ELLIPTIC CURVES WITH RATIONAL 2-TORSION AND RELATED TERNARY DIOPHANTINE EQUATIONS

Transcription

1 ELLIPTIC CURVES WITH RATIONAL -TORSION AND RELATED TERNARY DIOPHANTINE EQUATIONS by JAMIE THOMAS MULHOLLAND B.Sc. Simon Fraser University, 000 M.Sc. The University of British Columbia, 00 A THESIS SUBMITTED IN PARTIAL FULLFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Mathematics THE UNIVERSITY OF BRITISH COLUMBIA July 006 c Jamie Thomas Mulholland, 006

2 Abstract Our main result is a classification of ellitic curves with rational -torsion and good reduction outside, and a rime. This extends the work of Hadano and, more recently, Ivorra. A key factor in doing this is to have a method for efficiently comuting the conductor of an ellitic curve with -torsion. We secialize the work of Paadoolous to rovide such a method. Next, we determine all the rational oints on the hyer-ellitic curves y = x 5 ± a b. This information is required in roviding the classification mentioned above. We show how the commercial mathematical software ackage MAGMA can be used in solving this roblem. As an alication, we turn our attention to the ternary Diohantine equations x n + y n = a z and x + y = ± m z n, where denotes a fixed rime. In the first equation, we show that for = 5 or > 7 the equation is unsolvable in integers (x, y, z) for all suitably large rimes n. In the second equation, we show the same conclusion holds for an infinite collection of rimes. To do this, we use the connections between Galois reresentations, modular forms, and ellitic curves which were discovered by Frey, Hellegouarch, Serre, and Wiles. ii

3 Table of Contents Abstract Table of Contents List of Tables Acknowledgement Dedication ii iii vi vii viii Chater 1. Introduction Introduction to Diohantine Equations Generalized Fermat Equations Statement of Princial Results Overview of chaters Chater. The Conductor of an Ellitic Curve over Q with -torsion 11.1 Introduction Statement of Results The Proof of Theorem The case when v (a) = 1, v (b) = Proof of Theorem.1 art (vii) when v ( ) = Proof of Theorem.1 art (vii) when v ( ) = Proof of Theorem.1 art (vii) when v ( ) = Proof of Theorem.1 art (vii) when v ( ) = Proof of Theorem.1 art (vii) when v ( ) The Proof of Theorem The Proof of Theorem Chater. Classification of Ellitic Curves over Q with -torsion and conductor α β δ 0.1 Curves of Conductor α Statement of Results The Proof for Conductor α List of Q-isomorhism classes iii

4 Table of Contents iv.1.4 The end of the roof Curves of Conductor α β Curves of Conductor α β Proofs of α β and α β Chater 4. Diohantine Lemmata Useful Results Diohantine lemmata Chater 5. Rational oints on y = x 5 ± α β Introduction and Statement of Results Basic Theory of Jacobians of Curves Basic Setu Divisors Princial Divisors and Jacobian Geometric reresentation of the Jacobian torsion in the Jacobian Rational Points Structure of the Jacobian: The Mordell-Weil theorem Comuter Reresentations of Jacobians Some Examles (Using MAGMA) Chabauty s theorem Data for the curves y = x 5 ± α β The family of curves y = x 5 + A Proof of Theorem A = A = A = Rank cases Chater 6. Classification of Ellitic Curves over Q with -torsion and conductor α Statement of Results The Proof Chater 7. On the Classification of Ellitic Curves over Q with -torsion and conductor α Statement of Results iv

5 Table of Contents v 7. The Proofs Proof of Theorem Proof of Theorem Proof of Theorem Proof of Corollary Proof of Lemma Chater 8. On the equation x n + y n = α z Introduction Ellitic Curves Outline of the Proof of the main theorems Galois Reresentations and Modular Forms Useful Proositions Ellitic curves with rational -torsion Theorems 8.1 and Concluding Remarks Chater 9. On the equation x + y = ± m z n Introduction Frey Curve The Modular Galois Reresentation ρ a,b n Proof of Theorem Bibliograhy 6 Aendix A. On the Q-Isomorhism Classes of Ellitic Curves with - Torsion and Conductor α β δ 70 A.1 b > A. b < Aendix B. Tables of S-integral Points on Ellitic Curves. 11 B.1 S-integral oints on Ellitic Curves B. Comuting S-integral oints on Ellitic Curves B. Tables of S-integral oints on the curves y = x ± a b Aendix C. Tables of Q-Isomorhism Classes of Curves of Conductor α with Small. 17 v

6 List of Tables.1 Néron tye at of y = x + ax + bx Néron tye at of y = x + ax + bx (con t) Néron tye at of y = x + ax + bx Néron tye at of y = x + ax + bx Theorem 5.1: All oints on C : y = x 5 ± α β Data for y = x 5 + α β Data for y = x 5 + α β (con t) Data for y = x 5 + α β (con t) Data for y = x 5 + α β (con t) Data for y = x 5 α β Data for y = x 5 α β (con t) Data for y = x 5 α β (con t) Data for y = x 5 α β (con t) B.1 S-integral oints on y = x + a b B. S-integral oints on y = x a b C.1 Extraneous curves of conductor C. Extraneous curves of conductor C. Extraneous curves of conductor C.4 Extraneous curves of conductor C.5 Extraneous curves of conductor C.6 Extraneous curves of conductor C.7 Extraneous curves of conductor C.8 Extraneous curves of conductor vi

7 Acknowledgement It gives me great leasure to thank the many eole and organizations who have heled me to get where I am today. I am very grateful for financial suort from the University of British Columbia and from NSERC. To the many teachers who have guided me to where I am today thanks for your knowledge, wisdom, and insiration. I owe an enormous debt to my suervisor, Dr. Michael Bennett, for having guided me through my PhD, sharing his knowledge, wisdom and exerience of mathematics with me along the way. My fiancée, Heather, who has offered love and suort through thick and thin my deeest thanks. And my family, whose love and suort, encouragement and guidance, have always been comlete, and whose belief in me has enabled me to get to this oint, the warmest thanks of all. vii

