fibers are irreducible (in otherwords, those fibers have type I or II ). An isotrivial rational elliptic surface takes one of the following forms:1. y = x + aH ( u, v ) x + bH ( u, v ) where a, b ∈ Q ∗ are such that a + 27 b = 0 (if j ∈ Q \{ , } );2. y = x + A ( u, v ) x (if j = 0 );3. y = x + B ( u, v ) (if j = 1728 ),for polynomial A, B, H ∈ Z [ u, v ] such that deg H ≤ , deg A ≤ and deg B ≤ . To avoid thecase where the surface is trivial, we suppose also that H is not a square, A is not a th-powerand B is not a th-power.In each cases, the singular fibers have the following configuration:1. Every singular fiber has type I ∗ .2. The singular fibers have either type I ∗ , III or III ∗ .3. The singular fibers have either type I ∗ , II , II ∗ , IV or IV ∗ .The only case where an isotrivial rational elliptic surface has a del Pezzo surface of degree1 as a minimal model is the third one, when moreover the polynomial B is squarefree and hasdegree ≥ . The root number of an elliptic curve E is expressed as the product of the local factors W ( E ) = Y p ≤∞ W p ( E ) , where p runs through the finite and infinite places of Q , W p ( E ) ∈ {± } and W p ( E ) = +1 forall p except a finite number of them. The local root number of E in p , W p ( E ) , is defined in erms of the epsilon factors of the Weil-Deligne representations of Q p (see [Del73] and [Tat77]).Rohrlich [Roh93] gives an explicit formula for the local root numbers in terms of the reductionof the elliptic curve E at a prime p = 2 , and at p = 2 , in case E is semi-stable. Halberstadt[Hal98] gives tables (completed by Rizzo [Riz03]) for the local root number at p = 2 , accordingto the coefficients of E . Moreover we always have W ∞ ( E ) = − .The root number is hypothetically equal to the sign W ( E ) ∈ {± } of the conjectural func-tional equation of L ( E, s ) the L -function of E : N (2 − s ) / E (2 π ) s − Γ(2 − s ) L ( E, − s ) = W ( E ) N s/ E (2 π ) − s Γ( s ) L ( E, s ) . When we work on elliptic curves over Q , such a functional equation always exists (by Wiles’work [Wil95] and its generalisation by Breuil, Conrad, Diamond and Taylor [BCDT01]) and thevalues of the root number and the sign of the functional equation are indeed the same.The Birch and Swinnerton-Dyer conjecture implies that the root number is related to therank of the elliptic curve: Conjecture 3.1 (Parity Conjecture) . For all elliptic curve E over Q , we have W ( E ) = ( − rank E ( Q ) . As a consequence of this equality, it suffices that W ( E ) = − for the rank of E ( Q ) not to bezero and in particular for E ( Q ) to be infinite.Let E be a rational elliptic surface over P Q . The elliptic surface can be seen as a family ofelliptic curves, and admits a Weierstrass equation of the form E : y = x + A ( T ) x + B ( T ) , with A ( T ) , B ( T ) ∈ Z [ T ] have respectively degree less than or equal to 4 and 6.We denote by ∆( T ) = 4 A ( T ) − B ( T ) the discriminant and the corresponding homogenouspolynomial ∆ E ( X, Y ) = Y − deg ∆ ∆( X/Y ) . Let also M E ( X, Y ) the product of the polynomials associated to the places of multiplicativereduction, that is to say, polynomials dividing ∆ E , but not Y − deg A A ( X/Y ) .We consider the sets W + and W − given by W ± ( E ) = { t ∈ Q : E t is an elliptic curve and W ( E t ) = ± } . As a consequence of the parity conjecture, if W − ( E ) = ∞ , then there exist infinitely manyfibers of E that are non singular elliptic curves with positive rank, and this guarantees thedensity of the rational points on E .When the surface is isotrivial, it can happen that one of the set W − or W + is finite or empty.In [CS82], Cassels and Schinzel find a family of elliptic curves, such that j ( T ) = 1728 , on whichthe sign of the fibers always takes the value − : E T : y = x − (7 + 7 T ) x. Varilly-Alvarado gives more examples of elliptic surfaces with constant root number in [VA11],among them the following elliptic surface with j = 0 , given by the Weierstrass equation y = x + 6(27 T + 1) , whose fibers always have a root number of value +1 . .2 Local root number at 2 and 3 of y = x + αx and y = x + α We give here some formulas for the local root number at 2 and 3 of the elliptic curves y = x + αx and y = x + α for α ∈ Q . Lemma 3.2. [VA11, Lemme 4.7]Let t be a non-zero integer and let be the elliptic curve E t : y = x + tx . We denoteby W ( t ) and W ( t ) its local root numbers at 2 and 3. Put t and t the integers such that t = 2 v ( t ) t = 3 v ( t ) t . Then W ( t ) = − if v ( t ) ≡ or and t ≡ or or if v ( t ) ≡ and t ≡ , , , , or
15 mod 16; or if v ( t ) ≡ and t ≡ , , , , , or
15 mod 16;+1 otherwise. W ( t ) = ( − if v ( t ) ≡ otherwise. Lemma 3.3. [VA11, Lemme 4.1]Let t be a non-zero integer and the elliptic curve E t : y = x + t . We denote by W ( t ) and W ( t ) its local root numbers at 2 and 3. Put t and t the integers such that t = 2 v ( t ) t = 3 v ( t ) t .Then W ( t ) = − if v ( t ) ≡ or or if v ( t ) ≡ , , or and t ≡ , otherwise. W ( t ) = − if v ( t ) ≡ or and t ≡ or if v ( t ) ≡ or and t ≡ or if v ( t ) ≡ and t ≡ or or if v ( t ) ≡ and t ≡ or , otherwise. j ( T ) = 0 , Theorem 4.1. [KM14, Thm. 1] Let K be any field of characteristic = 2 and a ( t ) , . . . , a ( t ) ∈ K [ t ] polynomials of degree 2 giving a nontrivial family of elliptic curves. Then the surface S : y = a ( t ) x + a ( t ) x + a ( t ) x + a ( t ) ⊂ A xyt is unirational over K . In a first version of the article of Kollár-Mella [KM14], Theorem 4.1 excluded the isotrivialcase. The author wanted to complete this result, and obtained Theorem 4.2. However, it hadbeen completed by Kollár and Mella themselves by the time she submitted her ph.D thesis.Their technique is different from the one in this article.
Theorem 4.2.
Let E be a isotrivial rational elliptic surface given by the equation E : Y = X + aH ( T ) X + bH ( T ) , here a, b ∈ Z \{ } and H ( T ) is a degree ≤ polynomial that is not a square. Then the surface E is Q -unirational. In particular, its rational points are dense for Zariski topology.Remark . This result is proven by Rohrlich [Roh93, Theorem 3] under the a priori restrictiveassumption that there exists a fiber of positive rank. This assumption is removed here.
Proof.
