SOME ELEMENTARY PROPERTIES OF THE DISTRIBUTION OF THE NUMBERS OF POINTS ON ELLIPTIC CURVES OVER A FINITE PRIME FIELD

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1 SOME ELEMENTARY PROPERTIES OF THE DISTRIBUTION OF THE NUMBERS OF POINTS ON ELLIPTIC CURVES OVER A FINITE PRIME FIELD SAIYING HE AND J MC LAUGHLIN Abstract Lt 5 b a rim and for a, b F, lt E a,b dnot th llitic curv ovr F with quation y 2 = x 3 a x b As usual dfin th trac of Frobnius a, a, b by #E a,b F = 1 a, a, b W us lmntary facts about xonntial sums and known rsults about binary quadratic forms ovr finit filds to valuat th sums t F a, t, b, t F a 1, a, t, a2, t, b, 1 a2, a, t and 1 a3, t, b for rims in various congrunc classs As an xaml of our rsults, w rov th following: Lt 5 mod 6 b rim and lt b F Thn 1 a 3, t, b = b 1 Introduction Lt 5 b a rim and lt F b th finit fild of lmnts For a, b F, lt E a,b dnot th llitic curv ovr F with quation y 2 = x 3 a x b Dnot by E a, b F th st of F rational oints on th curv E a, b and dfin th trac of Frobnius, a, by th quation #E a, b F = 1 a A siml counting argumnt maks it clar that 11 a = x F x 3 a x b whr ṗ dnots th Lgndr symbol W rcall som of th arithmtic rortis of th distribution of a Th following thorm is du to Hass [4]: Thorm 1 Th intgr a satisfis 2 a 2 Dat: August 13th, Mathmatics Subjct Classification 11G20 Ky words and hrass Ellitic Curvs, Finit Filds 1,

2 2 SAIYING HE AND J MC LAUGHLIN Sinc w wish to look at how a varis as th cofficints a and b of th llitic curv vary, it is convnint for our uross to writ a for th llitic curv E a,b as a, a, b Th following rsult is wll known an asy consqunc of th rmarks on ag 36 of [3], for xaml Proosition 1 Lt th function f : Z N 0 b dfind by stting 12 fk = #{a, b F F : a, a, b = k} Thn for ach intgr k, 1 2 fk Th following rsult can b found in [2] ag 57 Proosition 2 Dfin th function f 1 : Z N 0 by stting 13 f 1 k = #{a, b F F \ {0, 0} : a, a, b = k} Thn for ach intgr k, f 1 k = f 1 k Th following rsult is also known [3], ag 37, for xaml Proosition 3 Lt v b a quadratic non-rsidu modulo Thn a, a, b = a, v 2 a, v 3 b To bttr undrstand th distribution of th a, a, b it maks sns to study th momnts Th j -invariant of th llitic curv E a,b is dfind by j = a 3 4a 3 27b 2, rovidd 4a 3 27b 2 0 Michl showd in [7] that if {E at, bt : t F } is a on-aramtr family of llitic curvs with at and bt olynomials in t such that at 3 jt := 4at 3 27bt 2, is non-constant, thn In [2] Birch dfind t F a 2, at, bt = 2 O 3/2 S R = 1 a, b=0 for intgral R 1, and rovd [ 1 x 3 ] 2R ax b x=0

3 POINTS ON ELLIPTIC CURVES OVER A FINITE PRIME FIELD 3 Thorm 2 1 For 5, S 1 = 1 2, S 2 = , S 3 = , S 4 = , S 5 = τ, whr τ is Ramanujan s τ-function Thorm 2 valuats sums of th form 1 a, b=0 a2r, a, b in trms of and ths rsults wr drivd by Birch as consquncs of th Slbrg trac formula In this rsnt ar w instad us lmntary facts about xonntial sums and known rsults about binary quadratic forms ovr finit filds to valuat th sums t F a, t, b, t F a, a, t, 1 a2, t, b, 1 a2, a, t and 1 a3, t, b, for rims in articular congrunc classs In articular, w rov th following thorms Thorm 3 Lt 5 b a rim, and a, b F Thn i t F a, t, b = b, ii t F a, a, t = 0 This rsult is lmntary but w rov it for th sak of comltnss Thorm 4 Lt 5 mod 6 b rim and lt b F Thn 14 1 a 2, t, b = 1 1 Thorm 5 Lt 5 b rim and lt a F Thn 15 1 a 2, a, t = 1 3 3a Thorm 4 and Thorm 5 could b dducd from Thorm 2, but w bliv it is of intrst to giv lmntary roofs that do not us th Slbrg trac formula Thorm 6 Lt 5 mod 6 b rim and lt b F Thn 1 2 a 3, t, b = 2 2 b 1 In [2], Birch incorrctly omittd th factor of 1 in his statmnt of Thorm 2

