Elliptic Curves Spring 2017 Problem Set #1

Transcription

1 Ellitic Curves Srig 017 Problem Set #1 These roblems are related to the material covered i Lectures 1-3. Some of them require the use of Sage; you will eed to create a accout at the SageMathCloud. You will fid eamles of Sage usage i the roblem descritios below, ad there is a wealth of iformatio to be foud o the Sage website, icludig tutorials. I may cases simly googlig how do I do X i sage? will quickly aswer a questio. Istructios: Do oe of Problems 1 ad ad both Problems 3 ad 4. Fially, comlete Problem 5, which is a short survey whose results will hel shae future roblem sets ad lectures. You ca use the late source for this roblem set as a temlate for writig u your solutios. SageMathCloud icludes a late editor, but feel free to use the late eviromet of your choice. Be sure to ut your ame o your solutio (you ca relace the due date i the header with your ame). Your solutios are to be writte u i late ad submitted as a df-file with a fileame of the form SuramePset1.df. Collaboratio is ermitted/ecouraged, but you must idetify your collaborators, ad ay refereces you cosulted that are ot listed i the course syllabus. The first erso to sot each o-trivial tyo/error i ay of the roblem sets or lecture otes will receive 1-5 oits of etra credit. Problem 1. Chebyshev s theorem (0 oits) Let π() deote the rime coutig fuctio, which for ay real umber couts the umber of rimes. The Prime Number Theorem is the asymtotic statemet π() log which meas lim π()/ = 1. We wo t rove the rime umber theorem i this course, as we are hay to make do with the weaker statemet that there eist c 1, c > 0 such that c 1 π() c (1) log log holds for all sufficietly large, as roved by Chebyshev. I fact, if we take c 1 = (log )/ ad c = 6 log the (1) holds for all, as you will ow rove. (a) Show that for all itegers 1 we have π() π() Y <, where rages over rimes. Coclude that π() π() ( log )/(log ). 1 (b) Usig (a), rove that π( ) 3 / for all itegers 1, ad use this to show for all 1. π() (6 log ) log 1 Whe you coclude somethig o a roblem set, you eed to rove it. 1

2 (c) For itegers 1, let v () deote the largest iteger e 0 such that e. Prove X v (!) =, ad the show that v ( e 1 e ) (log )/(log ). Usig this, rove that holds for all itegers 1. () π() (d) Usig (c), show that for we have log π(). log Problem. Edwards curves (0 oits) (a) Let d be a o-square i a field k of characteristic differet from, ad cosider the Edwards curve E : + y = 1 + d y with (0, 1) as the idetity elemet. Usig the grou law defied i class, show that the ratioal oit (1, 0) has order 4. Give a eamle of a ellitic curve over a fiite field of odd characteristic that is ot isomorhic to ay Edwards curve. (b) Elai how to modify the grou law o E so that (1, 0) is the idetity ad (0, 1) is a oit of order 4, ad show that this grou is isomorhic to the origial oe. (c) Let be the iteger formed by the last digits of your studet ID (but set = if you studet ID eds i 01), ad let 1 ( 1) ( + 1) 3 ( 4 + 1) 3 =, y 3 =, d = ( 1) 6 ( + 1) Show that P = ( 3, y 3 ) is a oit of order 3 o the curve + y = 1 + d y over Q. (d) Fid a oit of order 1 o the curve i art (c). Problem 3. Quadratic twists (40 oits) Let E/k be a ellitic curve i short Weierstrass form E : y = 3 + A + B. The quadratic twist of E by c k is the ellitic curve over k defied by the equatio E c : cy = 3 + A + B. (a) Usig a liear chage of variables, show that E c is isomorhic to a ellitic curve i 3 stadard Weierstrass form y = + A 0 + B 0, ad eress A 0 ad B 0 i terms of A ad B ad c. Verify that E c is ot sigular.

3 (b) For ay grou G ad ositive iteger, we use G[] to deote the -torsio subgrou of G, cosistig of all elemets whose order divides. Prove that E(k)[] = E c (k)[]. (c) Prove that if c is a square i k, the E ad E c are isomorhic over k (via a liear chage of variables with coefficiets i k). Coclude that E ad E c are always isomorhic over k( c), whether c is a square i k or ot (i geeral, curves defied over k are said to be twists if they are isomorhic over some etesio of k). (d) Now assume that k is a fiite field F ' Z/Z, for some odd rime, ad let t be the uique iteger for which #E(F ) = + 1 t, where #E(F ) is the cardiality of the grou of F -ratioal oits of E. Prove that c #E c (F ) = + 1 t, c where is the Legedre symbol, which is equal to +1 whe c is a square modulo ad 1 whe it is ot (ote that c F is ever zero modulo ). (e) Cotiuig with k = F, show that if t 6= 0 the E c ad E c 0 are isomorhic if ad c c oly if 0 = (this is also true whe t = 0 but you eed ot cosider this case). Problem 4. Sato-Tate for ellitic curves with comle multilicatio (40 oits) 3 Recall from Lecture 1 that the ellitic curve E/Q defied by y = + A + B has good reductio at a rime wheever does ot divide Δ(E) = 16(4A 3 + 7B ). For each rime of good reductio, let a = + 1 #E (F ) ad = a /, where E deotes the reductio of E modulo. To create a ellitic curve defied by a short Weierstrass equatio i Sage, you ca tye E=ElliticCurve([A,B]). To check whether the ellitic curve E has good reductio at, use E.has good reductio(), ad to comute a, use E.a(). I this roblem you will ivestigate the distributio of for some ellitic curves over Q to which the Sato-Tate cojecture does ot aly. These are ellitic curves with comle multilicatio (CM for short), a term we will defie later i the course. I Sage you ca check for CM usig E.has cm(). (a) Let E/Q be the curve defied by y = Comute a list of a values for the rimes 00 where E has good reductio (all but ad 3). The followig block of Sage code does this. E=ElliticCurve([0,1]) for i rimes(0,00): if E.has_good_reductio(): rit, E.a() 3