Observe that the surface E is endowed with many fibrations. E : H ( T ) Y = X + aX + b ✱ ϕ u u ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ ❴ ϕ (cid:15) (cid:15) ✒ ϕ ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ x y t The last two, ϕ and ϕ , are elliptic fibration (with section). Even if the fibration definedby ϕ is isotrivial, the one defined by ϕ is not. Indeed, if we write H ( T ) = α T + α T + α for the appropriate coefficients α i , the fibration ϕ has the fiber E y := α y T + α y T = X + aX + b − α y which can after a change of variables (first T ′ = α yT and x = α X , then t = T ′ + α α y ) bewritten E y : t = x − c ( y ) x − c ( y ) where c ( y ) = α a and c ( y ) = ( α α + α α ) y + α b . By computing the j -invariant, onesees that this curve is not isotrivial, except in the case where a = 0 ( c ( y ) is zero) and α = α α + α α , in other words when H is the square of a linear polynomial (in that case, E istrivial). These cases are excluded by our hypotheses. Hence, we can apply Theorem 4.1. Thisproves the unirationality of E endowed with the elliptic fibration ϕ . Remark . Another way to prove Theorem 4.2 would have been the use the work of Colliot-Thélène [CT90]. The second theorem of this article shows that for X , a conic bundle of degree4, the Brauer-Manin obstruction to the Hasse principle is the only obstruction. To deduce fromthis Theorem 4.2, one would have to check that the Brauer group of the surfaces that we consider(whose equation is h ( t ) y = x − ax where deg h = 2 ) is the Brauer-group of Q . j ( T ) = 1728 We study now the isotrivial rational elliptic surfaces of the form y = x + xA ( T ) where A ∈ Z [ T ] is such that deg A ≤ . The density of rational point is proven in the case where deg A ≤ by Ulas in [Ula07]. For this reason, we concentrate on surfaces such that deg A = 4 .Let a , a , a , a , a ∈ Z be the coefficients such that A ( T ) = a T + a T + a T + a T + a . First observe that we have F ( T ) = a (cid:16)(cid:0) T + g T + g (cid:1) + h T + h (cid:17) , where g = 4 a a − a a , g = a a , h = 2 a ( a + x ) − (4 a a − a ) a and h = 2 a a − a (4 a a − a )2 a . e make the change of variables T ′ = T − g / . Thus we can write A ( T ) = a ( T ′ + ( − g g ) T + ( − g g g ) T + ( − g + g + h )) . Replacing T and T by their expressions in terms of T ′ , we obtain the following equation: y = x + a x ( T ′ + A T ′ + A T ′ + A ) , where A = ( g − g , A = ( g a g ) , and A = ( − g + g + h ) . Hence, one can assume that a = 0 (or else we do the change of variable previously explained).The surface E has the following fibrations. E : Y = X + A ( T ) X ✳ ϕ v v ♥♥♥♥♥♥♥♥♥♥♥♥♥♥ ❴ ϕ (cid:15) (cid:15) ✏ ϕ ( ( PPPPPPPPPPPPPP x y t
The initial fibration is ϕ . The fiber ϕ : ( x, y, t ) x is a genus 1 curve, and this fibrationis a priori without section .The equation of the fiber at x can be written as: C x : y = a xt + a xt + a xt + ( a x + x ) . It is a genus 1 curve with two points at infinity, denoted by ∞ + and ∞ − , which are rationalif and only if x ∈ a Q ∗ . Proposition 5.1.
Let P x = cl (( ∞ + ) − ( ∞ − )) ∈ C x ( Q ) for x ∈ a Q ∗ .Then— if a = 0 , the point P x has order 2,— if a = 0 , P x has infinite order (except for finitely many x ).Proof. Explicitely, putting u = t and v = yt , one has in coordinate ( u, v ) : ∞ + = (0 , b ) , and ∞ − = (0 , − b ) . Suppose that b = a x for a certain rational number b . We write C x : y = b t + a xt + a xt + a x + x . A well-chosen change of variables gives the following general Weierstrass equation for C x . C x : S + h S = R − g R − h R, (2)where g = a a , h = 2 a ( a + x ) − a a and h = a a .
1. To ensure the existence of a section to ϕ : ( x, y, t ) x , it would be necessary to check that every ϕ − ( x ) admitsa rational point.2. We use here a very classical method, explained in particular in a book of Cassels’ [Cas91]. An interested readercan also find the details of the change of variables in the author’s phD thesis [Des16a, Section 1.1.3]. Observe that thecoefficients g , h , h in the general Weierstrass equation (2) correspond to the quantities previously defined in thisSection. he two points at infinity are send to the two obvious points of (2). We have: R ( ∞ + ) = ∞ and S ( ∞ + ) = ∞ .R ( ∞ − ) = 0 and S ( ∞ − ) = − h We put in a natural way the point obtain from ∞ + as marked point of the curve C x , thatis as the identity element of the group law of the set of rational points. With this configurationone has (0 , − h ) = [ − , . We deduce of this that if h = a a = 0 , then the point (0 , − h ) has order 2.In the case where h = 0 , let us find the order of Q = (0 , . Its order will be the same as theone obtained from ∞ − . We use a result proven simultaneously by Lutz and Nagell which canbe found [Sil94, p.240]: if E/ Q is an elliptic curve of Weierstrass equation y = x + Ax + B , A, B ∈ Z and that P ∈ E ( Q ) is a torsion point different for the point at infinity, then thefollowing properties hold:1. x ( P ) , y ( P ) ∈ Z .2. We have either [2] P = O or x ([2] P ) ∈ Z .To use this fact, we need to consider a curve with integer coefficients (denote these coefficientsby A i ). As the coefficients of C x might not be integers, we will choose a certain integer α forwhich the twist of the curve has integer coefficients. Let u and v be the coprime integers suchthat x = uv . If we put α = 2 · a v , the coefficients of the curve are integers C ′ x : S + α h S = R − α g R − α h R. (In fact, it is sufficient to put α = 2 a w where w is an integer such that v | w .) We now showthat if h = 0 , the point Q is not 2-torsion for infinitely many values of x . We first find thecondition for R ([2] Q ) to be an integer. We have: R ([2] Q ) = (cid:16) α h α h (cid:17) + 4 α g . For this coordinate to be an integer, we need α h to divide α h . Recall that x = uv where u, v ∈ Z are coprime. We have α h = A (cid:16) uv (cid:17) + B, where A = 2 a v and B = 2 a a u − a a v . As for the quantity α h , it is an integer ofvalue α h = 2 a a v . If α h | α h for every x ∈ Q ∗ a , then α h divides B (we obtain this taking for instance x = ( α h ) a ). Therefore, α h divides Ax for any choice of x . Choose v prime to a . Inthis case, we have a contradiction since Ax = 2 a v (2 a u ) should be divisible by a a v ,but v is assumed to be prime to a and to u . This contradiction shows that for every x ∈ Q ∗ a whose denominator is prime to a , the point Q is of infinite order on the curve C x .We conclude the proof by using Silverman’s specialization theorem (see [Sil83] and [Sil,Theorem 11.4, Chapter III]). A priori, the fibration ϕ : E → P Q ( x, y, t ) x. is not an elliptic surface over Q . However, let us consider the application φ : P Q → P Q z x = a z . nd the fibered product E ′ of E with respect to the fibration. By the previous argument, E ′ admits two sections ∞ + and ∞ − . It is thus an elliptic surface over Q . Let us choose as thecanonic section ∞ + .If there exists a linear change of variable such that A = A T ′ + A T + A , then ∞ − is atorsion point on every fiber at x = az of E . Therefore, the section ∞ − ( z ) is torsion for every z ∈ P Q (except finitely many, i.e. those defining a singular fiber).If there is no such change of variable, then the point ∞ − has infinite order for infinitely manyfibers of E . Therefore, Silverman’s specialization theorem guarantees that ∞ − ( z ) has infiniteorder on every fiber of E ′ except for finitely many z .We directly deduce from this proposition the following theorem : Theorem 5.2.
Let E be a rational elliptic surface given by the equation E : Y = X + A ( T ) X, where A ( T ) is a polynomial of degree 4 with integer coefficients.Suppose there exists no linear change of variable T → T ′ + b such that A is of the form A ( T ′ ) = A T ′ + A T ′ + A , where A , A , A ∈ Z .Then the rational points of E are Zariski-dense.Remark . The surfaces which are not treated by this theorem are of the form: y = x + x ( a T + 4 ba T + (6 b a + a ) T + (4 b + 2 ba ) T + a b + a b + a ) for a certain b ∈ Q and a , a , a ∈ Z such that p a − a a Q . Suppose E is of that form,then by Proposition 5.1 the points P x constructed previously is 2-torsion on C x for almost every x ∈ a Q ∗ Proof.
We can assume that a = a = 0 . For these surfaces, the application ( x, y, t ) x is afibration in genus 1 curves, infinitely many of which (in fact every fiber at x ∈ a Q ∗ except afinite number of them)) admits a structure of group and a point of infinite order. This showsthe density of rational points of E ( Q ) .In the two next sections, we give other arguments showing the density of the rational pointsin more generality. Theorem 5.3.
Let E be a rational elliptic surface of Weiestrass equation y = x + A ( T − α )( T − β ) x, where A, α, β ∈ Q . Then the rational points are Zariski-dense.Proof. By changing variables X = ( T − α ) x and Y = ( T − α ) y , one obtain the equation Y = ( T − α ) X + A ( T − β )( T − α ) X which is isomorphic to Y = ( T − α ) X + A ( T − β ) X. A reshuffle of the terms permits to obtain the following equation for E Y − T ( X − AX ) + ( αX + XAβ ) = 0 which is a conic bundle. This bundle has less than 6 singular fibers. Corollary 8 of the articleof Kollár and Mella [KM14] thus shows unirationnality of E . Therefore, the rational points aredense.
