THERE ARE NO ELLIPTIC CURVES DEFINED OVER Q WITH POINTS OF ORDER 11
|
|
- Brendan Lawrence
- 5 years ago
- Views:
Transcription
1 THERE ARE NO ELLIPTIC CURVES DEFINED OVER Q WITH POINTS OF ORDER 11 ALLAN LACY 1. Introduction If E is an elliptic curve over Q, the set of rational points E(Q), form a group of finite type (Mordell-Weil Theorem): E(Q) = Z r E(Q) tors where E(Q) tors is finite. In 1977, Mazur gave complete list of the only subgroups that can appear as E(Q) tors as E varies. Mazur [6] proved that there are finitely many possibilities (all the possibilities occur infinitely often) Theorem 1 (Mazur, 1977). Let E/Q be an elliptic curve. Then the torsion subgroup of E(Q) is one of the following 15 groups: E(Q) tors = { Z/nZ n = 1, 2,..., 10, 12; Z/2Z Z/2nZ n = 1, 2, 3, 4 In particular, Mazur s Theorem implies that there is no elliptic curve over Q with a point of order 11. This particular result was proved originally in 1939 by Billing and Mahler [1]. In this pape we present a proof of this fact, following the main idea on the original proof. 2. The curve X 1 (11) We start with some general considerations about torsion points on elliptic curves. Let E/Q be a elliptic curve with Weierstrass model E : y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 Lets assume that P E(Q) is point of exact order n 4. Since P has coordinates in Q, we can make a change of variables (over Q) to assume that P = (0, 0), and therefore that a 6 = 0. We have P = (0, a 3 ), and since we are assuming that P does not have order 2, it follows that a 3 0. Then the change of variable y y + a4 a 3 x allows us to write a WE for E with a 4 = 0. With a 4 = 0 we easily compute 2P = ( a 2, a 1 a 2 a 3 ), so in particular a 2 0, since otherwise 2P = P, and P would have order 3. Finally, if we set λ = a2 a 3, the change of variable x λ 2 x, y λ 3 y give us E : y 2 + a 1 λxy + λ 3 a 3 y = x 3 + λ 2 a 2 x 2 Note that λ 3 a 3 = λ 2 a 2, so setting b this common value, and c := 1 λa 1, we can write a model for E of the form 1
2 2 ALLAN LACY (1) E = E(b, c) : y 2 + (1 c)xy by = x 3 bx 2 The two-parameter Weierstrass equation (1) is called the Tate Normal Form for E with point P = (0, 0). Each point (b, c) Q Q for which (b, c), the discriminat of (1), is nonzero defines an elliptic curve over Q with P = (0, 0) E(Q) such that P, 2P, 3P. So far we have only used the fact that P is not a point of order 2 or 3. For any k Z, we can consider the point kp, which would have as coordinates rational functions of b and c. If we further assume that P has order n 4, then the equation np = defines a polynomial equation f n (b, c) = 0 over Z. Each point on the open affine curve f n (b, c) = 0, b 0, (b, c) 0 defines an elliptic curve E(b, c) together with a Q-rational point of order n. This is then an affine model of Y 1 (n), the parameter space (of isomorphism classes) of pairs (E, P ) of elliptic curves E with a point P of order n. The curve Y 1 (n) is not necessarily smooth, so we consider X 1 (n) to be its completion. The idea of the proof, roughly speaking, is to prove that the curve X 1 (11) does not contain points. Then, we start by computing an equation for X 1 (11). We follow the steps in [8] to find X 1 (11) : for n = 11, we compute X 1 (11) by looking at the condition imposed by 5P = 6P. We have 5P = (rs(s 1), r 2 s(s 1) 2 ); b = cr and c = s(r 1) 6P = ( mt, m 2 (m + 2t 1)); m(1 s) = s(1 r) and r s = t(1 s) so we get s(1 r) rs(s 1) = mt = (1 s) r s 1 s Clearing denominators we get r 2 4sr + 3s 2 r s 3 r + s = 0 If we further make the birational transformation r y + 1, s y x + 1 we get the following equation for X 1 (11) which is smooth (is a elliptic curve!): X 1 (11) : y 2 + y = x 3 x 2 We can trace back all the changes of variables and substitutions, to get b and c as a function of (x 0, y 0 ) X 1 (11): ( ) y0 (2) b = y 0 (y 0 + 1) + 1 x 0 and ( ) y0 c = y x 0 As we said roughly speakig, the idea is to prove that the set of points on the curve X 1 (11) is empty, since otherwise we can use (2) to define an elliptic curve E(b, c) with a point of order 11. Also, if we requiere that such E(b, c) is defined over Q, we have to restrict only to b, c Q, and therefore from ( ) 1 c b = c(y 0 + 1) and x 0 = y 0 1 y 0
3 THERE ARE NO ELLIPTIC CURVES DEFINED OVER Q WITH POINTS OF ORDER 11 3 we see that we also need make restrictions to the point (x 0, y 0 ) (in particular, it has to be Q-rational). Note that we can spot some rational points on X 1 (11): (0, 0), (1, 0), (0, 1) and (1, 1). Although, these 4 points produce b = 0 or b =, so have to be discarded (recall b = λ 3 a 3 0). We call these forbidden points cusps of X 1 (11). Now we can explain more rigorously the idea of the proof: we will prove that the only Q-rational points of X 1 (11) are these 4 cusps. We use the fact that X 1 (11) is an elliptic curve, and therefore X 1 (11)(Q) is a group. First of all, it not hard to check that these 4 points are torsion elements of X 1 (11). Moreover, these points (together with the point at infinity) form the whole torsion subgroup of X 1 (11)(Q). To see this, we use the following theorem, proved independently by Lutz and Nagell, to compute the torsion subgroup of an elliptic curve over Q: Theorem 2 (Lutz-Nagell). Let E/Q and elliptic curve with short Weierstrass equation y 2 = f(x) with f(x) Z[x]. If (x 0, y 0 ) E(Q) tors, then x 0, y 0 Z. Furthermore, either y 0 = 0 or y 2 0 D f, the discriminat of f. Then, in order to apply the