Elliptic Curves Spring 2015 Problem Set #1 Due: 02/13/2015

Transcription

1 Ellitic Curves Sring 2015 Problem Set #1 Due: 02/13/2015 Descrition These roblems are related to the material covered in Lectures 1-2. Some of them require the use of Sage, and you will need to create an account at the SageMathCloud. If you use the address listed on your student registration when you create the account, you should find an invitation to join the class roject; if not, lease let me know. Materials related to the roblem sets will be automatically be made available to you in this roject folder as the course roceeds. You will find examles of Sage usage in the roblem descritions below, and there is a wealth of information to be found on the Sage website, including tutorials. In many cases simly googling how do I do X in sage? will quickly answer a question. For this first roblem set, choose one of Problems 1 and 2, do both Problems 3 and 4. Be sure also to comlete Problem 5, which is a required survey whose results will hel shae future roblem sets and lectures. You can use the latex source for this roblem set as a temlate for writing u your solutions SageMathCloud includes a latex editor, but feel free to use the latex environment of your choice. Be sure to ut your name on your solution (you can relace the due date in the header with your name). If you should discover a tyo/error in the roblem set or lecture notes, lease let me know as soon as ossible the first erson to find each error will receive 1 3 oints of extra credit. Problem 1. Edwards curves (20 oints) (a) Show that (c, 0) is a oint of order 4 on the Edwards curve x 2 + y 2 = c 2 (1 + dx 2 y 2 ). (b) Modify the grou law so that (c, 0) is the identity and (0, c) is a oint of order 4. This defines a new grou on the same set of elements (rational oints on the curve). Show that this grou is isomorhic to the standard one. (c) Let n be the integer formed by the last 2 digits of your student ID, and let x 3 = n2 1 n 2 + 1, y (n 1)2 3 = n 2 + 1, d = (n2 + 1) 3 (n 2 4n + 1) (n 1) 6 (n + 1) 2. Show that P = (x 3, y 3 ) is a oint of order 3 on the curve x 2 + y 2 = 1 + dx 2 y 2 over Q. (d) Find a oint of order 12 on the curve in art (c). Problem 2. Automorhisms (20 oints) Recall that an endomorhism is a homomorhism from a grou to itself, and an automorhism is an endomorhism that is also an isomorhism. The automorhisms of a grou G form a grou Aut(G) under comosition, and the endomorhisms of an additive abelian grou G form a ring End(G) in which multilication is comosition (so the roduct of α, β End(G) is defined by (αβ)(g) = α(β(g))), and addition is addition in the grou (so the sum of α, β End(G) is defined by (α + β)(g) = α(g) + β(g)).

2 Let E : y 2 = x 3 + Ax + B be an ellitic curve defined over an algebraically closed field k whose characteristic is not 2 or 3. In each of the roblems below, assume that the secified ma sends the identity element (0 : 1 : 0) of E(k) to itself (a necessary requirement of any endomorhism). (a) Show that the ma (x, y) (x, y) is an automorhism of order 2. (b) Assume B = 0 and suose i k satisfies i 2 = 1. Show that α: (x, y) ( x, iy) is an automorhism of order 4, and that the equation α 2 +1 = 0 holds in End(E(k)). (c) Assume A = 0 and suose ζ k satisfies ζ 3 = 1 and ζ 1. Show that the ma β : (x, y) (ζx, y) is an automorhism of order 6 and that the equation β 2 β + 1 = 0 holds in End(E(k)). Problem 3. Quadratic twists (40 oints) Let E/k be an ellitic curve in short Weierstrass form E : y 2 = x 3 + Ax + B. The quadratic twist of E by c k is the ellitic curve over k defined by the equation E c : cy 2 = x 3 + Ax + B. (a) Using a linear change of variables, show that E c is isomorhic to an ellitic curve in standard Weierstrass form y 2 = x 3 + A x + B, and exress A and B in terms of A and B and c. Verify that E c is not singular. (b) For any grou G and ositive integer n, we use G[n] to denote the n-torsion subgrou of G, consisting of all elements whose order divides n. Prove that E(k)[2] = E c (k)[2]. (c) Prove that if c is a square in k, then E and E c are isomorhic over k (via a linear change of variables with coefficients in k). Conclude that E and E c are always isomorhic over k( c), whether c is a square in k or not (in general, curves defined over k are said to be twists if they are isomorhic over some extension of k). (d) Now assume that k is a finite field F Z/Z, for some odd rime, and let t be the unique integer for which #E(F ) = + 1 t, where #E(F ) is the cardinality of the grou of F -rational oints of E. Prove that ( ) c #E c (F ) = + 1 t, ( where c ) is the Legendre symbol, which is equal to +1 when c is a square modulo and 1 when it is not (note that c F is never zero modulo ). (e) Continuing with k = F, show that if t 0 then E c and E c ( ) ( ) are isomorhic if and only if c = c (this is also true when t = 0 but you need not consider this case).