4 4 SAIYING HE AND J MC LAUGHLIN 2 Proof of th Thorms W introduc som standard notation Dfin j/ := x2πij/, so that 1 { jt, j, 21 = 0, j, = 1 Dfin {, 1 mod 4, 22 G = i, 3 mod 4 Lmma 1 Lt ṗ dnot th Lgndr symbol, modulo Thn 23 z = 1 G P 1 d=1 d dz Proof S [1], Thorm 115 and Thorm 152 W will occasionally us th fact that if H is a subst of F, d 24 = d d H d F \H W will also occasionally mak us of som imlications of th Law of Quadratic Rcirocity s [5], ag 53, for xaml Thorm 7 Lt and q b odd rims Thn a 1 = 1 1/2 b 2 = 1 2 1/8 c q = 1 1/2q 1/2 q W now rov Thorms 3, 4, 5 and 6, Thorm 3 Lt 5 b a rim, and a, b F Thn i t F a, t, b = b, ii t F a, a, t = 0 Proof i From 11 and 23, it follows that 1 1 d dx 3 b a,t,b = G P t F x F d=1 t F a,t,b = G P d=1 t F tdx Th innr sum ovr t is zro unlss x = 0, in which cas it quals to Th lft sid thrfor can b simlifid to giv 1 d db b =

5 POINTS ON ELLIPTIC CURVES OVER A FINITE PRIME FIELD 5 Th last quality follows from 23 ii: From 11 and 23, it follows that 1 1 d dx 3 ax a,a,t = G t F x F d=1 t F dt = 0 Th innr sum ovr t is qual to 0, by 21, sinc 1 d 1 Th rsult at ii follows also, in th cas of rims 3 mod 4, from th fact that a, a, t = a, a, t Howvr, this is not th cas for rims 1 mod 4 For xaml, {a 13, 1, t : 0 t 12} = { 6, 4, 2, 1, 0, 5, 1, 1, 5, 0, 1, 2, 4} Th rsults in Thorm 3 ar almost crtainly known, although w hav not bn abl to find a rfrnc Thorm 4 Lt 5 mod 6 b rim and lt b F Thn 1 1 a 2, t, b = 1 Proof From 11 and 23 it follows that t F a 2,t,b = 1 G 2 1 d 1,d 2 =1 d1 d 2 d1 x 3 1 b d 2x 3 2 b x 1,x 2 F td1 x 1 d 2 x 2 t F Th innr sum ovr t is zro, unlss x 1 d 1 1 d 2x 2 mod, in which cas it quals Thus t F a 2,t,b = G 2 1 d 1,d 2 =1 d1 d 2 bd1 d 2 x 2 F d 2 1 d 2x 3 2 d2 1 d2 2 Sinc th ma x x 3 is on-to-on on F, whn 5mod 6, th x 3 2 in th innr sum can b rlacd by x 2 Thus th innr sum is zro unlss d 2 2 d2 1 0mod, in which cas it quals It follows that a 2,t,b = 2 1 d 2 1 b2d1 1 d 2 G 2 1 bd1 d 1 t F d 1 =1 d 1 =1 = 2 1 G = G 2 1 W hav onc again usd 23 to comut th sums, noting that th sums abov start with d 1 = 1 Th rsult now follows sinc /G 2 1 = 1 for all rims 3