Lutz-Nagell theorem to X 1 (11), we first need to write it in short form: completing the square for y and multiplying the resulting equation by 2 6 we get X 1 (11) : y 2 = x 3 4x = f(x); D f = First of all, the four (cusps) torsion points on y 2 + y = x 3 x 2 give us four torsion points on y 2 = f(x): (0, ±4) and (4, ±4). Now, we apply Lutz-Nagell theorem to prove these are the only torsion elements. Since the polynomial f(x) is irreducible over Q, we can discard the case y = 0. Then we have to look at the integers y such that y 2 2 8, that is y = ±2 i, i = 0, 1, 2, 3, 4. The case i = 2 give us the four points above. For i = 3 or i = 4, we need to solve x 2 (x 4) = 16(2 2i 4 1) which is not possible in Z: x must be divisible by 4, but if we write x = 4a, then 4a 2 (a 1) = 2 2i 4 1. The cases i = 0 and i = 1 can be ruled out easily too. We conclude that X 1 (11)(Q) tors = {, (0, 4), (0, 4), (4, 4), (4, 4)} = Z/5Z. We have proved that X 1 (11) has no more torsion points, so our next (and final) step is to prove that it has no points of infinite order. We make yet another change of variable to X 1 (11): completing the cube in f(x) and multiplying by 3 6 the resulting equation we get 1 X 1 (11) : y 2 = x 3 432x We will prove that the only rational points on this elliptic curve are ( 12, ±108) and (24, ±108) (which correspond to the cusps on y 2 + y = x 3 x 2 ). Assuming this result by the moment, we are ready: Lemma 1. The only rational points on y 2 + y = x 3 x 2 (0, 0), (0, 1), (1, ±1). are the four points 1 this is the curve appearing in [1], for which the authors prove to have only has 5 rational points (Lemma 2, [1])
4 4 ALLAN LACY Proof. The rational points on y 2 + y = x 3 x 2 are in bijection with the rational points of y 2 = x 3 432x , and there are only four such points. Theorem 3. There is no elliptic curve E/Q with a point of order 11. Proof. Suppose by contradiction that there is a elliptic curve E/Q with a point P E(Q) of order 11. We can assume that P = (0, 0) and that E is given by E = E(b, c) : y 2 + (1 c)xy by = x 3 bx 2 for some b, c Q, b 0. Since P has order 11, using formula (2) we define points x 0, y 0 Q satisfying y0 2 + y 0 = x 3 0 x 2 0. By construction (x 0, y 0 ) (0, 0), (0, 1), (1, ±1), since x 0 = 0, y 0 = 0, 1, x 0 are not allowed, according to (2). The previous Lemma gives us the contradiction. 3. The main tool The following is the key in [1], from where the authors conclude that there is no elliptic curve over Q with a point of order 11. Theorem 4. The elliptic curve E : y 2 = x 3 432x has no rational points of infinite order. Proof. The proof is by contradiction, so lets assume Q = (x 0, y 0 ) E(Q) is a point of infinite order. We start by recalling that the five points {, ( 12, ±108), (24, ±108)} = E(Q) tors form a cyclic group of order 5, generated say, by P. Then we have P = 6P and 3P = 8P, so every point of {, P, 2P, 3P, 4P } lie on 2E(Q). Therefore, if there is some Q E(Q) not in the subgroup 2E(Q), then Q is necessarily of infinite order. Conversely, if E(Q) contains a point of infinite order, we can find a point not in 2E(Q) (namely, a generator of E(Q)/E(Q) tors ). We are going to exploit this observation: we are going to prove that the quotient E(Q)/2E(Q) is trivial, by following the proof in [2] of the weak finite basis theorem, that in general, the quotient E(Q)/2E(Q) is finite. We start by factoring f(x) over Q: f(x) = x 3 432x := (x θ 1 )(x θ 2 )(x θ 3 ) Lets assume that θ = θ 1 R, and consider the number field K = Q(θ). We also consider a map given by { µ(q) = µ : E(Q) K /(K ) 2 1 if Q = (x θ) mod (K ) 2 if Q = (x, y) We use the following result, presented here without proof (for a proof, see [2]): Lemma 2. The map µ is a group homomorphism, and ker µ = 2E(Q).
5 THERE ARE NO ELLIPTIC CURVES DEFINED OVER Q WITH POINTS OF ORDER 11 5 Then our assumption that E(Q) contains a point Q of infinite order translates to Q / ker µ, that is, if Q = (x 0, y 0 ) then x 0 θ is not a square in K. If we write (x 0, y 0 ) = (r/t 2, s/t 3 ) Q 2 with (r, t) = (s, t) = 1, then y0 2 = f(x 0 ) is equivalent to s 2 = (r θt 2 )(r θ 2 t 2 )(r θ 3 t 2 ) Then µ(q) x 0 θ r θt 2 mod (K ) 2. We consider the principal ideal generated by r θt 2 in O K, the ring of integers of K. The bulk of the work is to prove that this ideal is a square. We also consider γ = (r θ 2 t 2 )(r θ 3 t 2 ) O K 2 and the factorization into primes of the ideals (r θt 2 ) and (γ): so (r θt 2 ) = p ai i and (γ) = p bi i (γ, r θt 2 ) = (γ) + (r θt 2 ) = p min(ai,bi) i Lets suppose that for p = p i in the above factorization we have a i = ord p (r θt 2 ) is odd. In particular a i > 0, and from s 2 = (r θt 2 )γ we have that b i = ord p (γ) > 0. Writing s 2 = (r θt 2 )γ = (r θt 2 ) (θ θ 2 )(θ θ 3 ) + (r θt 2 ) 2 (r + e 2 (θ θ 3 θ 3 )) we have = (r θt 2 ) f (θ) + (r θt 2 ) 2 (r + e 2 (θ θ 3 θ 3 )) ord p (f (θ)) = ord p ((r θt 2 )γ) min(a i, b i ) > 0 so we conclude that the ideal (γ, r θt 2 ) divides N K/Q (f (θ)) = (f) = , the discriminant of f. Therefore we can write (r θt 2 ) = IJ 2 for ideals I, J O K with I squarefree, divisible only by the primes above 2, 3 and 11. In O K we have the following factorization of these primes: so (2) = p 3 2, (3) = p 3, (11) = p 11 q 2 11 I = p a 2 p b 3 p c 11 q d 11, a, b, c, d {0, 1} We have prove that a = b = c = d = 0. Since s 2 = N K/Q (r θt 2 ) = N K/Q (I)N K/Q (J) 2 we see that N K/Q (I) = 2 a 3 b 11 c+d is a square. Then a = b = 0 and c = d = 0 or c = d = 1. Lets suppose that c = d = 1, so p 11 q 11 (r θt 2 ). Therefore p 2 11q 2 11 (r θt 2 ) 2 and since (11) = p 11 q 2 11 we see that 11 (r θt 2 ) 2, so (r θt2 ) 2 11 O K. But K has an integral basis { θ θ2 324, θ } 18 + θ2 54, θ2 36 so in particular, no element of O K has denominator 11. This ends the proof that the ideal (r θt 2 ) = J 2 is a square. 2 γ = r 2 rt 2 (θ 2 + θ 3 ) + θ 2 θ 3 t 4 = r 2 + rt 2 θ + t 4 bθ 1 O K