8 To Heather my love, my life viii

9 Chater 1 Introduction 1.1 Introduction to Diohantine Equations The study of Diohantine equations has a long and rich history, dating to the Arithmetica of Diohantus, written in the middle of the rd century, and dealing with the solution of algebraic equations and the theory of numbers. Much of modern number theory, as we know it, stems from tools develoed to solve Diohantine equations. By a Diohantine equation, we mean, intuitively, an equation where we are interested only in integer and/or rational solutions. For examle the equation x + y = z has the following solutions in ositive integers (x, y, z): (, 4, 5), (5, 1, 1), (8, 15, 17), (7, 4, 5). In fact, there are infinitely many solutions in ositive integers to this equation, and they can be arametrized: any solution (with y even, say) is of the form (d(u v ), uvd, d(u + v )) where u, v, d Z and gcd(u, v) = 1. On the other hand, the equations x + y = z, x 4 + y 4 = z 4, and x 5 + y 5 = z 5 have only the trivial solutions; solutions where one of the values is 0. Fermat

10 Chater 1. Introduction wrote in the margin of his coy of Arithmetica that, in fact, the equation x n + y n = z n has no nontrivial solutions for any n, and commented that he had a marvelous roof of this fact but the margin was too small to contain it. This became known as Fermat s Last Theorem. The quest to rove (or disrove, for that matter) Fermat s Last Theorem became the driving force for modern number theory over the last three hundred years. Amateurs and rofessionals alike all had their crack at a roof. Their attemts gave birth to many new beautiful ideas and tools that are used in number theory today, though, for more than three centuries, none were enough to resolve Fermat s enigma. After the work of Godel on undecidability in formal systems, many wondered whether the truth of Fermat s Last Theorem was even decidable. Ten years ago, Andrew Wiles announced a roof verifying Fermat s Last Theorem and finally utting to rest Fermat s challenge. Wiles attacked the roblem by treating a more general question regarding the connection between ellitic curves and modular forms. We ll say more on this in our final two chaters. Consider the Diohantine equation The only integer solutions are y = x + 1. ( 1, 0), (0, ±1), (, ±). These are, in fact, the only rational solutions. On the other hand, the Diohantine equation y = x + 17 has 16 integer solutions (, ±), ( 1, ±4), (, ±5), (4, ±9), (8, ±), (4, ±8), (5, ±75), (54, ±78661), Last because it was the remaining conjecture of his that needed resolving.

11 Chater 1. Introduction and infinitely many rational solutions. Both of these curves are examles of ellitic curves. A curve of the form E : y + a 1 xy + a y = x + a y = x + a x + a 4 x + a 6 with a i Z is called an ellitic curve (rovided it is nonsingular). For curves in Weierstrass form y = x + a x + a 4 x + a 6 the condition of being nonsingular is equivalent to the cubic on the right-hand side having distinct roots (i.e. nonzero discriminant). For ellitic curves it is known that the number of integral oints is finite (Siegel s theorem, see [69]), but the number of rational oints could ossibly be infinite. Though the roof of Siegel s theorem was not effective (i.e. did not give a method to find all the integral oints) de Weger [0], using Baker s work on bounding linear forms in logarithms, was able to give an algorithm for finding all the integral oints on an ellitic curve. The set of rational oints E(Q) on an ellitic curve carry an abelian grou structure, the identity being the oint at infinity which we denote by O (or sometimes ). That is, there is a natural way to add two rational oints P 1, P E(Q) to obtain a third rational oint P = P 1 + P. Geometrically, this is done by taking the (rational) line through P 1 and P and letting P 4 be the third oint of intersection of the line with E. Next, take the vertical line through P 4 (i.e. the line through P 4 and O) and let P be the other oint of intersection with E, and set P 1 + P = P. Mordell showed that E(Q) is finitely generated and abelian so it is of the form E(Q) E(Q) tors Z r where E(Q) tors is a finite grou consisting of the torsion elements and r is an integer called the rank of E. E(Q) tors is straightforward to comute; a theorem of Nagell and Lutz gives a method for comuting its oints. Moreover, a general result of Mazur ([50], [51]) states that it can only be one of 15 ossible grous (see for examle [69],. ). However, there is no known algorithm for comuting the rank of an ellitic curve. There are methods (i.e. a -descent) that work on bounding the rank. One can then hoe to find enough

12 Chater 1. Introduction 4 indeendent oints to meet this bound to obtain the rank exactly. In ractice this works quite well. For our two examles above we have E 1 : y = x + 1, E 1 (Q) Z/6Z E : y = x + 17, E (Q) Z. A hyerellitic curve is a curve of the form y = f(x) where, for our urose, f Z[x] of degree g + 1. The integer g is called the genus of the curve. For examle, an ellitic curve is a genus 1 hyerellitic curve. However, unlike the situation for ellitic curves, a celebrated theorem of Faltings states that C(Q) is finite when g. Unfortunately, Faltings theorem is not effective, but older work of Chabauty has recently been revived and in ractice often works very well in determining C(Q). For a hyerellitic curve C the set of rational oints do not form a grou, but C(Q) does embed into a finitely generated abelian grou called the Jacobian of C, denoted J(Q). The work of Chabauty requires calculation in the Jacobian. In Chater 5 we occuy ourselves with determining the rational oints on curves of the form y = x 5 ± a b. Chater 5 can be read indeendently of all other chaters. It rovides an introduction to the theory and ractice of comuting all rational oints on genus curves, with a heavy emhasis on using MAGMA as a comutational tool, something the current literature is somewhat lacking. The results of this chater are used in roofs of the Diohantine lemmata of Chater Generalized Fermat Equations In relation with Fermat s last theorem the equation x + y q = z r (1.1) has a long history. For a very fine survey on this toic see [45]. Here, we will rovide a very brief outline of what is known.