6 6 SAIYING HE AND J MC LAUGHLIN Rmarks: 1 It is clar that th rsults will rmain tru if at = t is rlacd by any function at which is on-to-on on F 2 It is mor difficult to dtrmin th valus takn by t F a 2,t,b for rims 1mod 6 This is rincially bcaus th ma x x 3 is not on-to-on on F for ths rims so that 21 cannot b usd so asily to simlify th summation, but also bcaus th answr dnds on which cost b blongs to in F /F 3 Bfor roving th nxt thorm, it is ncssary to rcall a rsult about quadratic forms ovr finit filds Lt q b a owr of an odd rim and lt η dnot th quadratic charactr on F q so that if q =, an odd rim, thn ηc = c/, th Lgndr symbol Th function v is dfind on F q by { 1, b F 25 vb = q, q 1, b = 0 Suos fx 1,, x n = n a ij x i x j, with a ij = a ji, i,j=1 is a quadratic form ovr F q, with associatd matrix A = a ij and lt dnot th dtrminant of A f is non-dgnrat if 0 Thorm 8 Lt f b a non-dgnrat quadratic form ovr F q, q odd, in an vn numbr n of indtrminats Thn for b F q th numbr of solutions of th quation fx 1,, x n = b in Fq n is 26 q n 1 vbq n 2/2 η 1 n/2 Proof S [6], Thorm 5 Lt 5 b rim and lt a F Thn 1 a 2, a, t = 1 3 Proof Onc again 11 and 23 giv that t F a 2, a, t = 1 G 2 1 d 1,d 2 =1 d1 d 2 x 1,x 2 F 3a d1 x 3 1 ax 1 d 2 x 3 2 ax 2 t bd1 d 2 t F

7 POINTS ON ELLIPTIC CURVES OVER A FINITE PRIME FIELD 7 Th innr sum ovr t is zro, unlss d 1 d 2 mod, in which cas it quals Thus a 2, a, t = 1 1 d1 x 3 1 G 2 a x 1 x 3 2 a x 2 27 t F x 1, x 2 F d 1 =1 = 1 d1 x 1 x 2 x 2 1 x 1x 2 x 2 2 a x 1, x 2 F d 1 =1 W hav usd th fact that /G 2 1 = 1 for all rims 3 Th innr sum ovr d 1 quals 1, unlss on of th factors x 1 x 2, x 2 1 x 1x 2 x 2 2 a quals 0, in which cas th sum is 1 Th quation x 1 = x 2 has solutions and, by 26 with q =, n = 2, fx 1, x 2 = x 2 1 x 1x 2 x 2 2 and A = 1 1/2 1/2 1 1, th quation x 2 1 x 1x 2 x 2 2 = a has /4 = 3 solutions Howvr, w nd to b carful to avoid doubl counting and to xamin whn x 2 1 x 1x 2 x 2 2 = a has a solution with x 1 = x 2 Th quation 3x 2 1 = a will hav two solutions if 3a = 1 and non if 3a = 1 1 Thus Hnc th numbr of solutions to th quation 3x 2 1 = a is 3a th numbr of solutions to x 1 x 2 x 2 1 x 1x 2 x 2 2 a = 0 is Thus t F a 2, a, t = a = 2 1 3a Th right sid now simlifis to giv th rsult 3 3 3a Bfor roving Thorm 6, w nd som rliminary lmmas Lmma 2 Lt 5 mod 6 b rim Thn 28 1 d,,f=1 f1 f y,z F = 1 3a 1 d y fz 3 y 3 fz 3 1 1

8 8 SAIYING HE AND J MC LAUGHLIN 1 d,,f=1 f f y,z F d fz f 2 y y 3 1 Proof If z = 0, th lft sid of 28 bcoms 1 f1 f dy S 0 : = d,,f=1 y F 1 f1 f y1 2 = 1,f=1 y F 1 = 1 f2 f 1 f 2 f=1 f=1 1 = 1 2f f=1 f=1 1 = Th scond quality follows sinc {y 3 : y F } = {y : y F } for th rims bing considrd, th third quality follows from 21 and th last quality follows from 24 If z 0, thn th lft sid of 28 quals 29 S 1 := 1 d,,f,z=1 1 d,,f,z=1 f1 f f1 f y F y F d y fz 3 y 3 fz 3 = dz 3 yz 1 f 3 yz 1 3 f Now rlac y by yz and thn z 3 by z justifid by th sam argumnt as abov and finally by f to gt this last sum quals 1 d,,f,z=1 1 f f y F dfz f 2 y 1 3 y 3 1 W wish to xtnd th last sum to includ z = 0 If w st z = 0 on th right sid of th last quation and dnot th rsulting sum by rs and sum ovr d and y w gt that rs = 1 1,f=1 1 f 1