6 6 ALLAN LACY But K has class number 1, so there is some α O K such that J = (α), so (r θt 2 ) = (α 2 ). Therefore, there is a unit η O K such that (3) r θt 2 = ηα 2 Note that η cannot be the square of another unit, for otherwise, r θt 2 would be a square in K. The trick in [1] is to note that K = Q(θ) = Q(ϑ) where θ is the real root of x 3 4x 4 = 0. Specifically we are going to use the following information about K = Q(ϑ) θ = 24 9ϑ 2 = 12(1 + 3/ϑ) (so, in fact, θ and ϑ generate the same field extension) {1, ϑ, 1 2 ϑ2 } is an integral basis of K The unit group of K has rank 1, generated by the fundamental unit ϑ2 We have s 2 = N K/Q (r θt 2 ) = N K/Q (η)n K/Q (α) 2 so N K/Q (η) must be a square. This leave us with the only possibility η = ϑ2. Multiplying (3) by η we have η(r θt 2 ) = (ηα) 2 ( ϑ2 )(r (24 9ϑ 2 )t 2 ) = (ηα) 2 = (a + bϑ + c ϑ2 2 )2 for some a, b, c Z. Using ϑ 4 = 4ϑ + 4ϑ 2, equating coefficients above, we get the system of equations r + 24t 2 = a 2 + 4bc 18t 2 = c 2 + 2ab + 4bc 1 2 r 3t2 = b 2 + c 2 + ac From the third and first equations, r and a must be even. Since (r, t) = 1, it follows t is odd, hence the second equation leads to the impossible congruence c 2 2(mod 2). This contradiction proves that E(Q) does not contain any point of infinite order. 4. Remarks The result in Theorem 3 is no longer valid if we replace Q for arbitrary number field. For example, if we fix 0, 4 x 1 Q and let y 1 Q be a solution of y 2 = x 3 1 4x then x 0 = x 1 4, y 0 = y is a solution of y 2 + y = x 3 x 2. Then, if we define b, c Q(y 1 ) using (2), the elliptic curve E(b, c) has P = (0, 0) as a point of order 11. This procedure allow us to define quadratic (or cubic, if we fix y 1 first) extensions of Q and elliptic curves over these extensions having points of order 11. It is natural to ask for some kind of generalization of Mazur s Theorem to arbitrary numbers fields; that is, given an number field K, to characterize the groups that can
7 THERE ARE NO ELLIPTIC CURVES DEFINED OVER Q WITH POINTS OF ORDER 11 7 appear as the torsion part of E(K) for arbitrary elliptic curves E/K. Historically some partial results led to Conjecture 1 (Uniform Boundness Conjecture (UBC)). Let d 1. For any number field K/Q of degree d and any elliptic curve E/K, there is constant C = C(d) such that E(K) tors C. So in particular, the UBC implies that for d 1, the set T (d) = {E(K) tors : [K : Q] d and E/K is an elliptic curve} is finite. For quadratic extensions we have the following result, due to S. Kamienny [4] Theorem 5 (Kamienny, 1992). If E is an elliptic curve defined over a quadratic number field K, then E(K) tors is isomorphic to one of the following 26 groups: Z/nZ n = 1, 2,..., 16, 18; Z/2Z Z/2nZ n = 1, 2,..., 6 E(K) tors = Z/3Z Z/3nZ n = 1, 2 Z/4Z Z/4Z From Mazur s Theorem, we see that the only possible primes that appears as divisors of E(Q) tors are 2, 3, 5 and 7; while from Kamienny s Theorem, if K/Q is a quadratic extension, the only possible primes diving E(K) tors are 2, 3, 5, 7, 11 and 13. For d 1 lets consider S(d) = {primes p : there is K/Q of degree d and an elliptic curve E/K such that p E(K) tors } An obvious necessary condition for the UBC to hold is that for all d, the set S(d) is finite. It happens that this is also sufficient: Theorem 6 (Frey, Faltings). If S(d) is finite, then T (d) is finite. In particular, if S(d) is finite, the UBC holds in degree d. The following theorem, due to Merel [7], proves affirmatively the UBC: Theorem 7 (Merel, 1996). S(d) d 3d2 for all d 2 Shortly after Merel s result, his bound was improved by Osterlé: Theorem 8 (Oesterlé). S(d) (3 d/2 + 1) 2 for all d 2 Finally, we state some known results: Theorem 9.. (Kamienny and Parent, 1999) S(3) = {2, 3, 5, 7, 11, 13} (Kamienny, Stein and Stoll, 2010) S(4) = {2, 3, 5, 7, 11, 13, 17}
8 8 ALLAN LACY References [1] Billing G. and Mahler K., On exceptional points on cubic curves J. London Math. Soc. 15, (1940) [2] J.W.S. Cassels, Lectures on elliptic curves London Mathematical Society Students Texts 24, Cambridege University Press, 1991 [3] Husemöller, D., Elliptic Curves, Graduate Texts in Mathematics, Springer-Verlag, New York- Berlin, 1987 [4] Kamienny, S. Torsion points on elliptic curves and q-coefficients of modular forms Invent. Math. 109 (1992), no. 2, [5] Kamienny, S., Stein, W. and Stoll, M. Torsion points on elliptic curves over quadratic numbers fields, available online [6] Mazur, B. Modular curves and the Eisenstein ideal Inst. Hautes tudes Sci. Publ. Math. No. 47 (1977), (1978). [7] Merel, Loc. Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124 (1996), no. 1-3, [8] Reichert, Markus A. Explicit determination of nontrivial torsion structures of elliptic curves over quadratic number fields Math. Comp. 46 (1986), no. 174,
THE NUMBER OF TWISTS WITH LARGE TORSION OF AN ELLITPIC CURVE
THE NUMBER OF TWISTS WITH LARGE TORSION OF AN ELLITPIC CURVE FILIP NAJMAN Abstract. For an elliptic curve E/Q, we determine the maximum number of twists E d /Q it can have such that E d (Q) tors E(Q)[2].