13 Chater 1. Introduction 5 The characteristic of equation (1.1) is defined to be χ(, q, r) = q + 1 r 1, and the study of these equations has been broken u into three cases: χ(, q, r) > 0 (sherical case), χ(, q, r) = 0 (euclidean case), and χ(, q, r) < 0 (hyerbolic case). Let S(, q, r) be the set of nontrivial roer solutions to equation (1.1). In the sherical case, S(, q, r) is infinite and there are in fact arametized solutions. In this case the ossible sets of {, q, r} are {,, r} with r, {,, }, {,, 4}, and {,, 5}, and the roer solutions corresond to rational oints on genus 0 curves. In the euclidean case, ossible sets of {, q, r} are {,, }, {, 4, 4}, and {,, 6}, and the oints in S(, q, r) corresonds to rational oints on genus 1 curves. It is known that the only roer nontrivial solution corresonds to the equality 1+ =. We have already mentioned that S(,, ) was emty and the fact that S(, 4, 4) is emty was first roven by Fermat using an argument of infinite descent. In the hyerbolic case there are only ten known solutions to date: =, = 4, = 9, = 71, = 1, = 10698, = 65 7, = 11 7, = , = Notice that an exonent of aears in each solution. This leads to the following conjecture. Conjecture 1.1 If min{, q, r} and S(, q, r) then min{, q, r} =. A number of names can be associated with this conjecture, including Beukers, Zagier (who incidently found the five larger solutions above in 199), Tijdeman, Granville and Beal. The first known result in the hyerbolic case is due to Darmon and Granville [7]. They used Faltings theorem to show that S(, q, r) is finite. Next was Wiles roof of Fermat s last theorem; S(n, n, n) =. Since then a number of secific cases have been tackled using the modularity of ellitic curves (Wiles, et al), and Chabauty techniques. Some cases are as follows.

14 Chater 1. Introduction 6 (, q, r) (n, n, ) Darmon, Merel (Poonen for n {5, 6, 9}) (n, n, ) Darmon, Merel (Lucas n = 4, Poonen for n = 5) (,, n) Kraus for 17 n 10000, Bruin for n = 4, 5 (, 4, n) Ellenberg for n 11, Bruin for n = 5, 6, Bennett, Ellenberg, Ng for n 7 (, n, 4) Bennett, Skinner (,, 7) Poonen, Schaefer, Stoll (,, 8) Bruin (,, 9) Bruin (, n, ) Chen for 7 n 1000, n 1 (5, 5, n), (7, 7, n) Darmon and Kraus (artial results) (n, n, 5) Bennett (4, n, ) Bennett, Chen 1. Statement of Princial Results Modularity techniques have since been alied to generalized Fermat equations with coefficients: Ax + By q = Cz r. Here, A, B, C,, q, and r are fixed integers and we are interested in integral solutions for x, y and z. If = q = r, then results have been obtained by Serre [64] for A = B = 1 and C = N α, α 1, with N {, 5, 7, 11, 1, 17, 19,, 9, 5, 59}, N, 11, Kraus [41] for ABC = 15, Darmon and Merel [8] for ABC =, and Ribet [61] for ABC = α, α. If (, q, r) = (,, ) then results have been obtained by Bennett and Skinner [5] for various A, B, C, Ivorra [6] for ABC = β, and Ivorra and Kraus [8] for various A, B, and C. If (, q, r) = (,, ) then Bennett, Vatsal and Yazdani [6] have shown Theorem 1. (Bennett, Vatsal, Yazdani) If and n are rime, and α is a nonnegative integer, then the Diohantine equation x n + y n = α z has no solutions in corime integers x, y and z with xy > 1 and n > 4.

15 Chater 1. Introduction 7 Their roof of this roceeds as follows. Attach to a suosed solution (a, b, c) an ellitic curve E = E a,b,c with a -torsion oint, and to this a Galois reresentation ρ E,n on the n-torsion oints. To ρ E,n there corresonds a cusidal newform f E,n of weight and level N n (E), where N n (E) can be exlicitly determined. It then remains to show that such a newform f cannot exist. In doing this, it is shown that the existence of f imlies either n is bounded by 4 or that there exists an ellitic curve over Q with rational -torsion and conductor τ ω. Hence a classification of such curves is needed to finish the argument. In Chater 8, we aly a similar argument to the equation and rove the following x n + y n = α z Theorem 1. (Bennett, Mulholland) Let 7 be rime. Then the equation x n + y n = α z has no solutions in corime nonzero integers x and y, ositive integers z and α, and rime n satisfying n > 7. A key ingredient in the roof is a classification of the ellitic curves with conductor M and ossessing a rational -torsion oint. In Chater 6, we rovide such a classification. In Chater 9, we study the equation x + y = ± m z n, where is rime and rove the following, Theorem 1.4 (Mulholland) Let T and m 1 an integer. Then the equation x + y = ± m z n has no solutions in corime nonzero integers x, y and z, and rime n satisfying n 8 and n m.

16 Chater 1. Introduction 8 Here T denotes the set of rimes for which there does not exist an ellitic curve with rational -torsion and conductor M, 1 M. Thus, in this case we need a classification of the ellitic curves with conductor M, 1 M, and ossessing a rational -torsion oint. In Chater 7, we rovide such a classification. Since we are interested in ellitic curves of conductor M or M L and ossessing a rational oint of order we start by considering the following more general question. Problem 1 Determine all the Q-isomorhism classes for ellitic curves over Q of conductor M L N and having at least one rational oint of order. As is well-known, there do not exist any ellitic curves defined over Q with conductor divisible by 9, 6, or q for q 5 rime (see e.g. Paadooulos [57]). Furthermore, as we show in Chater, the existence of rational - torsion imlies the conductor is not divisible by. Therefore, we can suose in the statement of roblem 1 that 0 M 8 and 0 L, N. In addition, a theorem of Shafarevich states that there are only finitely many isomorhism classes, for fixed (see [69]. 6). The first work on Problem 1 aears to have be done by Ogg in 1966, [55], [56]. He determined the ellitic curves defined over Q with conductor of the form M L or M. Coghlan in his dissertation [17] also studied the curves of conductor M L indeendently of Ogg. Vélu [78] classified curves of conductor 11, and in general Setzer [66] answers Problem 1 for any rime conductor. He shows that there are two distinct isomorhism classes when 64 is a square, and four when = 17. Hadano [4] begins treatment of conductors N and M N, and Ivorra, in his dissertion [7], classifies those of conductor M. There has been other work in classifying ellitic curves with conductors of a articular form and secified torsion structure. Most notable are the works of Hadano [5] and Miyawaki [5]. In Chater, we take u Problem 1 in general. In Section.1, we obtain results analogous to those of Ivorra for conductor N. In Sections. and.