9 POINTS ON ELLIPTIC CURVES OVER A FINITE PRIME FIELD 9 1 = f 1 f=1 =1 f=1 Rlac f by f 1 1 in th scond sum abov and thn f rs = 1 1 =1 f= f = 1 1 =1 f=1 1 1 = = It follows that th lft sid of 29 quals f f d,,f=1 y,z F and thus that th lft sid of 28 quals 210 S 0 S 1 = 1 1 = y,z F y,z F dfz f 2 y 1 3 y 3 1 d,,f=1, 1 1 f f dfz f 2 y 1 3 y d,,f=1 f f dfz f 2 y y 3 1 Th scond quality in 210 follows uon rlacing y by y 1 and thn by 1

10 10 SAIYING HE AND J MC LAUGHLIN Lmma 3 Lt 5 mod 6 b rim Thn 211 S := whr 1 d,,f=1 = 2 1 S := f f 1 1 2, f=2, 2 f 2 y,z F 1 dfz f 2 y y f1 f 1 f 1S, Proof Uon changing th ordr of summation slightly, w gt that 1 f f 1 dfz f S 2 y y 3 1 =,f=1 d=1 y,z F If y = 0, th innr doubl sum ovr d and z is zro, unlss f = ±1, if which cas it quals 1 and th right sid of 211 quals = =1 =1 By similar rasoning, if y = 1, th right sid of 211 also quals Thus S = f f 1 dfz f 2 y y 3 1 y=1,f=1 = 2 1 1,f=1 1 1 fy 1 f d=1 z F 1 2 yy 1 1 y=1 d=1 z F dfz 2 f 2 y 1 f 2 whr th last quality follows uon rlacing f by fy 1 1 and by y 1 Th innr sum ovr d and z is zro unlss 2 f 2 y 1 f 2 = 0,,

11 POINTS ON ELLIPTIC CURVES OVER A FINITE PRIME FIELD 11 in which cas th innr sum is 1 W distinguish th cass 2 = f 2 and 2 f 2 If 2 = f 2, thn ncssarily 2 = f 2 = 1 and th sum on th right sid of 212 bcoms yy 1 2y 1 1 y=1 If 2 f 2 thn 0 2 2y = y = f f 2, and sinc y 0, 1, w xclud f 2 = 1 and 2 = 1 Aftr substituting for y in th sum in th final xrssion in 212, w find that 214 S = 2 1 whr 215 S := 1 1 2, f=2, 2 f S, 1 f1 f 1 f Lmma 4 Lt 5 mod 6 b rim and lt S b as dfind in Lmma 3 Thn f1 f 1 f S = =0 f= Proof Clarly w can rmov th rstrictions f, f 1 and 1 frly If w st f =, w hav that 2 1 f1 f 1 f =, f=2, = f = = Th last quality follows from 24 Thus S = 2, f=2 1 f1 f 1 f

12 12 SAIYING HE AND J MC LAUGHLIN If f is st qual to 0 in th sum abov w gt 2 =2 = 1 If f is st qual to -1 in this sum w gt 2 22 = =2 Thus 4 2 = f1 f 1 f S = =2 f= If w st = 0 in this latst sum w gt 1 f1 f 1 f = f=0 If w st = 1 in this sum w gt 1 21 f2 f 1 f = f=0 Thus 1 f=0,f 1 1 f=0,f f1 f 1 f S = =0 f= f = f 2 = Lmma 5 Lt 5 mod 6 b rim Thn f1 f 1 f =0 f=0 = 2 1

13 POINTS ON ELLIPTIC CURVES OVER A FINITE PRIME FIELD 13 Proof If f is rlacd by f 1 and thn is rlacd by f, th valu of th doubl sum abov dos not chang Thus f1 f 1 f =0 f= f 1 ff = =0 f= f 1f = =0 f= = =0 f=0 W valuat th innr sum using ff = 1 G 2 f=0 1 d1 d 2 = 1 G 2 = G 2 = G 2 d 1,d 2 =1 1 d 2 = d 2 =1 1 1 d ff f=0 d1 d 2 f=0 d 1,d 2 =1 1 d2 1 d2 1 d1 f d 2 1 f fd1 d 2 Th nxt-to-last quality follows sinc th sum ovr f in th rvious xrssion is 0, unlss d 1 = d 2, in which cas this sum is Th sum ovr d 2 quals 1 if = 1 and quals 1 othrwis Hnc th sum at 216 quals G =0 = G = G 2 1 = 1, f=0 th last quality following from th rmark aftr 27