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #24 12/03/2013
18.78 Introduction to Arithmetic Geometry Fall 013 Lecture #4 1/03/013 4.1 Isogenies of elliptic curves Definition 4.1. Let E 1 /k and E /k be elliptic curves with distinguished rational points O 1 and
More informationTORSION AND TAMAGAWA NUMBERS
TORSION AND TAMAGAWA NUMBERS DINO LORENZINI Abstract. Let K be a number field, and let A/K be an abelian variety. Let c denote the product of the Tamagawa numbers of A/K, and let A(K) tors denote the finite
More informationIntroduction to Elliptic Curves
IAS/Park City Mathematics Series Volume XX, XXXX Introduction to Elliptic Curves Alice Silverberg Introduction Why study elliptic curves? Solving equations is a classical problem with a long history. Starting
More informationLECTURE 2 FRANZ LEMMERMEYER
LECTURE 2 FRANZ LEMMERMEYER Last time we have seen that the proof of Fermat s Last Theorem for the exponent 4 provides us with two elliptic curves (y 2 = x 3 + x and y 2 = x 3 4x) in the guise of the quartic
More informationPeriodic continued fractions and elliptic curves over quadratic fields arxiv: v2 [math.nt] 25 Nov 2014
Periodic continued fractions and elliptic curves over quadratic fields arxiv:1411.6174v2 [math.nt] 25 Nov 2014 Mohammad Sadek Abstract Let fx be a square free quartic polynomial defined over a quadratic
More informationTwists of elliptic curves of rank at least four
1 Twists of elliptic curves of rank at least four K. Rubin 1 Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA A. Silverberg 2 Department of Mathematics, University of
More informationarxiv: v1 [math.nt] 31 Dec 2011
arxiv:1201.0266v1 [math.nt] 31 Dec 2011 Elliptic curves with large torsion and positive rank over number fields of small degree and ECM factorization Andrej Dujella and Filip Najman Abstract In this paper,
More informationArithmetic Progressions Over Quadratic Fields
Arithmetic Progressions Over Quadratic Fields Alexander Diaz, Zachary Flores, Markus Vasquez July 2010 Abstract In 1640 Pierre De Fermat proposed to Bernard Frenicle de Bessy the problem of showing that
More informationTAMAGAWA NUMBERS OF ELLIPTIC CURVES WITH C 13 TORSION OVER QUADRATIC FIELDS
TAMAGAWA NUMBERS OF ELLIPTIC CURVES WITH C 13 TORSION OVER QUADRATIC FIELDS FILIP NAJMAN Abstract. Let E be an elliptic curve over a number field K c v the Tamagawa number of E at v and let c E = v cv.
More informationarxiv: v2 [math.nt] 23 Sep 2011
ELLIPTIC DIVISIBILITY SEQUENCES, SQUARES AND CUBES arxiv:1101.3839v2 [math.nt] 23 Sep 2011 Abstract. Elliptic divisibility sequences (EDSs) are generalizations of a class of integer divisibility sequences
More informationSome new families of positive-rank elliptic curves arising from Pythagorean triples
Notes on Number Theory and Discrete Mathematics Print ISSN 1310 5132, Online ISSN 2367 8275 Vol. 24, 2018, No. 3, 27 36 DOI: 10.7546/nntdm.2018.24.3.27-36 Some new families of positive-rank elliptic curves
More informationLECTURE 18, MONDAY
LECTURE 18, MONDAY 12.04.04 FRANZ LEMMERMEYER 1. Tate s Elliptic Curves Assume that E is an elliptic curve defined over Q with a rational point P of order N 4. By changing coordinates we may move P to
More informationAlgebraic Geometry: Elliptic Curves and 2 Theorems
Algebraic Geometry: Elliptic Curves and 2 Theorems Chris Zhu Mentor: Chun Hong Lo MIT PRIMES December 7, 2018 Chris Zhu Elliptic Curves and 2 Theorems December 7, 2018 1 / 16 Rational Parametrization Plane
More informationModern Number Theory: Rank of Elliptic Curves
Modern Number Theory: Rank of Elliptic Curves Department of Mathematics University of California, Irvine October 24, 2007 Rank of Outline 1 Introduction Basics Algebraic Structure 2 The Problem Relation
More informationCONGRUENT NUMBERS AND ELLIPTIC CURVES
CONGRUENT NUMBERS AND ELLIPTIC CURVES JIM BROWN Abstract. In this short paper we consider congruent numbers and how they give rise to elliptic curves. We will begin with very basic notions before moving
More informationON A FAMILY OF ELLIPTIC CURVES
UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLIII 005 ON A FAMILY OF ELLIPTIC CURVES by Anna Antoniewicz Abstract. The main aim of this paper is to put a lower bound on the rank of elliptic
More informationIntegral points of a modular curve of level 11. by René Schoof and Nikos Tzanakis
June 23, 2011 Integral points of a modular curve of level 11 by René Schoof and Nikos Tzanakis Abstract. Using lower bounds for linear forms in elliptic logarithms we determine the integral points of the
More informationThree cubes in arithmetic progression over quadratic fields
Arch. Math. 95 (2010), 233 241 c 2010 Springer Basel AG 0003-889X/10/030233-9 published online August 31, 2010 DOI 10.1007/s00013-010-0166-5 Archiv der Mathematik Three cubes in arithmetic progression
More informationElliptic curves and modularity
Elliptic curves and modularity For background and (most) proofs, we refer to [1]. 1 Weierstrass models Let K be any field. For any a 1, a 2, a 3, a 4, a 6 K consider the plane projective curve C given
More informationOn the Torsion Subgroup of an Elliptic Curve
S.U.R.E. Presentation October 15, 2010 Linear Equations Consider line ax + by = c with a, b, c Z Integer points exist iff gcd(a, b) c If two points are rational, line connecting them has rational slope.