17 Chater 1. Introduction 9 we obtain results for conductor N L and N L, resectively, thus comleting the remaining cases of Problem 1. As seen from glancing at the table of contents, the tables resented account for 10+ ages of this work (not to mention the 0+ ages of refined tables in Chater 6, and the 40+ ages of technical case by case analysis in Aendix A). We have tried to tidy this work u as best we can and make it readable but, unfortunately, there is no way to fully condense it; the tables are what they are long and technical. But we believe the determination of these tables rovides a useful ublic service. As seen from glancing at the tables in Chater, one is mainly confronted, as in [66] and [7], with the roblem of determining the integer solutions of certain ternary Diohantine equations. In Chater 4, we take u the roblem of resolving these Diohantine equations. We then come back the tables of Chater with these solutions at hand. This allows us to simlify the tables, these results aear in Chaters 6 and 7. Some of the works mentioned above regarding Problem 1 treat the following more general roblem, which we do not know how to attack in general. Problem Determine all the Q-isomorhism classes for ellitic curves over Q of conductor M L N. Let us note that Brumer and McGuinness have determined the ellitic curves of conductor < The definitive web source for tables of all the ellitic curves of conductor < is John Cremona s home age. These tables are constantly being exanded so the reader should check the web age to determine their extent at this time. The techniques Cremona uses for constructing his tables (and, indeed, a fine introduction to the arithmetic of ellitic curves) can be found in his excellent book [6] which is available for download from his web age. In addition, Cremona has reared tables for conductor k m with m rime and also m = 15 and Overview of chaters A brief outline of the contents of each chater is as follows. In Chater, we secialize the results of Paadoolous [57] to the roblem of comuting the conductor of an ellitic curve with a rational torsion oint,

18 Chater 1. Introduction 10 i.e. curves of the form y = x + ax + bx. There we resent an easy criterion for comuting the conductor. The results of this section are used throughout the rest of this work. Chater is the first ste toward our classifying roblem. Here we resent twenty-seven theorems, one for each value of M L N, listing the Q-isomorhism classes of the ellitic curves with that conductor. The roof is long and tedious but not that technical, it deends on two main lemmata which are roven in Aendix A. It is in these tables that we are confronted with the roblem of determining the integer solutions to certain ternary Diohantine equations. In order to get a useful classification theorem we need to resolve these Diohantine equations. This is taken u in Chater 4. In order to solve some of the Diohantine equations, it is sufficient to find all {,, }-integral oints on the genus 1 curves y = x ± α β, and the genus curves y = x 5 ± α β. We deal with the former in Aendix B and the latter in Chater 5. Having these Diohantine results at hand, we come back to the tables of Chater. In Chater 6, we resent nine theorems classifying ellitic curves of conductor M ossessing a rational -torsion oint. These table are analogous to those of Ivorra [7]. In Chater 7, we investigate the admissible for which there exist curves of conductor M, 1 M, with rational -torsion. These results will be used in Chater 9. In Chaters 8 and 9, we look at what can be said about the generalized Fermat equations x n + y n = α z and x + y = ± m z n, resectively. A modified version of Chater 8 has aeared in rint [4].

19 Chater The Conductor of an Ellitic Curve over Q with -torsion In this chater, we secialize the work of Paadoolous [57] to ellitic curves over Q with nontrivial -torsion: y = x + ax + bx, and show that the exonent of in the conductor of the curve is determined by the values v (a) and v (b) and some simle congruences of a and b modulo, 4 and 8. Here v denotes the -adic valuation on Q..1 Introduction Let E be the ellitic curve over Q defined by E : y + a 1 xy + a y = x + a x + a 4 x + a 6, with a i Z. Let b, b 4, b 6, b 8, c 4, c 6, and be the standard invariants associated with E: b = a 1 + 4a, b 4 = a 1 a + a, b 6 = a + 4a 6 (.1) b 8 = a 1a 6 a 1 a a 4 + 4a a 6 + a a a 4 (.) c 4 = b 4b 4, c 6 = b + 6b b 4 16b 6 (.) = b b 8 8b 4 7b 6 + 9b b 4 b 6. (.4) The conductor of an ellitic curve over Q is defined to be N = f 11

20 Chater. The Conductor of an Ellitic Curve over Q with -torsion 1 where f = v ( )+1 n. Here n is the number of irreducible comonents of the secial fibre of the minimal Néron model at the rime (see [69]). Essentially, N is an encoding of the rimes for which E has bad reduction and the reduction tyes at these rimes. E has bad reduction at a rime if and only if N, and the reduction tye of E at is multilicative (E has a node over F ) or additive (E has a cus over F ) deending on whether f = 1 or, resectively. It is well known that for,, the value of f is comletely determined by the values of v (c 4 ), v (c 6 ) and v ( ). This is not always the case when = or. Paadoolous [57] has determined when the trile (v (c 4 ), v (c 6 ), v ( )) (res. (v (c 4 ), v (c 6 ), v ( ))) is not sufficient to determine the value of f (res. f ) and in these cases he has given sulementary conditions involving the values of a 1, a, a, a 4, a 6, b, b 4, b 6 and b 8. In the case of the rime these sulementary conditions involve checking a single congruence involving c 4 and c 6 modulo 9. However, for the rime the sulementary conditions are a little more comlicated. One usually needs to check a number of congruences in sequence for solutions. Furthermore, in the case when (v (c 4 ), v (c 6 ), v ( )) = (6, 9, 1) one is unable to decide from Table IV in [57] whether f is 5 or 6 (whereby one is forced to aly Tate s algorithm directly). If E is an ellitic curve over Q with nontrivial -torsion then E is isomorhic to a curve of the form y = x + ax + bx, where a, b Z are such that v (a) and v (b) 4 do not both hold for all. The discriminant in this case is = 4 b (a 4b). In this chater, we show that for such curves the conditions one needs to check in [57] simlify greatly. In fact, the value of f is comletely determined by the values of v (a), v (b) and the congruence classes of a and b modulo 4, with one excetion. In this excetional case, v ( ) = 8, one needs to check a congruence involving a and b modulo 8 (see Theorem.1).. Statement of Results. Let denote a rime 5. We will rove the following theorems.