4 You will otice that may of the a values are zero. Give a cojectural criterio for the rimes for which a = 0. Verify your cojecture for all rimes 10 where E has good reductio. (b) Give a boud B, the th momet statistic M of is defied as the average value of over rimes B where E has good reductio. I Lecture 1 we saw that for a ellitic curve over Q without comle multilicatio, the sequece of momet statistics M 0, M 1, M,... aear to coverge to the iteger sequece 1, 0, 1, 0,, 0, 5, 0, 14, 0, 4,..., whose odd terms are 0 ad whose eve terms are the Catala umbers. Your goal is to determie a aalogous sequece for ellitic curves over Q with comle multilicatio. To do this efficietly, use the E.alist() method i Sage. The followig block of code comutes the momet statistics M 0,..., M 10 of usig the boud B = k. k=1 E=ElliticCurve([0,1]) A=E.alist(ˆk) P=rime_rage(0,ˆk) X=[A[i]/sqrt(RR(P[i])) for i i rage(0,le(a))] M=[sum([aˆ for a i X])/le(X) for i [0..10]] rit M (ote that use of RR(P[i]) to coerce the rime P[i] to a real umber before takig its square root without this Sage will use a symbolic reresetatio of the square root as a algebraic umber, which is ot what we wat). With this aroach we are also icludig a few a values at bad rimes (which will yield 0), but this is harmless as log as we make B = k large eough. By comutig momet statistics usig bouds B = k with k = 1, 16, 0, 4, determie the itegers to which the first te momet statistics aear to coverge, ad come u with a cojectural formula for the th momet (if you get stuck o this, look at arts (e) ad (f) below). The test your cojecture by comutig the 1th ad 14th momet statistics ad comarig the results. (c) Reeat the aalysis i arts (a) ad (b) for the followig ellitic curves over Q: y = , y = , y = You will robably eed to look at more a values tha just u to 00 i order to formulate a criterio for the a that are zero. Do the momet statistics for these ellitic curves aear to coverge to the same sequece you cojectured i art (b)? (d) Pick oe of the three curves from art (c) ad take its quadratic twist by the last four digits of your studet ID. Does this chage the sequece of a values? Does it chage the momet statistics of? 4

5 (e) Recall that the secial orthogoal grou SO() cosists of all matrices of the form R cos θ si θ θ =. To geerate a radom matri i SO(), oe simly icks θ si θ cos θ uiformly at radom from the iterval [0, π); this is the Haar measure o SO(), the uique robability measure that is ivariat uder the grou actio. Derive a formula for the th momet of the trace of a radom matri i SO() by itegratig the th ower of the trace of R θ over all θ [0, π). Be sure to ormalize by 1/(π) so that M 0 = 1. (f) The ormalizer N(SO()) of SO() i the secial uitary grou SU() cosists of all matrices of the form R i 0 θ ad JR θ, where J = 0 i. Derive a formula for the th momet of the trace of a radom matri i N(SO()) (uder the Haar measure o N(SO()) oe icks θ [0, π) uiformly at radom ad the takes R θ or JR θ with equal robability). Comare the results to the formula you cojectured i art (b). Problem 5. Survey Comlete the followig survey by ratig each of the roblems you solved o a scale of 1 to 10 accordig to how iterestig you foud the roblem (1 = mid-umbig, 10 = mid-blowig ), ad how difficult you foud the roblem (1 = trivial, 10 = brutal ). Also estimate the amout of time you set o each roblem to the earest half hour. Problem 1 Problem Problem 3 Problem 4 Iterest Difficulty Time Set Please rate each of the followig lectures that you atteded o a scale of 1 to 10, accordig to the quality of the material (1= oitless, 10= riceless ), the quality of the resetatio (1= eic fail, 10= erfectio ), the ace (1= watchig ait dry, 10= head still siig ), ad the ovelty of the material (1= old hat, 10= all ew ). Date Lecture Toic Material Presetatio Pace Novelty /8 Itroductio /13 The grou law /15 Fiite field arithmetic Feel free to record ay additioal commets you have o the roblem sets or lectures; i articular, how you thik they might be imroved. 5

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