More informationLECTURE 7, WEDNESDAY
LECTURE 7, WEDNESDAY 25.02.04 FRANZ LEMMERMEYER 1. Singular Weierstrass Curves Consider cubic curves in Weierstraß form (1) E : y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6, the coefficients a i
More informationOutline of the Seminar Topics on elliptic curves Saarbrücken,
Outline of the Seminar Topics on elliptic curves Saarbrücken, 11.09.2017 Contents A Number theory and algebraic geometry 2 B Elliptic curves 2 1 Rational points on elliptic curves (Mordell s Theorem) 5
More informationA FAMILY OF INTEGER SOMOS SEQUENCES
A FAMILY OF INTEGER SOMOS SEQUENCES BETÜL GEZER, BUSE ÇAPA OSMAN BİZİM Communicated by Alexru Zahărescu Somos sequences are sequences of rational numbers defined by a bilinear recurrence relation. Remarkably,
More informationTHE MODULAR CURVE X O (169) AND RATIONAL ISOGENY
THE MODULAR CURVE X O (169) AND RATIONAL ISOGENY M. A. KENKU 1. Introduction Let N be an integer ^ 1. The affine modular curve Y 0 (N) parametrizes isomorphism classes of pairs (E ; C N ) where E is an
More informationInfinite rank of elliptic curves over Q ab and quadratic twists with positive rank
Infinite rank of elliptic curves over Q ab and quadratic twists with positive rank Bo-Hae Im Chung-Ang University The 3rd East Asian Number Theory Conference National Taiwan University, Taipei January
More informationAbstracts of papers. Amod Agashe
Abstracts of papers Amod Agashe In this document, I have assembled the abstracts of my work so far. All of the papers mentioned below are available at http://www.math.fsu.edu/~agashe/math.html 1) On invisible
More informationArithmetic Progressions over Quadratic Fields
uadratic Fields ( D) Alexer Díaz University of Puerto Rico, Mayaguez Zachary Flores Michigan State University Markus Oklahoma State University Mathematical Sciences Research Institute Undergraduate Program
More information6.5 Elliptic Curves Over the Rational Numbers
6.5 Elliptic Curves Over the Rational Numbers 117 FIGURE 6.5. Louis J. Mordell 6.5 Elliptic Curves Over the Rational Numbers Let E be an elliptic curve defined over Q. The following is a deep theorem about
More informationMath 120 HW 9 Solutions
Math 120 HW 9 Solutions June 8, 2018 Question 1 Write down a ring homomorphism (no proof required) f from R = Z[ 11] = {a + b 11 a, b Z} to S = Z/35Z. The main difficulty is to find an element x Z/35Z
More informationIntroduction to Arithmetic Geometry
Introduction to Arithmetic Geometry 18.782 Andrew V. Sutherland September 5, 2013 What is arithmetic geometry? Arithmetic geometry applies the techniques of algebraic geometry to problems in number theory
More informationPoints of Finite Order
Points of Finite Order Alex Tao 23 June 2008 1 Points of Order Two and Three If G is a group with respect to multiplication and g is an element of G then the order of g is the minimum positive integer
More informationAlgorithm for Concordant Forms
Algorithm for Concordant Forms Hagen Knaf, Erich Selder, Karlheinz Spindler 1 Introduction It is well known that the determination of the Mordell-Weil group of an elliptic curve is a difficult problem.
More informationComputing a Lower Bound for the Canonical Height on Elliptic Curves over Q
Computing a Lower Bound for the Canonical Height on Elliptic Curves over Q John Cremona 1 and Samir Siksek 2 1 School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7
More informationOn a Problem of Steinhaus
MM Research Preprints, 186 193 MMRC, AMSS, Academia, Sinica, Beijing No. 22, December 2003 On a Problem of Steinhaus DeLi Lei and Hong Du Key Lab of Mathematics Mechanization Institute of Systems Science,
More informationRank-one Twists of a Certain Elliptic Curve
Rank-one Twists of a Certain Elliptic Curve V. Vatsal University of Toronto 100 St. George Street Toronto M5S 1A1, Canada vatsal@math.toronto.edu June 18, 1999 Abstract The purpose of this note is to give
More informationProjects on elliptic curves and modular forms
Projects on elliptic curves and modular forms Math 480, Spring 2010 In the following are 11 projects for this course. Some of the projects are rather ambitious and may very well be the topic of a master
More informationCONSTRUCTION OF HIGH-RANK ELLIPTIC CURVES WITH A NONTRIVIAL TORSION POINT
MATHEMATICS OF COMPUTATION Volume 66, Number 217, January 1997, Pages 411 415 S 0025-5718(97)00779-5 CONSTRUCTION OF HIGH-RANK ELLIPTIC CURVES WITH A NONTRIVIAL TORSION POINT KOH-ICHI NAGAO Abstract. We
More informationIsogeny invariance of the BSD conjecture
Isogeny invariance of the BSD conjecture Akshay Venkatesh October 30, 2015 1 Examples The BSD conjecture predicts that for an elliptic curve E over Q with E(Q) of rank r 0, where L (r) (1, E) r! = ( p
More informationElliptic Curves: An Introduction
Elliptic Curves: An Introduction Adam Block December 206 Introduction The goal of the following paper will be to explain some of the history of and motivation for elliptic curves, to provide examples and
More informationElliptic Curves. Akhil Mathew (Department of Mathematics Drew UniversityElliptic MathCurves 155, Professor Alan Candiotti) 10 Dec.