21 Chater. The Conductor of an Ellitic Curve over Q with -torsion 1 Theorem.1 If a, b Z are such that not both v (a) and v (b) 4 hold, then the Néron tye at of the ellitic curve y = x + ax + bx is given by Tables.1 and. on ages 15 and 16. In the cases where f = 0 or 1 the model y = x + ax + bx is non-minimal at, this is indicated in Table.1 by the aearance of non-minimal in the corresonding column. In the cases where f 0, 1 the model y = x + ax + bx is minimal at. During the course of the roof of Theorem.1 we will also deduce the following. Corollary. In the case that the model E(a, b) : y = x + ax + bx is nonminimal at we have the following: 1. If v (a) = 0, v (b) 4 and a 1 (mod 4) then ( a 1 y + xy = x + 4 is a minimal model for E(a, b) at. ) x + ( ) b x 16. If v (a) = 1, v (b) = 0, v ( ) 1 and a 1 (mod 4) then ( ) ( (a + ) (a y + xy = x + x ) ( 4b) a(a ) 4b) + x is a minimal model for E(a, b) at. Theorem. If a, b Z are such that not both v (a) and v (b) 4 hold, then the Néron tye at of the ellitic curve y = x + ax + bx is given by Table. on age 16. In all cases the model y = x + ax + bx is minimal at. Theorem.4 Let be a rime 5. If a, b Z are such that not both v (a) and v (b) 4 hold, then the Néron tye at of the ellitic curve y = x + ax + bx is given by Table.4 on age 16. In all cases the model y = x + ax + bx is minimal at. We have the following corollary to Theorems. and.4. Corollary.5 Let q be an odd rime. If a, b Z such that not both v q (a) and v q (b) 4 hold and N (a,b) is the conductor of the ellitic curve y = x + ax + bx then:

22 Chater. The Conductor of an Ellitic Curve over Q with -torsion 14 (i) q N (a,b) if and only if q = 4 b (a 4b), (ii) if q N (a,b) then q does not divide a, (iii) q N (a,b) if and only if q divides a and b.

23 Chater. The Conductor of an Ellitic Curve over Q with -torsion 15 v(a) v(b) 0 1 v( ) Sulementary a 1 (4) a 1 (4) a 1 (4) a 1 (4) a 1 (4) a 1 (4) a 1 (4) a 1 (4) conditions b 1 (4) b 1 (4) b 1 (4) b 1 (4) Kodaira symbol II IV III II III I 1 I 0 III I Case of Tate Exonent v(n) of conductor v(a) v(b) v( ) 7 8 Sulementary a 1 (4) a 1 (4) a 1 (4) a 1 (4) a b 1 (16) a b 5 (16) conditions non-minimal non-minimal Kodaira symbol I0 I 4 I v ( ) 1 I v( ) 8 I 0 I 1 II Case of Tate Exonent v(n) of conductor v(a) v(b) v( ) 8 (con t) a Sulementary a b 1 (16) a b 9 (16) 1 (4) a 1 (4) a 1 (4) a conditions Kodaira symbol I 0 IV I 0 I III I II Case of Tate Exonent v(n) of conductor (4) v(a) v(b) v( ) 1 1 a Sulementary 1 (4) a 1 (4) a 1 (4) a 1 (4) b 1 (4) b 1 (4) conditions non-minimal non-minimal Kodaira symbol I0 I v ( ) 1 I 4 I v( ) 8 III I 0 I v( ) 10 II III III Case of Tate Exonent v(n) of conductor Table.1: Néron tye at of y = x + ax + bx.

24 Chater. The Conductor of an Ellitic Curve over Q with -torsion 16 v(a) v(b) v( ) b 4 1 (4) b Sulementary 1 (4) 4 conditions Kodaira symbol I I v( ) 10 III I I III 4 1 (4) b 4 1 (4) b Case of Tate Exonent v(n) of conductor Table.: Néron tye at of y = x + ax + bx (con t). v(a) v(b) Sulementary b 1 () b 1 () conditions Kodaira symbol I v ( ) I0 I v ( ) I0 III I v( ) 6 I 0 III Case of Tate Exonent v(n) of conductor Table.: Néron tye at of y = x + ax + bx. v(a) v(b) Sulementary a 4b () a 4b () a 4b ( ) a 4b ( ) conditions Kodaira symbol I0 I v( ) I 0 I v( ) 6 I v(b) I0 III I v(b) 4 I 0 III Case of Tate Exonent v(n) of conductor Table.4: Néron tye at of y = x + ax + bx.

25 Chater. The Conductor of an Ellitic Curve over Q with -torsion 17. The Proof of Theorem.1. We rove this theorem using the work of Paadoolous [57] 1 excet in cases (ix) v (a) = 1, v (b) = and (xiv) v (a) =, v (b) = where we will need to aly Tate s algorithm directly. The seventeen cases we consider are labeled as follows: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) v (a) v (b) (x) (xi) (xii) (xiii) (xiv) (xv) (xvi) (xvii) v (a) 1 1 v (b) The standard invariants for the curve y = x + ax + bx are (see (.1)) a 1 = 0, a = a, a = 0, a 4 = b, a 6 = 0, b = 4a, b 4 = b, b 6 = 0, b 8 = b, c 4 = 4 (a b), c 6 = 5 a(9b a ), = 4 b (a 4b). Some of the cases immediately follow from Table IV of [57] so we quickly deal with these first. We have the following table. Case of v (a) v (b) v (c 4 ) v (c 6 ) v ( ) Tate Kodaira f (viii) III 7 (ix) I 0 6 (xi) I v ( ) 10 6 (xiii) III 8 (xv) III 7 (xvii) III 8 1 Errata: In the column labeled Equation non minimale of table IV in [57] the first column should read [4, 6, 1] not [4, 6, 1].

26 Chater. The Conductor of an Ellitic Curve over Q with -torsion 18 As for the remaining cases, we must check the sulementary conditions in [57]. (i) When v (a) = 0 and v (b) = 0 we have v (c 4 ) = { 5 if b 1 (mod 4), 6 if b 1 (mod 4), v (c 6 ) = 5, v ( ) = 4. If b 1 (mod 4) then from Table IV of [57] we are in case or 4 of Tate. We use Proosition 1 of loc. cit. with r = t = 1. The congruence a 4 + a = b + a { (mod 4) if a 1 (mod 4), 0 (mod 4) if a 1 (mod 4), imlies that if a 1 (mod 4) we are in case of Tate and f = 4. So assume a 1 (mod 4), whence we are in case 4 of Tate. Using Proosition of loc. cit. with r = 1 and since b 8 + rb 6 + r b 4 + r b + r 4 (1 + a) 0 (mod 8), we are in case 4 of Tate and f =. On the other hand, if b 1 (mod 4) then from Table IV of [57], we are in case or 5 of Tate. Take r = t = 1 in Proosition 1 of loc. cit.. It follows from the congruence a 4 + a = b + a { 0 (mod 4) if a 1 (mod 4), (mod 4) if a 1 (mod 4), that if a 1 (mod 4), we are in case of Tate and f = 4, whereas if a 1 (mod 4), we are in case 5 of Tate and f =. (ii) When v (a) = 0 and v (b) = 1 we have v (c 4 ) = 4, v (c 6 ) 7, v ( ) = 6, so, from Table IV of [57], we are in case or 4 of Tate. Using r = t = 0 in Proosition 1 of loc. cit., it follows that we are in case 4 of Tate and f = 5. (iii) When v (a) = 0 and v (b) = we have v (c 4 ) = 4, v (c 6 ) = 6, v ( ) = 8.