3. Note that the case a = 4 a a is already excluded by assumption that E is non-trivial. .3 Density on isotrivial elliptic surfaces with j = 1728 As a conclusion for this section, we show the density of rational points on every isotrivialrational elliptic surfaces with j -invariant 1728. Theorem 5.4.
Let E be an isotrivial rational elliptic surface with j ( T ) = 1728 . Then therational points E ( Q ) are Zariski-dense.Proof. Let E be an isotrivial rational elliptic surface with j -invariant j ( T ) = 1728 .Recall Theorem 2.2 due to Iskovskikh that says that a rational elliptic surface has a minimalmodel which is either a conic bundle of degree 1 or a del Pezzo surface.Let X be a minimal model of E . As a corollary of Lemma 2.3, X is never a del Pezzo surfaceof degree 1. Indeed, the discriminant of E is ∆( T ) = − · A ( T ) so every fiber at a factor of A ( T ) has a reduction of Kodaira type III , I ∗ or III ∗ , and such singular fibers are reducible.In the case where X is a conic bundle of degree 1, [KM14] show unirationality of X , andthus the density of its rational points. Therefore, it is also the case for E . In the case where X is a del Pezzo surface of degree ≥ , [Man74] shows unirationality of X and E .We only have to consider the case where X is a del Pezzo surface of degree 2. In this case,we have the following sections on E :1. The section of the points at infinity [0 , y, , .2. The section of Proposition 5.1 [ x, − bu , u , where u = 0 , b = √ a x and x ∈ a Q ∗ .3. The section [0 , , t, .If the contraction two of them gives a del Pezzo surface of degree 2, then the image of thethird is a rational curve. If it is an exceptional curve, we can contract is to obtain a del Pezzoof degree 3, on which the rational point are dense. If the image is not an exceptional curve, itallows all the same to find an infinity of points on X . Therefore, some of them are not on anexceptional curve nor on a distinguished quartic. We can thus use the work of Salgado, Testaand Várilly-Alvarado [STVA14] which shows unirationality and density of rational points on X and E . As seen in Section 3, the root number is conjecturally equal to the parity of the geometricrank. It can thus be used as a substitute and becomes useful in the study of the density ofrational point, especially when no geometric argument is known.Let E be an isotrivial elliptic surface . We study here the variation of the root number ofthe fibers E t , more precisely the cardinality of the sets W ± ( E ) = { t ∈ Q | W ( E t ) = ± } . If ♯W − = ∞ , we can conclude the density of the rational points conditionally to the parityconjecture.We restrict ourselves to specific surfaces of quartic (j-invariant = 1728 ) and sextic twists( j = 0 ), since the quadratic twist case is detailled in another paper [Des18].
4. The non-isotrivial case is already studied in [Des16b] .1 Case j ( T ) = 0 Let E be a rational elliptic surface described by the Weierstrass equation E : y = x + f ( T ) , where f ∈ Q [ t ] is such that deg f ≤ and is sixth-power free.A general geometric argument to show the density of the rational points, like those presentedin the previous sections, is still not known. However, there are partial results. In [Ula08] and[Ula07], Ulas gives various conditions for the rational points on E f to be dense: 1) when E f isrelated to a del Pezzo surface of degree 1 and that a certain section on E f is non-torsion, 2)when f is a monic polynomial of degree six and f is not even. Jabara generalizes this secondUlas’ work in [Jab12, Theorem C] and treats the case with f ( T ) monic.We chose to study a surface given by an equation of the form y = x + AT + B (3)as a sequel of [VA11, Theorem 2.1] which shows that the variation of the root number of thefibers of a rational elliptic surface of the form y = x + F ( T ) where F has a primitive factor f i such that µ Q [ T ] /f i where µ is the group of the thirdroots of unity. A natural example of a polynomial not obeying this condition is F ( T ) = C (3 A T + B ) . (4)Our Theorem 6.1 is thus the natural continuation of the work of Várilly-Alvarado, in particularof [VA11, Theorem 1.1].In broad terms, the proof used in Várilly-Alvarado’s article is based on the fact that the rootnumber of the fiber E t = mn , m, n ∈ Z coprime, is given by the formula ([VA11, Prop. 4.8]) : W ( E t ) = − R ( t ) Y p | F ( m,n ) p ≥ ( if v p ( F ( m, n )) ≡ , , , (cid:16) − p (cid:17) if v p ( F ( m, n )) ≡ , (5)where R ( t ) = W ( E t ) (cid:18) − F ( m, n ) (cid:19) W ( E t )( − v ( F ( m,n )) , where F ( m, n ) = F ( mn ) n deg F . It relies essentially on making the product over p ≥ vary.Families of sextic twists with F ( T ) of the form (4) have the property that whenever p | F ( m, n ) and p ∤ C for p ≥ then F ( m, n ) = C (3 A m + B n ) ≡ p and thus (cid:18) Bn Am (cid:19) ≡ − p, forcing the terms in the product in the formula (5) to be always equal to +1 except maybe for p | C . This allows to prove the following: Theorem 6.1.
Let E A,B,C be an elliptic surface described by the Weierstrass equation E A,B,C : y = x + aT + b, where a, b ∈ Z .Then, the function of the root number of the fibers is constant, except for the surfaces of theform E A,B,C : y = x + C (3 A T + B ) such that the integers A, B coprime and C fulfill one ofthe conditions of Lemma A.1, and one of Lemma A.2. roof. Put C = pgcd( a, b ) . If a/b is not a rational square, then by [VA11, Thm 2.1] the twosets W ± = { t ∈ P | W ( E t ) = ± } are both infinite, or in other words, the root number of the fibers of E is non-constant. Supposethus A/B is a rational square, that i.e. there exists
A, B ∈ Z such that A = aC and B = bC .For each t ∈ Q , let ( m, n ) be the pair of coprime integers such that t = mn , and let E m,n denote the elliptic curve E m,n : y = x + C (3 A m + B n ) , which is isomorphic to E t . Observethat E m,n and E t must then have the same root number. Put P ( m, n ) = C (3 A m + B n ) .Thereafter, we will use the following notations to put together similar terms in the formula(5): ω ( t ) := W ( E t ) (cid:18) − P ( m, n ) (cid:19) and ω ( t ) := W ( E t )( − v ( P ( m,n )) . and P ( t ) := Y p | αp ≥ ( if v p ( P ( m, n )) ≡ , , , (cid:16) − p (cid:17) if v p ( P ( m, n )) ≡ , First, note that for any choice of
A, B, C , the function P ( t ) is a constant. Indeed, for anyprime p dividing a certain value A m + B n , one has (cid:18) Bn Am (cid:19) ≡ − p. We thus have P = ( − σ , where σ = { p such that p | C , v p ( C ) ≡ , and p ≡ } , Moreover, note that the three functions are independent to each other namely: the function ω depends on v ( P ( m, n )) mod 4 and P ( m, n ) mod 4 the function ω depends on v ( P ( m, n )) mod 6 , and P ( m, n ) mod 9 .Therefore, if one of the values ω or ω non-constant, then the global root number is non-constant.This proves that the root number is non constant, except for the surfaces such that A, B, C fulfill one of the conditions of Lemma A.2, and one of Lemma A.1.
Remark . The independance of ω and of ω is also given by the Helfgott’s formula for theaverage root number [Hel03, Proposition 7.2].This allows us to compute the value of the constant root number in each of the special cases. Example 1.