Elliptic Curves Akhil Mathew Department of Mathematics Drew University Math 155, Professor Alan Candiotti 10 Dec. 2008 Akhil Mathew (Department of Mathematics Drew UniversityElliptic MathCurves 155, Professor
More informationON THE FAMILY OF ELLIPTIC CURVES y 2 = x 3 m 2 x+p 2
Manuscript 0 0 0 ON THE FAMILY OF ELLIPTIC CURVES y = x m x+p ABHISHEK JUYAL, SHIV DATT KUMAR Abstract. In this paper we study the torsion subgroup and rank of elliptic curves for the subfamilies of E
More informationCongruent Number Problem and Elliptic curves
Congruent Number Problem and Elliptic curves December 12, 2010 Contents 1 Congruent Number problem 2 1.1 1 is not a congruent number.................................. 2 2 Certain Elliptic Curves 4 3 Using
More informationDensity of rational points on Enriques surfaces
Density of rational points on Enriques surfaces F. A. Bogomolov Courant Institute of Mathematical Sciences, N.Y.U. 251 Mercer str. New York, (NY) 10012, U.S.A. e-mail: bogomolo@cims.nyu.edu and Yu. Tschinkel
More informationCONGRUENT NUMBERS AND ELLIPTIC CURVES
CONGRUENT NUMBERS AND ELLIPTIC CURVES JIM BROWN Abstract. These are essentially the lecture notes from a section on congruent numbers and elliptic curves taught in my introductory number theory class at
More informationarxiv: v1 [math.nt] 11 Aug 2016
INTEGERS REPRESENTABLE AS THE PRODUCT OF THE SUM OF FOUR INTEGERS WITH THE SUM OF THEIR RECIPROCALS arxiv:160803382v1 [mathnt] 11 Aug 2016 YONG ZHANG Abstract By the theory of elliptic curves we study
More informationSelected exercises from Abstract Algebra by Dummit and Foote (3rd edition).
Selected exercises from Abstract Algebra by Dummit and Foote (3rd edition). Bryan Félix Abril 12, 2017 Section 2.1 Exercise (6). Let G be an abelian group. Prove that T = {g G g < } is a subgroup of G.
More information14 Ordinary and supersingular elliptic curves
18.783 Elliptic Curves Spring 2015 Lecture #14 03/31/2015 14 Ordinary and supersingular elliptic curves Let E/k be an elliptic curve over a field of positive characteristic p. In Lecture 7 we proved that
More informationElliptic Curves over Q
Elliptic Curves over Q Peter Birkner Technische Universiteit Eindhoven DIAMANT Summer School on Elliptic and Hyperelliptic Curve Cryptography 16 September 2008 What is an elliptic curve? (1) An elliptic
More informationAn example of elliptic curve over Q with rank equal to Introduction. Andrej Dujella
An example of elliptic curve over Q with rank equal to 15 Andrej Dujella Abstract We construct an elliptic curve over Q with non-trivial 2-torsion point and rank exactly equal to 15. 1 Introduction Let
More informationTorsion subgroups of rational elliptic curves over the compositum of all cubic fields
Torsion subgroups of rational elliptic curves over the compositum of all cubic fields Andrew V. Sutherland Massachusetts Institute of Technology April 7, 2016 joint work with Harris B. Daniels, Álvaro
More informationCongruent number elliptic curves of high rank
Michaela Klopf, BSc Congruent number elliptic curves of high rank MASTER S THESIS to achieve the university degree of Diplom-Ingenieurin Master s degree programme: Mathematical Computer Science submitted
More informationHow many elliptic curves can have the same prime conductor? Alberta Number Theory Days, BIRS, 11 May Noam D. Elkies, Harvard University
How many elliptic curves can have the same prime conductor? Alberta Number Theory Days, BIRS, 11 May 2013 Noam D. Elkies, Harvard University Review: Discriminant and conductor of an elliptic curve Finiteness
More informationINTRODUCTION TO ELLIPTIC CURVES
INTRODUCTION TO ELLIPTIC CURVES MATILDE LALÍN Abstract. These notes correspond to a mini-course taught by the author during the program Two Weeks at Waterloo - A Summer School for Women in Math. Please
More information#A20 INTEGERS 11 (2011) ON CONGRUENT NUMBERS WITH THREE PRIME FACTORS. Lindsey Reinholz
#A20 INTEGERS 11 (2011) ON CONGRUENT NUMBERS WITH THREE PRIME FACTORS Lindsey Reinholz Department of Mathematics and Statistics, University of British Columbia Okanagan, Kelowna, BC, Canada, V1V 1V7. reinholz@interchange.ubc.ca
More informationELLIPTIC CURVES BJORN POONEN
ELLIPTIC CURVES BJORN POONEN 1. Introduction The theme of this lecture is to show how geometry can be used to understand the rational number solutions to a polynomial equation. We will illustrate this
More informationFORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS
Sairaiji, F. Osaka J. Math. 39 (00), 3 43 FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS FUMIO SAIRAIJI (Received March 4, 000) 1. Introduction Let be an elliptic curve over Q. We denote by ˆ
More information1.6.1 What are Néron Models?