27 Chater. The Conductor of an Ellitic Curve over Q with -torsion 19 and, from Table IV of [57], we are in case 6, 7 or 8 of Tate. We use Proosition of [57]. The integer r = satisfies the congruence b 8 + rb 6 + r b 4 + r b + r 4 0 (mod ). The integer t = satisfies the congruence a 6 + ra 4 + r a + r ta t rta 1 0 (mod 8). Moreover, for r = t = we have the congruence a 6 + ra 4 + r a + r ta t rta 1 b + 4a (mod 16). if and only if a 1 (mod 4). It follows from Proosition of loc. cit. that if a 1 (mod 4) we are in case 6 of Tate and f = 4, whereas if a 1 (mod 4) then we are in case 7 of Tate. So assume a 1 (mod 4) and that we are in case 7 of Tate. Take r = in Proosition 4 of loc. cit.. The congruence 0 a + r ta 1 t t (mod 4) has no solutions for t thus it follows that we are in case 7 of Tate and f =. (iv) When v (a) = 0 and v (b) = we have v (c 4 ) = 4, v (c 6 ) = 6, v ( ) = 10, and, from Table IV of [57], we are in case 7 or 9 of Tate. The integer r = 0 satisfies the congruence b 8 + rb 6 + r b 4 + r b + r 4 0 (mod ). Moreover, we have the congruence 0 a + r ta 1 t { 1 t (mod 4) if a 1 (mod 4), t (mod 4) if a 1 (mod 4), has a solution for t if and only if a 1 (mod 4). It follows from Proosition 4 of loc. cit. that if a 1 (mod 4), we are in case 7 of Tate and f = 4, whereas if a 1 (mod 4), we are in case 9 of Tate and f =.

28 Chater. The Conductor of an Ellitic Curve over Q with -torsion 0 (v) When v (a) = 0 and v (b) = 4 we have v (c 4 ) = 4, v (c 6 ) = 6, v ( ) = 1, and, from Table IV of [57], we are in case 7 of Tate or the model is nonminimal. The integer r = 0 satisfies the congruence Moreover, the congruence b 8 + rb 6 + r b 4 + r b + r 4 0 (mod ). 0 a + r ta 1 t { 1 t (mod 4) if a 1 (mod 4), t (mod 4) if a 1 (mod 4), has a solution for t if and only if a 1 (mod 4). It follows from Proosition 4 of loc. cit. that if a 1 (mod 4), we are in case 7 of Tate and f = 4, whereas if a 1 (mod 4) the model is non-minimal. In the latter case, consider the change of variables x = 4X, y = 8Y + 4X. We obtain the new model with integer coefficients (a 1, a, a, a 4, a 6) = (1, a 1 4, 0, b 16, 0), and such that v (c 4 ) = 0, v (c 6 ) = 0, and v ( ) = 0. Hence we are in case 1 of Tate and f = 0. (vi) When v (a) = 0 and v (b) 5 we have v (c 4 ) = 4, v (c 6 ) = 6, v ( ) 14, and, from Table IV of [57], we are in case 7 of Tate or the model is nonminimal. The integer r = 0 satisfies the congruence Moreover, the congruence b 8 + rb 6 + r b 4 + r b + r 4 0 (mod ). 0 a + r ta 1 t { 1 t (mod 4) if a 1 (mod 4), t (mod 4) if a 1 (mod 4),

29 Chater. The Conductor of an Ellitic Curve over Q with -torsion 1 has a solution for t if and only if a 1 (mod 4). It follows from Proosition 4 of loc. cit. that if a 1 (mod 4), we are in case 7 of Tate and f = 4, whereas if a 1 (mod 4) the model is non-minimal. In the latter case, take the change of variables x = 4X, y = 8Y + 4X to obtain the new model with integer coefficients (a 1, a, a, a 4, a 6) = (1, a 1 4, 0, b 16, 0). Then v (c 4 ) = 0, v (c 6 ) = 0, and v ( ), whence we are in case of Tate and f = 1. (vii) When v (a) = 1 and v (b) = 0 we have v (c 4 ) =, v (c 6 ) = 6, v ( ) 7. We consider the cases v ( ) = 7, 8, 9, 10, 11, 1, 1 searately. If v ( ) = 7 then from Table IV of [57] we are in case of Tate and f = 7. If v ( ) = 9 then from Table IV of [57] we are in case 6 of Tate and f = 5. In the remaining cases; v ( ) = 8, 10, 11, 1, 1, some work is required to determine f. We defer the roof for these cases until Section.4. (x) When v (a) = 1 and v (b) = we have v (c 4 ) = 6, v (c 6 ) 9, v ( ) = 1, and from Table IV of [57] we are in case 7 of Tate. There are, however, two ossibilities for f. We need to aly Tate s algorithm directly in this case. We will use the seudocode for Tate s algorithm given in [6]. It is straightforward to check that we may ass directly to line 4 in loc. cit. without having to make any changes to our model. Furthermore, in the notation of loc. cit. since xa = a 4 = 0 is even, xa6 = a 6 16 = 0 is even, and xa4 = a 4 8 = b 8 is odd we exit the loo after line 54, with m =. Thus f = v ( ) 6 = 6 and the Kodaira symbol is I. (xii) When v (a) and v (b) = 0 we have v (c 4 ) = 4, v (c 6 ) 7, v ( ) = 6,