Suppose that A = B = 1 . Let E , ,C be the elliptic surface defined by the equation E : y = x + C ( T + 1) . By looking at the tables 1, 2, 3 and 4, we have that the function ω ( t ) is constant when t runsthrough Q if and only if1. (for ω ( t ) = − )(a) v ( C ) ≡ and C ≡ ,(b) v ( C ) ≡ and C ≡ ,(c) or v ( C ) ≡ and C ≡ , ,2. (for ω ( t ) = +1 ) a) v ( C ) ≡ and C ≡ , (b) v ( C ) ≡ and C ≡ (c) v ( C ) ≡ and C ≡ and that the function ω ( t ) is constant when t runs through Q (and ω ( t ) = +1 ) if and only if v ( C ) ≡ , , .If we suppose that C is less or equal to , we find only the following values for which theroot number is constant:— if C = 10 , , , the root number is -1.— if C = 90 the root number is +1.When the surface has negative root number on every non-singular fiber, the parity conjecturestates that the rank of the fibers of this elliptic surface should be always positive. For thesurfaces on which the root number is +1 , however, it is not possible to conclude anything aboutthe density of rational points from the study of the variation of the root number.In the case of C = 90 , the surface E , , has no section defined over Q , and so as far as weknow the density of the rational points is still an open question. j ( T ) = 1728 The density of rational points on certain elliptic surfaces of the form E : y = x + g ( T ) x isgaranteed by the construction of a section for E done in Section 5.1. However, there are surfacessuch that this section is not of infinite order. This happens in particular when g ( T ) = AT + B .This case fails as well to satisfy the hypotheses of [VA11, Theorem 2.3] and it is thus possiblethat the root number is constant. By the parity conjecture, an elliptic surface with constant rootnumber always equal to +1 is such that every fiber has even rank, thus although the followingresult doesn’t give new density result, it still give us some interesting (conditional) informationabout the distribution of the rank in the family of the fibers. Theorem 6.2.
Let F A,B,C be an elliptic surface represented by the Weierstrass equation F A,B,C : y = x + C ( A T + B ) x, where A, B, C ∈ Z and gcd( A, B ) = 1 .Then, the function t → W ( E t ) of the root number of the fibers is constant, except for thespecific surfaces such that A, B, C fulfill one of the conditions of Lemma A.4, and one of LemmaA.3.Proof.
Let
A, B, C ∈ Z be such that gcd( A, B ) = 1 . Let us write F = F A,B,C . For each t ∈ Q ,let ( m, n ) be the pair of coprime integers such that t = mn , and let F m,n denote the elliptic curve F m,n : y = x + C (3 A m + B n ) x, isomorphic to F t .Put P ( m, n ) = C (3 A m + B n ) . It is not very difficult to see that the root number isgiven by the formula W ( F t ) = − W ( t ) W ( t ) (cid:18) − P ( m, n ) (cid:19) Y p | P ( m,n ) p ≥ ( if v p ( P ( m, n )) ≡ , , , (cid:16) − p (cid:17) if v p ( P ( m, n )) ≡ , . Thereafter, we will use the following notations to congregate similar terms: ω ( t ) := W ( F m,n ) (cid:18) − P ( m, n ) (cid:19) and ω ( t ) := W ( F m,n )
5. This formula is shown in [Des18]. nd P ( t ) := Y p | P ( m,n ) p ≥ ( if v p ( P ( m, n )) ≡ , , , (cid:16) − p (cid:17) if v p ( P ( m, n )) ≡ , First, note that for any choice of
A, B, C , the function P ( t ) is a constant. Indeed, for anyprime p dividing a certain value A m + B n , one has (cid:18) Bn Am (cid:19) ≡ − p. We have thus P = ( − σ , where σ = { p such that p | C , v p ( C ) ≡ , and p ≡ } , Moreover, note that the three functions depends on independent parameters, namely: the function ω depends on v ( P ( m, n )) mod 4 and P ( m, n ) mod 4 the function ω depends on v ( P ( m, n )) mod 6 , and P ( m, n ) mod 9 .Therefore, if one of the values ω or ω non-constant, then the global root number is non-constant.Therefore, the root number is non constant, except for the surfaces such that A, B, C fulfillone of the conditions of Lemma A.2, and one of Lemma A.1.
Example 2.
Suppose that A = B = 1 . Let F , ,C be the elliptic surface given by the equation F , ,C : y = x + C ( T + 1) x. By looking at the formula of Lemma A.3 as well as Tables 5 and 6, we find that the function ω ( t ) = W ( t ) is always constant when t runs through Q and its values is1. W ( E t ) = +1 if v ( C ) ≡ , , W ( E t ) = − if v ( C ) ≡ ,and the function ω ( t ) is constant and equal to − if and only if1. v ( C ) ≡ and C ≡ , , ,
11 mod 16 v ( C ) ≡ and C ≡ v ( C ) ≡ and C ≡ , , ,
15 mod 16 v ( C ) ≡ and C ≡ This makes quite a lot of possibilities for F , ,C : for C ≤ we have the following:1. the root number of every fiber is +1 if C = 1 , , , , ,
2. the root number of every fiber is − if C = 9 , .However, the density of rational points holds all the same on every surface F A,B,C regardless ofthe variation of the root number by Theorem 5.4.
In the ph.D thesis of the author [Des16a] and in [Des16b], we prove the following theorem.This work is based on a preprint of Helfgott [Hel03], revisited and completed with a differentapproach. heorem 7.1. Let E be a rational elliptic surface given by the equation E : y = x + F ( u, x + G ( u, , where F and G are homogeneous polynomials of degree respectively 4 and 6 defining a minimalmodel. We suppose that E is non-isotrivial, and thus in particular F G = 0 . Define the twofollowing polynomials associated to E :— ∆ E ( U, V ) = 4 F ( U, V ) + 27 G ( U, V ) = Q si =0 P m i i ( U, V ) (the homogeneous discriminant of E )— and M E ( U, V ) = Q i ∈ M E P i ( U, V ) where M E = { i such that P i ∤ F } (the product of polyno-mials coming from places of multiplicative reduction).Suppose that every P | ∆ E verifies the squarefree conjecture and every P | M E verifiesChowla’s conjecture.Then the sets W ± are both infinite. This means that ∆ E needs to verify the squarefree conjecture, and that M E needs to verifyChowla’s conjecture. Those conjectures are known to hold in the following cases: Theorem 7.2.
Let P ∈ Z [ X, Y ] be a homogeneous polynomial.1. (Greaves [Gre92]) The squarefree conjecture holds if deg P i ≤ .2. (Helfgott [Hel05], Lachand [Lac14]) Chowla’s conjecture holds if deg P ≤ or (Green-Tao[GT10]) if P is a product of linear factors; The following proposition classifies all the rational elliptic surfaces on which Theorem 7.1 isunconditional.
Proposition 7.3.
Let E be a non-isotrivial rational elliptic surface given by the equation: E : y = x + F ( T, X + G ( T, , where F and G are homogeneous polynomials of degree respectively and .We suppose that E respects one of the following properties:1. M = ∅ ;2. the places in M are all rational;3. M = { P } where P ∈ Z [ T ] is a polynomial of degree 3;4. M = { P , P } where P , P ∈ Z [ T ] are polynomials of degree respectively 1 and 2;5. M = { T , P } where P ∈ Z [ T ] is a polynomial of degree 2.Then the sets W ± are both infinite.Remark . There are examples of rational elliptic surfaces of each of the case of the list.When M = ∅ , the surface obtained by the contraction of the canonical section never is a delPezzo surface of degree 1. Indeed, an elliptic surface with no place of multiplicative reductionadmits automatically a place of potentially multiplicative reduction. In this case, Corollary 2.3gives us that E does not come from a degree 1 del Pezzo surface. Each of the four last classesof rational elliptic surfaces contains del Pezzo surfaces of degree 1. Remark . The geometric arguments presented in the section 7.3 prove the density in certaincases on which it is not possible to apply unconditionnally the work of Helfgott. In particular,Proposition 7.5 requires that there exists a rational place of type I ∗ m , II ∗ , III ∗ IV ∗ or I ∗ . Proof. (of Proposition 7.3)Let B E and M E be the polynomials such that1. B E is the product of the polynomials associated to the places of bad reduction of E thatare not of type I ∗ , . M E the product of the polynomials associated to the places of multiplicative reduction of E .Theorem 7.1 and the parity conjecture show the variation of the root number on the fiberswhen E is a non-isotrivial surface whose polynomial B E and M E are such that1. B E verifies the squarefree conjecture,2. M E verifies Chowla’s conjecture.If these exists no place of multiplicative reduction on E , we have M E = 1 . Thus there is noneed to consider Chowla’s conjecture. Moreover, the irreducible factors of ∆ appear with theexponent ≥ . They are of degree ≤ . Therefore, squarefree conjecture holds.Suppose now that E admits a place of multiplcative reduction on E .Let be the following minimal Weierstrass model for E : y = x + F ( T, x + G ( T, , where F, G ∈ Z [ U, V ] are homogeneous polynomials of degree and respectively. Let C , be thelargest primitive polynomial such that C | F and C | G . We write F = aCF and G = bC G where F and G are primitive polynomial and a, b ∈ Z are constants. Let R := pgcd( F , G ) .Observe that the polynomial R splits by construction. We write F = aCRF and G = bC RG where F , G ∈ Z [ X, Y ] are suitable polynomials. The discriminant can be written ∆ = C R (4 a RF + 27 b CG ) . When the surface is non-isotrivial, if there exists P a polynomial such that P | F , then P ∤ G ,and thus ord P C ≤ . Observe that R , C , F and G verify squarefree conjecture as their degreesare ≤ .We define M o = (4 RF − CG ) and we observe that ∆ = C R M . The polynomial M o isa product of powers of polynomials associated to places of multiplicative or additive reduction.It is possible that M o is divisible by the polynomials associated to places of additive reduction:the factors of C or R . For F = P α . . . P α r r (the decomposition of F in irreducible factors) thereexist integers β i ∈ N such that M = M o P β . . . P β r r . Proposition 7.4.