3 Problem 4. (40 oints) Sato-Tate for ellitic curves with comlex multilication Recall from Lecture 1 that the ellitic curve E/Q defined by y 2 = x 3 + Ax + B has good reduction at a rime whenever does not divide (E) = 16(4A B 2 ). For each rime of good reduction, let a = + 1 #E (F ) and x = a /, where E denotes the reduction of E modulo. To create an ellitic curve defined by a short Weierstrass equation in Sage, you can tye E=ElliticCurve([A,B]). To check whether the ellitic curve E has good reduction at, use E.has good reduction(), and to comute a, use E.a(). In this roblem you will investigate the distribution of x for some ellitic curves over Q to which the Sato-Tate conjecture does not aly. These are ellitic curves with comlex multilication (CM for short), a term we will define later in the course. In Sage you can check for CM using E.has cm(). (a) Let E/Q be the curve defined by y 2 = x Comute a list of a values for the rimes 200 where E has good reduction (all but 2 and 3). The following block of Sage code does this. E=ElliticCurve([0,1]) for in rimes(0,200): if E.has_good_reduction(): rint, E.a() You will notice that many of the a values are zero. Give a conjectural criterion for the rimes for which a = 0. Verify your conjecture for all rimes 2 10 where E has good reduction. (b) Given a bound B, the nth moment statistic M n of x is defined as the average value of x n over rimes B where E has good reduction. In Lecture 1 we saw that for an ellitic curve over Q without comlex multilication, the sequence of moment statistics M 0, M 1, M 2,... aear to converge to the integer sequence 1, 0, 1, 0, 2, 0, 5, 0, 14, 0, 42,..., whose odd terms are 0 and whose even terms are the Catalan numbers. Your goal is to determine an analogous sequence for ellitic curves over Q with comlex multilication. To do this efficiently, use the E.alist() method in Sage. The following block of code comutes the moment statistics M 0,..., M 10 of x using the bound B = 2 k. k=12 E=ElliticCurve([0,1]) A=E.alist(2ˆk) P=rime_range(0,2ˆk) X=[A[i]/sqrt(RR(P[i])) for i in range(0,len(a))] M=[sum([aˆn for a in X])/len(X) for n in [0..10]] rint M

4 (note that use of RR(P[i]) to coerce the rime P[i] to a real number before taking its square root without this Sage will use a symbolic reresentation of the square root as an algebraic number, which is not what we want). With this aroach we are also including a few a values at bad rimes (which will yield x 0), but this is harmless as long as we make B = 2 k large enough. By comuting moment statistics using bounds B = 2 k with k = 12, 16, 20, 24, determine the integers to which the first ten moment statistics aear to converge, and come u with a conjectural formula for the nth moment (if you get stuck on this, look at arts (e) and (f) below). Then test your conjecture by comuting the 12th and 14th moment statistics and comaring the results. (c) Reeat the analysis in arts (a) and (b) for the following ellitic curves over Q: y 2 = x 3 595x , y 2 = x 3 608x , y 2 = x x You will robably need to look at more a values than just u to 200 in order to formulate a criterion for the a that are zero. Do the x moment statistics for these ellitic curves aear to converge to the same sequence you conjectured in art (b)? (d) Pick one of the three curves from art (c) and take its quadratic twist by the last four digits of your student ID. Does this change the sequence of a values? Does it change the moment statistics of x? (e) Recall that the secial orthogonal grou SO(2) consists of all matrices of the form R θ = ( ) cos θ sin θ sin θ cos θ. To generate a random matrix in SO(2), one simly icks θ uniformly at random from the interval [0, 2π); this is the Haar measure on SO(2), the unique robability measure that is invariant under the grou action. Derive a formula for the nth moment of the trace of a random matrix in SO(2) by integrating the nth ower of the trace of R θ over all θ [0, 2π). Be sure to normalize by 1/(2π) so that M 0 = 1. (f) The normalizer N(SO(2)) of SO(2) in the secial unitary grou SU(2) consists of all matrices of the form R θ and JR θ, where J = ( ) i 0 0 i. Derive a formula for the nth moment of the trace of a random matrix in N(SO(2)) (under the Haar measure on N(SO(2)) one icks θ [0, 2π) uniformly at random and then takes R θ or JR θ with equal robability). Comare the results to the formula you conjectured in art (b). Problem 5. Survey Comlete the following survey by rating each of the roblems you solved on a scale of 1 to 10 according to how interesting you found the roblem (1 = mind-numbing, 10 = mind-blowing ), and how difficult you found the roblem (1 = trivial, 10 = brutal ). Also estimate the amount of time you sent on each roblem to the nearest half hour. Problem 1 Problem 2 Problem 3 Problem 4 Interest Difficulty Time Sent

5 Please rate each of the following lectures that you attended on a scale of 1 to 10, according to the quality of the material (1= ointless, 10= riceless ), the quality of the resentation (1= eic fail, 10= erfection ), and the novelty of the material to you (1= old hat, 10= all new ). Date Lecture Toic Material Presentation Novelty 2/3 Introduction 2/5 Grou Law Feel free to record any additional comments you have on the roblem sets or lectures; in articular, how you think they might be imroved.