30 Chater. The Conductor of an Ellitic Curve over Q with -torsion so, from Table IV of [57], we are in case or 4 of Tate. Take r = 1 and t = 0 in Proosition 1 of loc. cit.. It follows from the congruence { (mod 4) if b 1 (mod 4), a 6 +ra 4 +r a +r ta t rta 1 = b+a+1 0 (mod 4) if b 1 (mod 4), that if b 1 (mod 4), we are in case of Tate and f = 6, whereas if b 1 (mod 4), we are in case 4 of Tate and f = 5. (xiv) When v (a) = and v (b) = we have v (c 4 ) = 6, v (c 6 ) = 9, v ( ) 1, so, from Table IV of [57], we are in case 7 of Tate. There are, however, two ossibilities for f deending on whether v ( ) = 1 or v ( ) 14. We claim v ( ) = 1 if and only if b/4 1 (mod 16). Indeed, since = 16b (a 4b), the hyothesis on a and b imly ( (a ) ) b v ( ) = 1 v = But (a/4) 1 (mod 4), from which the claim follows. Thus, f = 7 if b/4 1 (mod 4) and f = 6 if b/4 1 (mod 4). (xvi) When v (a) and v (b) = we have v (c 4 ) = 6, v (c 6 ) 10, v ( ) = 1, and, from Table IV of [57], we are in case 7 of Tate. There are, however, two ossibilities for f. We need to aly Tate s algorithm directly in this case. Again, we will use the seudocode for Tate s algorithm given in [6]. We consider the cases b/4 1 (mod 4) and b/4 1 (mod 4) searately. Suose b/4 1 (mod 4). Before starting the algorithm let us first make the change of variables x = X +, y = Y so our new model has coefficients a 1 = 0, a = a + 6, a = 0, a 4 = b + 4a + 1, a 6 = b + 4a + 8. It follows that v (a ) = 1, v (a 4 ) =, v (a 6 ) 5,

31 Chater. The Conductor of an Ellitic Curve over Q with -torsion where we ve used the fact that b/4 1 (mod 4). It is straight forward to check that we may ass directly to line 4 in loc. cit. without having to make any changes to our model. Furthermore, in the notation of loc. cit. since xa = a 4 = 0 is even, xa6 = a 6 16 is even, and xa4 = a 4 8 is odd, we exit the loo after line 54, with m =. Thus f = v ( ) 6 = 6 and the Kodaira symbol is I. Suose b/4 1 (mod 4). Similar to above, we first make the change of variables x = X + 6, y = Y + 4 to obtain a new model with coefficients a 1 = 0, a = a + 18, a = 8, a 4 = b + 1a + 108, a 6 = 6b + 6a + 00, and find v (a ) = 1, v (a ) =, v (a 4 ) 4, v (a 6 ) 5, (here we ve used the fact that b/4 1 (mod 4)). Moreover, v (b ) =, v (b 4 ) 5, v (b 6 ) =, v (b 8 ) 7. It is straightforward to check that we may ass directly to line 4 in loc. cit. without having to make any changes to our model. Furthermore, in the notation of loc. cit. since xa = a 4 = 0, xa6 = a 6 16, and xa4 = a 4 8 are all even, we have from line 56 that r = { 4 if a 6 is odd, 0 if a 6 is even. We then must aly the change of variables transcoord(r, 0, 0, 1) at line 59. In either case the change of variables leads to a curve (a 1, a, a, a 4, a 6 ) such that a 1 = 0, v (a ) = 1, v (a ) =, v (a 4) 4, v (a 6) 5. We have now reached the end of the loo and return back to line 45. Since xa = a 8 is odd we exit the loo after line 47 with m =. Thus f = v ( ) 7 = 5 and the Kodaira symbol is I. To finish the roof it remains to verify the cases when v (a) = 1, v (b) = 0, and v ( ) = 8, 10, 11, 1, and 1. We do this in the next section.

32 Chater. The Conductor of an Ellitic Curve over Q with -torsion 4.4 The case when v (a) = 1, v (b) = 0. We have already determined in art (iv) of the roof of Theorem.1 the values of f when v ( ) = 7 or 9. In this section we determine the value of f for the remaining cases: v ( ) = 8, 10, 11, 1, 1. First we make two observations. Lemma.6 If a, b Z such that v (a) = 1, v (b) = 0 and v ( ) = v (16b (a 4b)) 8 then b 1 (mod 4). Furthermore, if v ( ) = 8 then b 5 (mod 8). Proof. If v ( ) = v (16b (a 4b)) 8 then v (( a ) b)). It follows that b 1 (mod 4) since a/ is odd. Moreover, if v ( ) = 8 then v (( a ) b) = thus b 1 (mod 8). We will use the next lemma when alying Proosition 4 of [57]. Lemma.7 For a, b Z such that v (a) = 1 and v (b) = 0 the congruence b + 6r b + 4r a + r 4 0 (mod ) has no solutions for r if b 1 (mod 4), whereas for b 1 (mod 4) it has solutions r = { if a (mod 8), 1 if a 6 (mod 8). Proof. If b 1 (mod 4) then the congruence has no solutions mod 8 (when a is even). So, it certainly can t have any solutions mod. Assume b 1 (mod 4) and write b = 4k + 1 for some k Z. If a (mod 8) then we may write a = 8l + for some l Z. Taking r = we have b + 6r b + 4r a + r 4 16k(k + 1) 0 (mod ). Similarly, one can easily show r = 1 is a solution when a 6 (mod 8).

33 Chater. The Conductor of an Ellitic Curve over Q with -torsion Proof of Theorem.1 art (vii) when v ( ) = 8 It follows from Lemma.6 that b 5 (mod 8). Since v (c 4 ) = 4, v (c 6 ) = 6 and v ( ) = 8 it follows from Table IV of [57] that we are in case 6, 7 or 8 of Tate. We use Proosition of loc. cit.. By Lemma.7 the congruence has solutions b 8 + rb 6 + r b 4 + r b + r 4 0 (mod ) r = { if a (mod 8), 1 if a 6 (mod 8). In either case the integer t = satisfies the congruence a 6 + ra 4 + r a + r ta t rta 1 rb + r a + r t 4 t 0 (mod 8). Fix t = and r as above. Suose a (mod 8). We have the congruence a 6 + ra 4 + r a + r ta t rta 1 b + 9a (mod 16) if and only if a b 5 (mod 16). Thus, we are in case 6 of Tate (and f = 4) if a b 1 (mod 16), and in case 7 of Tate if a b 5 (mod 16). So, suose the latter holds. Taking r = in Proosition 4 of loc. cit. the congruence a + r sa 1 s s 0 (mod 4) has no solution for s, whereby we are in case 7 of Tate and f =. In the statement of the theorem we do not need to include the condition a (mod 8) since this automatically follows from the congruences b 5 (mod 8) and a b 5 or 1 (mod 16). Now suose a 6 (mod 8). We have the congruence a 6 + ra 4 + r a + r ta t rta 1 b + a 0 (mod 16) if and only if a b 9 (mod 16). Thus, we are in case 6 of Tate (and f = 4) if a b 1 (mod 16) and in case 7 of Tate if a b 9 (mod 16). So, suose the latter holds. Taking r = 1 in Proosition 4 of loc. cit. the congruence a + r sa 1 s 1 s 0 (mod 4)