Let X be a non-isotrivial rational elliptic surface with no place of reductionof type I m . Then X can be described by one of the following equations: E : y = x + aL Qx + bL QM, (6) where Q = cL − b M a ; and E : y = x + aL L L x + bL L L , (7) where L = 4 a L − b L . We have a, b ∈ Z , L , L , L and M linear polynomials and Q aquadratic polynomial.Remark . The homogeneous and one-variable versions of the conjectures hold on the surfaces E a and E b . Indeed, Chowla’s conjecture is true since M E = 1 , and as every irreducible factors ofthe coefficients are linear, squarefree conjecture also holds. emark . In the first case, the places of bad reduction are those associated to L (of type I ∗ ),and those associated to the irreducible factors of Q (of type II ).In the second case, we have three rational places of bad reduction: the one associated to a L − b L has type I ∗ , the one associated to L has type III and the one associated to L has type II . Proof.
Let E be the rational elliptic surface associated to X , given by the Weierstrass equation: E : y = x − c ( T ) x − c ( T ) , where c ( T ) , c ( T ) ∈ Z [ T ] have degree respectively less than or equal to and . Let ∆ be thediscriminant of E . This surface has a place of reduction of type I ∗ m because the invariant j = c ∆ admits necessarily a pole (at a irreducible polynomial P ∈ Z [ T ] or at T ).Each fiber at t = mn is given by the equation : E t = E m,n : y = x + n − deg c c (cid:16) mn (cid:17) x + n − deg c c (cid:16) mn (cid:17) . Suppose the P is the polynomial associated to a place of reduction type I ∗ m , then we write F = P F and G = P G for F , G polynomials. We have that P | ( F − G ) . We need to have deg P = 1 . Indeed, if deg P = 2 , then G and F are constant and thus E is isotrivial. The casewhere deg P > is not possible because we would have deg G > . Therefore, a non-isotrivialrational elliptic surface with no place of type I m admits a rational place of reduction I ∗ m . Wehave deg P = 1 , deg F = 2 and deg G = 3 .The case where ( F , G ) = 1 is not possible. Indeed, we would have P | F − G and thesurface would be isotrivial. Therefore, F and G have a common factor, that we will denoteby A . We write F = AF and G = AG for convenient polynomial F and G . We have ∆ = P A ( AF − G ) . The reduction at A is thus additive.Suppose deg A = 2 . In this case, if ( A, G ) = 1 , we have A = γP + G . If ( A, G ) = A for a linear polynomial A , then we have P = A − A γ . Suppose deg A = 1 . If A | G , we must have A = F − γP G . However, there exist no polynomial
A, P, G , F ∈ Z [ T ] with this property. Indeed, by imposinga linear change v = P ( t ) , and puting ν = uv , we are lead to solve a F ( ν ) + 27 b A ( ν ) M ( ν ) = c. As F = P , F ( ν ) is non-constant. Let u such that F ( u ) = 0 . We have b A ( u ) G ( u ) = c = 0 . By deriving at u , we obtain : A ( u ) G ( u ) G ′ ( u ) + A ′ ( u ) G ( u ) = 0 . When we derive another time, we have : A ( u ) G ′ ( u ) + 2 A ( u ) + 4 A ′ ( u ) G ′ ( u ) G ( u ) + A ′′ ( u ) G ( u ) = 0 . e observe that G is linear. We have: A ( u ) G ′ ( u ) + 4 A ′ ( u ) G ( u ) = 0 . Therefore, A is proportionnal to G . For all P ∈ Z [ T ] linear, the polynomial P ( T ) − c has nodouble root. Thus, F has to be constant. Therefore G , F , A and P are proportional to eachother and the surface E is isotrivial.If A ∤ G , we must have the equality A = γP + G F . By a similar argument as in the previous case, this is not possible either.
In this section we prove unconditionally the density on many more elliptic surfaces, notnecessarily isotrivial. Moreover Helfgott’s paper does not prove unconditionally the variation ofthe root number for those surfaces.
Proposition 7.5.
Let E be a elliptic surface given by the equation E : y = x + L Qx + L C, (8) where L, Q, C ∈ Z [ u, v ] have respective degree 1, 2 and 3. Then the surface is Q -unirational. Inparticular, E ( Q ) is Zariski-dense.Remark . The polynomial L of the surface E in this proposition is such that L | ∆ . As wechose a minimal Weiestrass model for E , this means that the reduction at L has type I ∗ , II ∗ , III ∗ , IV ∗ or I ∗ m . Conversly, if we consider a surface with a rational place of one of these types,we can find an equation of the form (8). We deduce directly the following corollary: Corollary 7.6.
If a rational elliptic surface E has a rational place of type I ∗ , II ∗ , III ∗ , IV ∗ or I ∗ m , then the rational points of X are Zariski-dense.In particular, if E is a non-isotrivial elliptic surface with no place of multiplicative reduction,then its rational points are dense.Proof. Let S be an elliptic surface given by the equation S : y = x + L ( t, Q ( t, x + L ( t, C ( t, , where L, Q, C ∈ Z [ u, v ] have respective degree 1, 2 and 3. Note that this surface is rational. Westudy the surface which is birational (cid:16) yL (cid:17) = (cid:16) xL (cid:17) + QL (cid:16) xL (cid:17) + (cid:16) CL (cid:17) . (9)We can suppose that L ( u, v ) = v (otherwise, we do a linear change on u, v ). Put t = uv , x ′ = xv and y ′ = yv , whose inverse transformation is x = x ′ v , y = y ′ v , u = tv . By this changeof variables, (9) becomes S ′ : y ′ = x ′ + q ( t ) x ′ + c ( t ) ⊂ P , with Q ( t,
1) = q ( t ) and C ( t,
1) = c ( t ) , which is a cubic surface with a finite number of singularpoints.Note that on a cubic surface which is not a cone on a cubic curve, the existence of a rationalpoint is equivalent to the density of the rational points. This is shown by Kollar [Kol02],generalizing the work of Segre and Manin [Man74].From a geometric point of view, this surface is obtained by the contraction of two exceptionalcurves. For a surface obtained by the successive blow-down of two disjoint exceptional curves(which is the case of S ′ ), we are guaranteed to have a rational point: the one associated to thepoint [0 , , , (which is not singular). n the previous section, we show that a rational elliptic surface with no place of multiplicativereduction has one of the two following forms: E : y = x + aL Qx + bL QM (10)and E : y = x + a (4 a L − b L ) L L x + b (4 a L − b L ) L L , (11)where a, b ∈ Z , L , L , L and M linear polynomials and Q a quadratic polynomial. In the firstcase, we impose moreover that M is such that M = (cid:16) L − a Q b (cid:17) .On surface E , the places of bad reduction are those associated to L , of type I ∗ , and thoseassociated to the irreducible factors of Q , of type II .On surface E , we have three rational places of bad reduction - the one associated to a L +27 b L has type I ∗ , the one associated to L has type III and the one associated to L has II .Therefore, the results previously presented prove the density of rational points on thesesurfaces. The work of Helfgott proves in those cases the density of rational points althoughunder the parity conjecture which we are not using here.There is a fourth method to show the density, at least for surface E . Let E be an ellipticsurface and E its generic fiber (that is to say E seen as an elliptic curve over Q [ T ] ). By theShioda-Tate formula, we have rg N S ( E Q ) = 2 + rg E ( Q ( T )) + X v ( m v − . The surfaces that we consider are obtained by blowing-up P at 9 points in general position, theNéron-Severi rank is rg N S ( E ) = 10 .In the first case, Shioda-Tate formula says that rg E ( Q ( T )) = 4 . Unfortunately, although itgives an interesting majoration: rg( E ( Q ( T ))) ≤ , this is not precise enough to conclude on thedensity. There is indeed an uncertainty, except in the case where we can bound it this way : rg( E ( Q ( T ))) ≥ . It is just what happen in the second case. Indeed the Shioda-Tate formulagives rg E ( Q ( T )) = 1 .We have E ( Q ( T )) = Z · P o , for a certain point P o . Therefore there exists K a quadraticextension of Q such that P o ∈ E ( K ( T )) . Indeed, if for every σ ∈ Gal( Q / Q ) = G Q we put σP o := ε ( σ ) · P o where ε : G Q → ± , then1. either ε is trivial and P o ∈ E ( Q ( T )) ,2. or ε is non-trivial and in this case, Q Ker ε = K , the subfield of Q stabilised by ε , is aquadratic field such that P o ∈ E ( K ( T )) .One can remark, similarly as in the proof of Proposition 7.5, that E is birational to a cubicsurface. We use the following proposition to end the argument : Proposition 7.7.