18 1. Abelian Varieties: 10/20/03 notes by W. Stein 1.6.1 What are Néron Models? Suppose E is an elliptic curve over Q. If is the minimal discriminant of E, then E has good reduction at p for all p, in
More informationLECTURE 22, WEDNESDAY in lowest terms by H(x) = max{ p, q } and proved. Last time, we defined the height of a rational number x = p q
LECTURE 22, WEDNESDAY 27.04.04 FRANZ LEMMERMEYER Last time, we defined the height of a rational number x = p q in lowest terms by H(x) = max{ p, q } and proved Proposition 1. Let f, g Z[X] be coprime,
More informationQuadratic points on modular curves
S. Alberts Quadratic points on modular curves Master thesis Supervisor: Dr. P.J. Bruin Date: November 24, 2017 Mathematisch Instituut, Universiteit Leiden Contents Introduction 3 1 Modular and hyperelliptic
More informationTorsion Points of Elliptic Curves Over Number Fields
Torsion Points of Elliptic Curves Over Number Fields Christine Croll A thesis presented to the faculty of the University of Massachusetts in partial fulfillment of the requirements for the degree of Bachelor
More informationOn the polynomial x(x + 1)(x + 2)(x + 3)
On the polynomial x(x + 1)(x + 2)(x + 3) Warren Sinnott, Steven J Miller, Cosmin Roman February 27th, 2004 Abstract We show that x(x + 1)(x + 2)(x + 3) is never a perfect square or cube for x a positive
More informationImproving Lenstra s Elliptic Curve Method
Oregon State University Masters Paper Improving Lenstra s Elliptic Curve Method Author: Lukas Zeller Advisor: Holly Swisher August 2015 Abstract In this paper we study an important algorithm for integer
More informationCounting points on elliptic curves: Hasse s theorem and recent developments
Counting points on elliptic curves: Hasse s theorem and recent developments Igor Tolkov June 3, 009 Abstract We introduce the the elliptic curve and the problem of counting the number of points on the
More informationGALOIS GROUPS ATTACHED TO POINTS OF FINITE ORDER ON ELLIPTIC CURVES OVER NUMBER FIELDS (D APRÈS SERRE)
GALOIS GROUPS ATTACHED TO POINTS OF FINITE ORDER ON ELLIPTIC CURVES OVER NUMBER FIELDS (D APRÈS SERRE) JACQUES VÉLU 1. Introduction Let E be an elliptic curve defined over a number field K and equipped
More informationOn a Sequence of Nonsolvable Quintic Polynomials
1 3 47 6 3 11 Journal of Integer Sequences, Vol. 1 (009), Article 09..8 On a Sequence of Nonsolvable Quintic Polynomials Jennifer A. Johnstone and Blair K. Spearman 1 Mathematics and Statistics University
More informationVisibility and the Birch and Swinnerton-Dyer conjecture for analytic rank one
Visibility and the Birch and Swinnerton-Dyer conjecture for analytic rank one Amod Agashe February 20, 2009 Abstract Let E be an optimal elliptic curve over Q of conductor N having analytic rank one, i.e.,
More informationElliptic curves and Hilbert s Tenth Problem
Elliptic curves and Hilbert s Tenth Problem Karl Rubin, UC Irvine MAA @ UC Irvine October 16, 2010 Karl Rubin Elliptic curves and Hilbert s Tenth Problem MAA, October 2010 1 / 40 Elliptic curves An elliptic
More informationVojta s conjecture and level structures on abelian varieties
Vojta s conjecture and level structures on abelian varieties Dan Abramovich, Brown University Joint work with Anthony Várilly-Alvarado and Keerthi Padapusi Pera ICERM workshop on Birational Geometry and
More informationON POONEN S CONJECTURE CONCERNING RATIONAL PREPERIODIC POINTS OF QUADRATIC MAPS
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 43, Number 1, 2013 ON POONEN S CONJECTURE CONCERNING RATIONAL PREPERIODIC POINTS OF QUADRATIC MAPS BENJAMIN HUTZ AND PATRICK INGRAM ABSTRACT. The purpose of
More informationOn some congruence properties of elliptic curves
arxiv:0803.2809v5 [math.nt] 19 Jun 2009 On some congruence properties of elliptic curves Derong Qiu (School of Mathematical Sciences, Institute of Mathematics and Interdisciplinary Science, Capital Normal
More informationMT5836 Galois Theory MRQ
MT5836 Galois Theory MRQ May 3, 2017 Contents Introduction 3 Structure of the lecture course............................... 4 Recommended texts..................................... 4 1 Rings, Fields and
More informationOn Partial Lifting and the Elliptic Curve Discrete Logarithm Problem
On Partial Lifting and the Elliptic Curve Discrete Logarithm Problem Qi Cheng 1 and Ming-Deh Huang 2 1 School of Computer Science The University of Oklahoma Norman, OK 73019, USA. Email: qcheng@cs.ou.edu.
More informationON MATCHINGS IN GROUPS
ON MATCHINGS IN GROUPS JOZSEF LOSONCZY Abstract. A matching property conceived for lattices is examined in the context of an arbitrary abelian group. The Dyson e-transform and the Cauchy Davenport inequality
More informationLECTURE 15, WEDNESDAY
LECTURE 15, WEDNESDAY 31.03.04 FRANZ LEMMERMEYER 1. The Filtration of E (1) Let us now see why the kernel of reduction E (1) is torsion free. Recall that E (1) is defined by the exact sequence 0 E (1)
More informationMATH 431 PART 2: POLYNOMIAL RINGS AND FACTORIZATION
MATH 431 PART 2: POLYNOMIAL RINGS AND FACTORIZATION 1. Polynomial rings (review) Definition 1. A polynomial f(x) with coefficients in a ring R is n f(x) = a i x i = a 0 + a 1 x + a 2 x 2 + + a n x n i=0
More informationThe Birch & Swinnerton-Dyer conjecture. Karl Rubin MSRI, January
The Birch & Swinnerton-Dyer conjecture Karl Rubin MSRI, January 18 2006 Outline Statement of the conjectures Definitions Results Methods Birch & Swinnerton-Dyer conjecture Suppose that A is an abelian
More informationAnatomy of torsion in the CM case
Anatomy of torsion in the CM case (joint work with Abbey Bourdon and Pete L. Clark) Paul Pollack Illinois Number Theory Conference 2015 August 14, 2015 1 of 27 This talk is a report on some recent work
More informationFactorization in Integral Domains II
Factorization in Integral Domains II 1 Statement of the main theorem Throughout these notes, unless otherwise specified, R is a UFD with field of quotients F. The main examples will be R = Z, F = Q, and
More informationGALOIS GROUPS OF CUBICS AND QUARTICS (NOT IN CHARACTERISTIC 2)
GALOIS GROUPS OF CUBICS AND QUARTICS (NOT IN CHARACTERISTIC 2) KEITH CONRAD We will describe a procedure for figuring out the Galois groups of separable irreducible polynomials in degrees 3 and 4 over
More informationTotal 100
Math 542 Midterm Exam, Spring 2016 Prof: Paul Terwilliger Your Name (please print) SOLUTIONS NO CALCULATORS/ELECTRONIC DEVICES ALLOWED. MAKE SURE YOUR CELL PHONE IS OFF. Problem Value 1 10 2 10 3 10 4
More informationThe Congruent Number Problem and the Birch and Swinnerton-Dyer Conjecture. Florence Walton MMathPhil
The Congruent Number Problem and the Birch and Swinnerton-Dyer Conjecture Florence Walton MMathPhil Hilary Term 015 Abstract This dissertation will consider the congruent number problem (CNP), the problem
More informationAlgebra Homework, Edition 2 9 September 2010
Algebra Homework, Edition 2 9 September 2010 Problem 6. (1) Let I and J be ideals of a commutative ring R with I + J = R. Prove that IJ = I J. (2) Let I, J, and K be ideals of a principal ideal domain.