34 Chater. The Conductor of an Ellitic Curve over Q with -torsion 6 has solution s = 1, whereby we are in case 8 of Tate and f =. Again, we do not need to include the condition a 6 (mod 8) in the statement of the theorem since it follows automatically from b 5 (mod 8) and a b 1 or 9 (mod 16)..4. Proof of Theorem.1 art (vii) when v ( ) = 10 In this case we have v (c 4 ) = 4, v (c 6 ) = 6, v ( ) = 10, so from Table IV of [57] we are in case 7 or 9 of Tate. We use Proosition 4 of loc. cit. to distinguish between these two cases. By Lemma.7, the congruence has solutions b 8 + rb 6 + r b 4 + r b + r 4 0 (mod ) r = Furthermore, the congruence 0 a + r t { if a (mod 8)), 1 if a 6 (mod 8)). { t (mod 4) if a (mod 8), 1 t (mod 4) if a 6 (mod 8), has solution t = 1 if a 6 (mod 8) and no solution for t otherwise. Thus, we are in case 9 of Tate if a 6 (mod 8) and in case 7 of Tate if a (mod 8). The assertion follows..4. Proof of Theorem.1 art (vii) when v ( ) = 11 In this case v (c 4 ) = 4, v (c 6 ) = 6, v ( ) = 11, so, from Table IV of [57], we are in case 7 or 10 of Tate. By exactly the same argument as in Section.4., if a 6 (mod 8), we are in case 10 of Tate and if a (mod 8), we are in case 7 of Tate. The assertion follows.

35 Chater. The Conductor of an Ellitic Curve over Q with -torsion Proof of Theorem.1 art (vii) when v ( ) = 1 In this case v (c 4 ) = 4, v (c 6 ) = 6, v ( ) = 1, so, from Table IV of [57], we are in case 7 of Tate or the model is non-minimal. By exactly the same argument as in section.4., if a (mod 8), we are in case 7 of Tate and if a 6 (mod 8), the model is non-minimal. In the case that the model is non-minimal, we make the change of variables x = 4X a/, y = 8Y + 4X. (.5) The new model has coefficients ( (a 1, a, a, a 4, a 6) = 1, a + ) 8, 0, 4b a, a(a 4b), (.6) which are all integers (by assumtions on a and b). Also, v (c 4 ) = 0, v (c 6 ) = 0, and v ( ) = 0, whence we are in case 1 of Tate and f = Proof of Theorem.1 art (vii) when v ( ) 1 In this case v (c 4 ) = 4, v (c 6 ) = 6, v ( ) 1, so, from Table IV of [57], we are in case 7 of Tate or the model is non-minimal. By exactly the same argument as in section.4., if a (mod 8), we are in case 7 of Tate and if a 6 (mod 8), the model is non-minimal. In the case that the model is non-minimal we take the change of variables (.5) which gives us a new integral model with coefficients as in (.6). Since v (c 4 ) = 0, v (c 6 ) = 0, and v ( ) 1, we are in case of Tate and f = 1. This comletes the roof of Theorem.1..5 The Proof of Theorem.. We can quickly deal with the following cases by using Table II of [57].

36 Chater. The Conductor of an Ellitic Curve over Q with -torsion 8 Case of v (a) v (b) v (c 4 ) v (c 6 ) v ( ) Tate Kodaira f I v ( ) I or 7 I v ( ) III I III There are only three remaining cases to check: (1) v (a) = 0, v (b) = 0; () v (a) = 1, v (b) = 1; () v (a) =, v (b) =. (1) Suose v (a) = 0 and v (b) = 0. Then v (c 4 ) = 0, v (c 6 ) = 0, and divides if and only if b 1 (mod ). It follows that f = { 1 if b 1 (mod ), 0 if b 1 (mod ), and the Néron tye at is I v ( ) if b 1 (mod ) and I 0 if b 1 (mod ). () Suose v (a) = 1 and v (b) = 1. Then v (c 4 ), v (c 6 ) =, and v ( ) =. We consider the intervening condition P in Table II of [57]. P is decided if we have ( 5 ( a or equivalently ) ( ( a ) )) ( (a b + 4 ( a ) ) b ) ( 4 b ( a ) ( b (mod 9), ) ) (mod 9), Since v (a) = v (b) = 1, this is certainly the case. Therefore f = and the Néron tye at is III. () Suose v (a) = and v (b) =. Then v (c 4 ) 4, v (c 6 ) = 6, and v ( ) = 9. We consider the intervening condition P 5 in Table II of [57]. P 5 is decided if we have ( 5 ( a 9 ) ( b ( a 9 9) )) ( (a ) ) + 4 b 9 7 (mod 9),

37 Chater. The Conductor of an Ellitic Curve over Q with -torsion 9 or equivalently ( a 9 ) ( ) 4 b ( a ) ( ) b (mod 9), Since v (a) = and v (b) =, this is the case. Therefore f = and the Néron tye at is III..6 The Proof of Theorem.4. We can quickly deal with the following cases by using Table I of [57]. Case of v (a) v (b) v (c 4 ) v (c 6 ) v ( ) Tate Kodaira f I v(b) I III I v (b) I III There are only two remaining cases to check: (1) v (a) = 0, v (b) = 0; () v (a) = 1, v (b) =. (1) Suose v (a) = 0, v (b) = 0. In this case, can divide at most one of c 4, c 6 and. If does not divide then f = 0. If then does not divide c 4 or c 6, whence f = 1 and the Néron tye at is I v( ). () Suose v (a) = 1, v (b) =. Then v (c 4 ), v (c 6 ), and v ( ) 6. Moreover, in this case, can divide at most one of a b, 9b a and a 4b. If v ( ) 7, i.e. a 4b 0 (mod ), then we are in case 7 of Tate, f =, and the Néron tye at is I v ( ) 6. On the other hand, if v ( ) = 6, i.e. a 4b 0 (mod ), then we are in case 6 of Tate, f =, and the Néron tye at is I 0. This roves Theorem.4.