Let S be a non-singular cubic surface on a number field k . Suppose S is nota cone on a cubic curve.1. If S ( k ) = ∅ , then S ( k ) is Zariski-dense.2. Let k be a quadratic extension of k . If S ( k ) = ∅ is Zariski-dense, then S ( k ) is Zariski-dense.Proof. The first statement of the proposition is shown by Segre and Manin [Man74] and by Kollár[Kol02]. They actually prove a stronger result : if k is an arbitrary field and that S ( k ) = ∅ ,then S est k -unirational. When k is infinite, this implies the Zariski-density of rational points.We now show the second point of the proposition. Let P ∈ S ( k ) . If P ∈ S ( k ) , then therational points are dense. Suppose the that P S ( k ) . Consider D the line passing through P and P σ where σ is the automorphism of k fixing k . If D ⊂ S , then D ( k ) ⊂ S ( k ) and thus theset of rational points of S is not empty. Otherwise, the intersection D ∩ S contains three points: P , P σ , and a third point which is necessarily in S ( k ) . e end the section with a result concerning smooth rational elliptic surfaces, associated toa del Pezzo surface of degree 1. Let X be a del Pezzo surface of degree 1. In general, if thereexists C and C a pair of exceptional curves defined over Q on X such that their intersection isempty, one can contract those curves to obtain X a del Pezzo surface of degree 3. On X , theexistence of a rational point garantees the Zariski-density of X ( k ) . In what follows, we use thisidea to prove the density one some other surfaces on which we find two exceptional curves withnon empty intersection. Proposition 7.8.
Let X a del Pezzo surface of degree 1 on which lie C and C two distinctexceptional curves defined over k with possibly points in common. Then X ( k ) is Zariski-dense.Proof. The contraction of C gives X ′ a del Pezzo surface of degree 2. We know that on thesesurfaces, the rational points of X ′ are dense if X ′ ( k ) contains a point which is neither on anexceptional curve nor on a distinguished quartic. Put E the union of the points of this quarticand of the exceptional curves. The contraction sends C on a rational curve of X ′ that we willdenote by C . Note that C is not an exceptional curve on X ′ because it is the blow-down of acurve which has a point in common with C . In the case where C ∩ E is finite, one can find arational point outside of E , and this proves the density of the rational points. A Local root number
Let E α be the (sextic or quartic) twist by a non-zero α ∈ Q of an elliptic curve with j = 0 or j = 1728 . In this appendix we study in more details the two functions ω and ω definedin Section 6 appearing in the decomposition of the root number (of Equation 5). The values ofthose functions depend only on α and v ( α ) or respectively on α and v ( α ) . Remember ournotation: for any prime number p , α p is the integer such that t = p v p ( α ) α p .As in section 6, we restrict our attention to: α = C (3 A m + B n ) (in case j = 0 ) and α = C ( A m + B n ) (in case j = 1728 )and thus study the surfaces E A,B,C or F A,B,C given by the equations: E A,B,C : y = x + C (3 A T + B ) and F A,B,C : y = x + C ( A T + B ) x (12)because those are the natural cases where the root number is likely to be constant according to[VA11]. In those equation A, B, C ∈ Z are such that gcd( A, B ) = 1 .Let us briefly recapitulate what was done in Section 6 before we state the result. We use aformula of Varilly-Alvarado splitting the root number into three functions, ω ( t ) , ω ( t ) , P ( t ) corresponding to the contribution of respectively the prime numbers p = 2 , p = 3 and p ≥ .While P ( t ) is constant for a given surface of one of the forms of (12), the functions ω and ω varies independently from each other, and hence each of them must be constant for the globalroot number to take always the same value over the fibers of the surface. A.1 The elliptic surface E A,B,C : y = x + C (3 A T + B ) Let E be an elliptic surface given by the Weierstrass equation E : y = x + C (3 A T + B ) , (13)where A, B, C ∈ Z and pgcd( A, B ) = 1 .Put P ( m, n ) = C (3 A m + B n ) and define the two functions, for p = 2 or , ω p ( t ) : t = mn ∈ Q p → {− , +1 } s the following ω ( t ) = W ( E t )( − v ( P ( m,n )) . Define the function ω ( t ) := W ( E t ) (cid:18) − P ( m, n ) (cid:19) . Lemma A.1.
The function ω is constant if and only if one option is satisfied1. A, B, C are in Table 1 (in which case ω ( t ) = +1 )2. A, B, C are in Table 2 (in which case ω ( t ) = − ) Lemma A.2.
The function ω is constant if and only if one option is satisfied:1. A, B, C are in Table 3 (in which case ω ( t ) = +1 )2. A, B, C are in Table 4 (in which case ω ( t ) = − ) v ( A )[3] v ( B )[3] v ( C )[6] Additional conditions0 0 0 C B ≡ C A ≡ C A ≡ , C A ≡ , , C B ≡ , , , C ≡ C A ≡ , , , , C B ≡ , C ≡ C B ≡ , C A ≡ C B ≡ C B ≡ C B ≡ , C A ≡ C ≡ C ≡ C A ≡ , C B ≡ , Table 1 – Cases where ω ( t ) = +1 for every fiber E t of the surface 13. Proof. of Lemma A.1. Let A , B , C be integers such that gcd( A, B ) = 1 . To ease the notation,let us simply write E = E A,B,C . For each fiber E t , we study instead the curve E m,n : y = x + C ( A m + B n ) which is Q -isomorphic. Let P ( m, n ) = C (3 A m + B n ) . For every ( m, n ) ∈ Z one has P ( m, n ) = 3 v ( C ) C (cid:16) v ( A )+6 v ( m )+1 A m + 3 v ( B )+6 v ( n ) B n (cid:17) , . Remark that according to the 3-valuations of A and B different situations occur. We willtreat in details the case where v ( A ) = 0 and v ( B ) = 0 . In that case, v ( P ( m, n )) is at least v ( P ( m, n )) = v ( C ) + min (cid:0) v ( A ) + 6 v ( m ) + 1 , v ( B ) + 6 v ( n ) (cid:1) . (14)We make the distinction between three properties for the coprime integers m, n : ( A )[3] v ( B )[3] v ( C )[6] Additional conditions0 0 2 C A ≡ , C B ≡ C A ≡ C A ≡ , , , , C B ≡ , C ≡ C ≡ C A ≡ , , C B ≡ , , , C B ≡ C B ≡ , C A ≡ C B ≡ , C A ≡ C B ≡ C ≡