More informationMATH 3030, Abstract Algebra FALL 2012 Toby Kenney Midyear Examination Friday 7th December: 7:00-10:00 PM
MATH 3030, Abstract Algebra FALL 2012 Toby Kenney Midyear Examination Friday 7th December: 7:00-10:00 PM Basic Questions 1. Compute the factor group Z 3 Z 9 / (1, 6). The subgroup generated by (1, 6) is
More informationMathematics for Cryptography
Mathematics for Cryptography Douglas R. Stinson David R. Cheriton School of Computer Science University of Waterloo Waterloo, Ontario, N2L 3G1, Canada March 15, 2016 1 Groups and Modular Arithmetic 1.1
More informationALGEBRA QUALIFYING EXAM SPRING 2012
ALGEBRA QUALIFYING EXAM SPRING 2012 Work all of the problems. Justify the statements in your solutions by reference to specific results, as appropriate. Partial credit is awarded for partial solutions.
More informationINFINITELY MANY ELLIPTIC CURVES OF RANK EXACTLY TWO. 1. Introduction
INFINITELY MANY ELLIPTIC CURVES OF RANK EXACTLY TWO DONGHO BYEON AND KEUNYOUNG JEONG Abstract. In this note, we construct an infinite family of elliptic curves E defined over Q whose Mordell-Weil group
More informationA Diophantine System and a Problem on Cubic Fields
International Mathematical Forum, Vol. 6, 2011, no. 3, 141-146 A Diophantine System and a Problem on Cubic Fields Paul D. Lee Department of Mathematics and Statistics University of British Columbia Okanagan
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #23 11/26/2013
18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #23 11/26/2013 As usual, a curve is a smooth projective (geometrically irreducible) variety of dimension one and k is a perfect field. 23.1
More informationELLIPTIC CURVES OVER FINITE FIELDS
Further ELLIPTIC CURVES OVER FINITE FIELDS FRANCESCO PAPPALARDI #4 - THE GROUP STRUCTURE SEPTEMBER 7 TH 2015 SEAMS School 2015 Number Theory and Applications in Cryptography and Coding Theory University
More informationFall 2004 Homework 7 Solutions
18.704 Fall 2004 Homework 7 Solutions All references are to the textbook Rational Points on Elliptic Curves by Silverman and Tate, Springer Verlag, 1992. Problems marked (*) are more challenging exercises
More informationRECIPES FOR TERNARY DIOPHANTINE EQUATIONS OF SIGNATURE (p, p, k)
RECIPES FOR TERNARY DIOPHANTINE EQUATIONS OF SIGNATURE (p, p, k) MICHAEL A. BENNETT Abstract. In this paper, we survey recent work on ternary Diophantine equations of the shape Ax n + By n = Cz m for m
More informationFINITE GROUPS AND EQUATIONS OVER FINITE FIELDS A PROBLEM SET FOR ARIZONA WINTER SCHOOL 2016
FINITE GROUPS AND EQUATIONS OVER FINITE FIELDS A PROBLEM SET FOR ARIZONA WINTER SCHOOL 2016 PREPARED BY SHABNAM AKHTARI Introduction and Notations The problems in Part I are related to Andrew Sutherland
More informationClass numbers of cubic cyclic. By Koji UCHIDA. (Received April 22, 1973)
J. Math. Vol. 26, Soc. Japan No. 3, 1974 Class numbers of cubic cyclic fields By Koji UCHIDA (Received April 22, 1973) Let n be any given positive integer. It is known that there exist real. (imaginary)
More information15 Elliptic curves and Fermat s last theorem
15 Elliptic curves and Fermat s last theorem Let q > 3 be a prime (and later p will be a prime which has no relation which q). Suppose that there exists a non-trivial integral solution to the Diophantine
More informationWhy Should I Care About Elliptic Curves?
Why Should I Care About? Edray Herber Goins Department of Mathematics Purdue University August 7, 2009 Abstract An elliptic curve E possessing a rational point is an arithmetic-algebraic object: It is
More informationTHE JOHNS HOPKINS UNIVERSITY Faculty of Arts and Sciences FINAL EXAM - SPRING SESSION ADVANCED ALGEBRA II.
THE JOHNS HOPKINS UNIVERSITY Faculty of Arts and Sciences FINAL EXAM - SPRING SESSION 2006 110.402 - ADVANCED ALGEBRA II. Examiner: Professor C. Consani Duration: 3 HOURS (9am-12:00pm), May 15, 2006. No
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics MOD p REPRESENTATIONS ON ELLIPTIC CURVES FRANK CALEGARI Volume 225 No. 1 May 2006 PACIFIC JOURNAL OF MATHEMATICS Vol. 225, No. 1, 2006 MOD p REPRESENTATIONS ON ELLIPTIC
More informationQuasi-reducible Polynomials
Quasi-reducible Polynomials Jacques Willekens 06-Dec-2008 Abstract In this article, we investigate polynomials that are irreducible over Q, but are reducible modulo any prime number. 1 Introduction Let
More information