1, 4 C ≡ C B ≡ , C A ≡ , Table 2 – Cases where ω ( t ) = − for every fiber E t of the surface 13. (a) if v ( A ) + 6 v ( m ) = 2 v ( B ) + 6 v ( n ) ,then v ( m, n ) = a and P ( m, n ) ≡ C (3 A + B )mod 9 ,(b) if v ( A ) + 6 v ( m ) < v ( B ) + 6 v ( n ) then v ( m, n ) = a and P ( m, n ) ≡ C B mod 9 (c) if v ( A ) + 6 v ( m ) > v ( B ) + 6 v ( n ) ,then v ( m, n ) = a + 1 and P ( m, n ) ≡ C A In those subcases, we obtain a different formula for the function ω , as follows.(a) Suppose that v ( A ) + 6 v ( m ) = 2 v ( B ) + 6 v ( n ) . One has v ( P ( m, n )) = 2 v ( B ) + 6 v ( n ) ≡ v ( B ) mod 6 and P ( m, n ) ≡ C (3 A + B ) mod 9 . By [VA11, Lemma 4.1], the local root number at is equal to W ( E P ( m,n ) ) = − if v ( C ) ≡ , and C ≡ or if v ( C ) ≡ , and C ≡ or if v ( C ) ≡ and P ( m, n ) ≡ , or if v ( C ) ≡ and P ( m, n ) ≡ , otherwise. (15)and thus ω ( t ) = +1 if v ( C ) + 2 v ( B ) ≡ , and C ≡ or if v ( C ) + 2 v ( B ) ≡ , and C ≡ , or si v ( C ) + 2 v ( B ) ≡ and P ( m, n ) ≡ , , , or si v ( C ) + 2 v ( B ) ≡ and P ( m, n ) ≡ , − otherwise. (16)(b) Suppose that the coprime integers ( m, n ) are such that v ( B )+6 v ( n ) < v ( A )+6 v ( m ) .One has v ( P ( m, n )) ≡ v ( C ) mod 6 and P ( m, n ) ≡ C B mod 9 ( A ) mod 3 v ( B ) mod 3 v ( C ) mod 6 Additional conditions0 0 1,3,51 0 1,3,52 C ≡ C ≡ C ≡ C ≡ C ≡ C ≡ C ≡ C ≡ Table 3 – Cases where ω ( t ) = +1 for every fiber E t of the surface 13. v ( A ) mod 3 v ( B ) mod 3 v ( C ) mod 6 Additional conditions1 0 2 C ≡ C ≡ C ≡ Table 4 – Cases where ω ( t ) = − for every fiber E t of the surface 13. Note that B will take values among the congruence classes , , , or , and so P ( m, n ) ≡ C mod 3 . Moreover, in case v ( A ) + 3 v ( m ) = v ( B ) + 3 v ( n ) , one has P ( m, n ) ≡ C if B ≡ C if B ≡ C if B ≡ . Thus we have ω ( t ) = +1 if v ( C ) + 2 v ( B ) ≡ , and C ≡ or if v ( C ) + 2 v ( B ) ≡ , and C ≡ , or if v ( C ) + 2 v ( B ) ≡ and P ( m, n ) ≡ , , , or if v ( C ) + 2 v ( B ) ≡ and P ( m, n ) ≡ , − otherwise. (17)(c) Suppose now that v ( n ) + 2 v ( B ) > v ( A ) + 6 v ( m ) + 1 (and that in particular, since | n , then ∤ m ). In this case one has v ( P ( m, n )) = v ( C ) + 2 v ( A ) + 1 and P ( m, n ) ≡ C A .As previously, we find that with this choice of ( m, n ) , the value of ω is ω ( t ) = +1 if v ( C ) + 2 v ( A ) ≡ , and C ≡ or if v ( C ) + 2 v ( A ) ≡ , and C ≡ , or if v ( C ) + 2 v ( A ) ≡ and P ( m, n ) ≡ , or if v ( C ) + 2 v ( A ) ≡ and P ( m, n ) ≡ , − otherwise. (18)We deduce that the function ω is constant in the cases listed in the lemma (and only inthose cases). To achieve this, we compare the two formulas for each value of k mod 3 . hen v ( A ) ≡ v ( B ) ≡ , then the equation 18 compared with 17 gives: ω ( t ) = +1 if v ( C ) ≡ and P ( m, n ) ≡ , or if v ( C ) ≡ , and C ≡ , or if v ( C ) ≡ and P ( m, n ) ≡ , or if v ( C ) ≡ and P ( m, n ) ≡ or if v ( C ) ≡ and P ( m, n ) ≡ − if v ( C ) ≡ and P ( m, n ) ≡ or if v ( C ) ≡ , and P ( m, n ) ≡ , or if v ( C ) ≡ and P ( m, n ) ≡ or if v ( C ) ≡ and P ( m, n ) ≡ , or if v ( C ) ≡ and C A ≡ , non-constant otherwise.When v ( A ) ≡ , , we proceed in a similar way and obtain that the cases where theroot number is constant are those listed in the Tables. This is the same method for v ( A ) = 0 and v ( B ) ≡ , . However, we have different subcases, for instance : v ( A ) = 0 and v ( B ) = 1
1. if ∤ m, n , then v ( m, n ) = a + 1 and P ( m, n ) ≡ C ( A + 3 B )mod 9 ,2. if ∤ m and | n , then v ( m, n ) = a + 1 and P ( m, n ) ≡ C A mod 9
3. if | m and ∤ n , then v ( m, n ) = a + 2 and P ( m, n ) ≡ C B Proof. of Lemma A.2There is only one of A or B at a time that may be divisible by 2. According to which ofthem is (or isn’t), the formula for w ( t ) is different.Let ( m, n ) ∈ Z × Z ≤ be a pair of coprime integers. We have P ( m, n ) = 2 v ( C ) C (2 v ( A )+6 v ( m ) · A m + 2 v ( n )+2 v ( B ) · B n ) . So, except if v ( A ) + 6 v ( m ) = 2 v ( B ) + 6 v ( n ) , we have that v ( P ( m, n )) = v ( C ) + min(2 v ( A ) + 6 v ( m ) , v ( B ) + 6 v ( n )) . a) If v ( B ) + 6 v ( n ) < v ( A ) + 6 v ( m ) , then v ( P ( m, n )) = v ( C ) + 6 v ( n ) ≡ v ( C ) mod 6 and moreover P ( m, n ) ≡ B C ≡ C mod 4 . By [VA11, Lemma 4.1], we have W ( E t ) = − v ( C ) ≡ , v ( C ) ≡ , , , and C ≡ otherwise.and thus ω ( t ) = +1 if v ( C ) + 2 v ( B ) ≡ , , , or if v ( C ) + 2 v ( B ) ≡ , and C ≡ − if v ( C ) + 2 v ( B ) ≡ , and C ≡ (19)b) If v ( n ) > v ( A ) , then in particular, ∤ m . Hence, v ( P ( m, n )) ≡ v ( C ) + 2 v ( A )mod 6 and P ( m, n ) ≡ C mod 4 . In this case we have ω ( t ) = +1 if v ( C ) + 2 v ( A ) ≡ , , , or if v ( C ) + 2 v ( A ) ≡ , and C ≡ − if v ( C ) + 2 v ( A ) ≡ , and C ≡ . rom these formulas, we now deduce the behavior of the function ω ( t ) when v ( n ) = 2 v ( A ) .For instance when v ( A ) ≡ v ( B ) ≡ , then ω ( t ) = ( +1 v ( C ) ≡ , , , non-constant otherwise. (20)We proceed in a similar way for the case v ( A ) ≡ , .c) If v ( A ) ≡ v ( B ) ≡ , we need to proceed to a more raffined selection. Let ( m, n ) be a pair such that v ( n ) = 2 v ( A ) . In this case, one has v ( P ( m, n )) ≡ and P ( m, n ) ≡ ( C mod 16 if A m ≡ B n mod 163 C mod 16 otherwise . Observe moreover that replacing n by n ′ such that n ′ ≡ n + 8 mod 16 , then the value of P ( m, n ) mod 4 passes from C to C and vice-versa. Thus we get the formula ω ( t ) = ( non-constant if v ( C ) ≡ , otherwise . . Therefore, comparing with the formula (20), we get that the function ω ( t ) is constant and equalto +1 in the case where v ( C ) ≡ , , . The method is similar for the case v ( A ) = 0 and v ( B ) ≡ , . A.2 The elliptic surface F A,B,C : y = x + C ( A T + B ) x Let F be an elliptic surface given by the Weierstrass equation F A,B,C : y = x + C ( A T + B ) x, (21)where A, B, C ∈ Z and gcd( A, B ) = 1 . Lemma A.3.
The local root number at 3 is W ( E t ) = +1 if v ( C ) ≡ , if v ( C ) ≡ and v ( A ) , v ( B ) even − if v ( C ) ≡ and v ( A ) , v ( B ) evennon-constant otherwise.Proof. Let A , B , C integers such that gcd( A, B ) = 1 . Let us write simply F = F A,B,C . Forany t ∈ Q consider the pair of coprime integers ( m, n ) ∈ Z × Z < such that t = mn . For eachfiber F t , let F m,n be the curve given by the equation F m,n : y = x + C ( A m + B n ) x whichis Q -isomorphic to F t and thus have the same local root number (at any prime p ). Put P ( m, n ) = C ( A m + B n ) . The local root number at 3 of F m,n only depends of v ( P ( m, n )) . For every m, n ∈ Z coprime,we have P ( m, n ) = 3 v ( C ) C (cid:16) v ( A )+4 v ( m ) A m + 3 v ( B )+4 v ( n ) B n (cid:17) , and thus v ( P ( m, n )) = v ( C ) + min(2 v ( A ) + 4 v ( m ) , v ( B ) + 4 v ( n )) . In case where v ( A ) + 2 v ( m ) < v ( B ) + 2 v ( n ) , we get the formula: W ( F t ) = − if v ( C ) ≡ and v ( C ) + 2 v ( A ) ≡ ≡ or if v ( C ) ≡ and v ( A ) is odd, +1 otherwise. (22) he case v ( A ) + 2 v ( m ) ≥ v ( B ) + 2 v ( n ) is similar, but with the condition v ( C ) + 2 v ( B ) ≡ ≡ . Comparing those formulas, we obtain the conclusion of the Lemma.Define the function ω ( t ) := W ( F t ) (cid:16) − P ( m,n ) (cid:17) . Lemma A.4.
The value of the function ω ( t ) is constant when t ∈ Q varies if and only if oneoption is satisfied:1. A, B, C are in Table 5 (in which case ω ( t ) = +1 )2. A, B, C are in Table 6 (in which case ω ( t ) = − ). v ( A )[2] v ( B )[2] v ( C )[4] Additional conditions1 0 0 C A ≡ , ,
15 mod 16 , C B ≡ , ,
15 mod 16 C A ≡ , ,
15 mod 16 , C B ≡ , ,
15 mod 16 C B ≡ , ,
15 mod 16 , C A ≡ , ,
15 mod 16 C B ≡ , ,
15 mod 16 , C A ≡ , ,
15 mod 16
Table 5 – Cases where ω ( t ) = +1 for every fiber at t ∈ Q of the surface F A,B,C . v ( A )[2] v ( B )[2] v ( C )[4] Additional conditions0 0 0 C A , C B ≡ , , ,
11 mod 16 C ≡ C A , C B ≡ , , ,
15 mod 16 C ≡ C A ≡ , , ,
15 mod 16 , C B ≡ , , ,
11 mod 16 C A ≡ , , ,
15 mod 16 , C B ≡ , , ,
11 mod 16 C B ≡ , , ,
15 mod 16 , C A ≡ , , ,
11 mod 16 C B ≡ , , ,
11 mod 16 , C A ≡ , , ,
15 mod 16
Table 6 – Cases where ω ( t ) = − for every fiber F t of the surface F A,B,C . Proof.
For every choice of m, n ∈ Z coprime, let E m,n : y = x + C ( A m + B n ) x be anelliptic curve Q -isomorphic to E mn . We know the formula of the local root number at 2 by[VA11, Lemme 4.7] (that we recall at Lemma 3.2). Moreover, recall that if t is an odd integer,one has (cid:18) − t (cid:19) = ( +1 if t ≡ , , − otherwise.Put, for every m, n ∈ Z coprime integers, P ( m, n ) = C ( A m + B n ) . We have P ( m, n ) = 2 v ( C ) C (cid:16) v ( A )+4 v ( m ) A m + 2 v ( B )+4 v ( n ) B n (cid:17) and thus, when v ( A ) + 2 v ( m ) = v ( B ) + 2 v ( n ) , one has v ( P ( m, n )) = v ( C ) + 2 min( v ( A ) + 2 v ( m ) , v ( B ) + 2 v ( n )) , nd P ( m, n ) ≡ C (4 A + B ) mod 4 if v ( A ) oddC ( A + 4 B ) mod 4 if v ( B ) oddC B if v ( A ) ≥ and even C A if v ( B ) ≥ and evenWhen both v ( A ) and v ( B ) are even, it is possible that v ( A ) + 2 v ( m ) = v ( B ) + 2 v ( n ) ,and in this case one has v ( P ( m, n )) = v ( C ) + 2 v ( A ) + 4 v ( m ) + 2 , and P ( m, n ) ≡ C (cid:18) A + B (cid:19) mod 16 . Suppose first v ( A ) + 2 v ( m ) < v ( B ) + 2 v ( n ) . In this case, we have v ( P ( m, n )) = v ( C ) + 2 v ( A ) + 4 v ( n ) ≡ v ( C ) + 2 v ( A ) mod 4 and P ( m, n ) ≡ B C mod 16 if v ( A ) even C if v ( A ) odd and B ≡ . C if v ( A ) odd and B ≡ .. We have, W ( E m,n ) = − if v ( C ) + 2 v ( A ) ≡ , and P ( m, n ) ≡ , , or if v ( C ) + 2 v ( A ) ≡ and P ( m, n ) ≡ , , , , ,
15 mod 16 , or if v ( C ) + 2 v ( A ) ≡ and P ( m, n ) ≡ , , , , ,
15 mod 16 , +1 otherwise. (23)and thus ω ( E t ) = − if v ( C ) + 2 v ( A ) ≡ , or if v ( C ) + 2 v ( A ) ≡ and P ( m, n ) ≡ , , ,
15 mod 16 or if v ( C ) + 2 v ( A ) ≡ and P ( m, n ) ≡ , , ,
11 mod 8+1 otherwise.The equation for v ( A ) + 2 v ( m ) > v ( B ) + 2 v ( n ) is identical, with the role of A and B swapped since the equation of the surface is symmetric.Moreover, note that as we supposed that A and B are coprime, at most one of them isdivisible by .Comparing formulas for v ( A ) + 2 v ( m ) > v ( B ) + 2 v ( n ) and v ( A ) + 2 v ( m ) < v ( B ) +2 v ( n ) , we get some of the entries in Tables 5 and 6. Observe moreover that, given that A , B ≡ , , some cases describe by the conditions of one line of each formula are not possible.The only work left is to study more into details the case where both v ( A ) and v ( B ) are botheven.For these exceptions, we proceed to a more raffined sorting.According to the values of A m , B n (among , , ,
25 mod 32 ), we find the possiblevalues of P ( m, n ) . We always have in this case v ( P ( m, n )) = v ( C ) + 1 mod 4 . Hence Thus e have ω ( t ) = − (cid:16) − C (cid:17) if v ( C ) ≡ , and P ( m, n ) ≡ , or if v ( C ) ≡ and P ( m, n ) ≡ , , , , ,
15 mod 16 or if v ( C ) ≡ and P ( m, n ) ≡ , , , , ,
15 mod 16 (cid:16) − C (cid:17) otherwise.Observe that m and n can take the values ,
17 mod 32 . Therefore, choosing a value of n ′ such that n ′ ≡ n mod 32 , we have P ( m, n ′ ) ≡ P ( m, n ) mod 16 . Therefore, we have P ( m, n ) ∈ { , } if A ≡ B mod 16 , and P ( m, n ) ∈ { , } if A B mod 16 , .This means that ω ( t ) is non-constant when v ( C ) ≡ and C ≡ ,
15 mod 16 ,and when v ( C ) ≡ and C ≡ , .We obtain thus that in that case ω ( t ) = − if v ( C ) ≡ , or if v ( C ) ≡ and C ≡ or if v ( C ) ≡ and C ≡ if v ( C ) ≡ and C ≡ or if v ( C ) ≡ and C ≡ non-constant otherwise.Comparing with the formula when v ( A ) + 2 v ( m ) = v ( B ) + 2 v ( n ) , we complete the Tables 5and 6. In particular, there is no cases were w ( t ) = +1 for all t ∈ Q when v ( A ) ≡ v ( B ) ≡ . References [BCDT01] Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor. On the modu-larity of elliptic curves over Q : wild 3-adic exercises. J. Amer. Math. Soc. , 14(4):843–939 (electronic), 2001.[BDD16] S. Bettin, C. David, and C. Delaunay. Families of elliptic curves with non-zeroaverage root number. arXiv:1612.03095, 12 2016.[Cas91] J. W. S. Cassels.
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