14 14 SAIYING HE AND J MC LAUGHLIN Corollary 1 Lt S and S b as dfind in Lmma 3 Thn i S = , 1 2 ii S = Proof Lmmas 4 and 5 and th fact that 3 = 1 if 5 mod 6 giv i Lmma 3 and art i giv ii Thorm 6 Lt 5 mod 6 b rim and lt b F Thn 1 2 b 217 a 3, t, b = 2 2 Proof Lt g b a gnrator of F It is a siml mattr to show, using 11, that 1 1 a 3, t, b = a 3, t, bg b=1 Thus th statmnt at 217 is quivalnt to th statmnt 1 1 b a 3, t, b = Lt S dnot th lft sid of 218 From 11 and 23 it follows that 1 1 x 3 t x b y 3 t y b z 3 t z b b S = = 1 G 3 = 1 G 2 = 1 G 2 b=1 x,y,z F 1 d,,f=1 1 d,,f=1 1 d,,f=1 df df x,y,z,t F dx 3 tx y 3 ty fz 3 tz b F b bd f d f dx 3 tx y 3 ty fz 3 tz df x,y,z,t F d f x,y,z F dx 3 y 3 fz 3 tdx y fz t F

15 POINTS ON ELLIPTIC CURVES OVER A FINITE PRIME FIELD 15 Th innr sum is zro, unlss dx y fz = 0 in F, in which cas it quals Uon ltting x = d 1 y fz, rlacing by d and f by f, w gt that S = G 2 1 d,,f=1 = 2 1 G 2 G 2 = 2 1 G 2 f1 f d,,f=1 = 2 1 G 2 = 1 1 f f y,z F y,z F G 2 S 2 2, d y fz 3 y 3 fz 3 dfz f 2 y y which was what ndd to b shown, by 218 Th scond quality abov follows from Lmma 2 Abov S is as dfind in Lmma 3 and in th nxtto-last quality w usd Corollary 1, art ii In th last quality w usd onc again th fact that /G 2 1 = 1 3 Concluding Rmarks Lt 5 mod 6 b rim, b F and k b an odd ositiv intgr Dfin 1 b f k = a k, t, b It is not difficult to show that th right sid is indndnt of b F By Thorm 6 2 f 3 = 2 2 W hav not bn abl to dtrmin f k for k 5 W do not considr vn k, sinc a formula for ach vn k can b drivd from Birch s work in [2] W conclud with a tabl of valus of f k and small rims 5 mod 6, with th ho of ncouraging othrs to work on this roblm Rfrncs [1] Brndt, Bruc C; Evans, Ronald J; Williams, Knnth S Gauss and Jacobi sums Canadian Mathmatical Socity Sris of Monograhs and Advancd Txts A Wily- Intrscinc Publication John Wily & Sons, Inc, Nw York, 1998 xii583

16 16 SAIYING HE AND J MC LAUGHLIN k Tabl 1 f k for small rims 5 mod 6 and small odd k [2] Birch, B J How th numbr of oints of an llitic curv ovr a fixd rim fild varis J London Math Soc [3] Blak, I F; Sroussi, G; Smart, N P Ellitic curvs in crytograhy Rrint of th 1999 original London Mathmatical Socity Lctur Not Sris, 265 Cambridg Univrsity Prss, Cambridg, 2000 xvi204 [4] Hass, H Bwis ds Analogons dr Rimannschn Vrmutung für di Artinschn und F K Schmidtschn Kongrunzztafunktionn in gwissn litischn Fälln Vorläufig Mittilung, Nachr Gs Wiss Göttingn I, Math-hys Kl Fachgr I Math Nr , [5] Irland, Knnth; Rosn, Michal A classical introduction to modrn numbr thory Scond dition Graduat Txts in Mathmatics, 84 Sringr-Vrlag, Nw York, 1990 xiv389 [6] Lidl, Rudolf; Nidrritr, Harald Finit filds With a forword by P M Cohn Scond dition Encyclodia of Mathmatics and its Alications, 20 Cambridg Univrsity Prss, Cambridg, 1997 xiv755 [7] Michl, Phili Rang moyn d famills d courbs llitiqus t lois d Sato-Tat Monatsh Math , no 2, [8] Millr, Stvn J On- and two-lvl dnsitis for rational familis of llitic curvs: vidnc for th undrlying grou symmtris Comos Math , no 4, Trinity Collg, 300 Summit Strt, Hartford, CT addrss: SaiyingH@trincolldu Mathmatics Dartmnt, Trinity Collg, 300 Summit Strt, Hartford, CT addrss: jamsmclaughlin@trincolldu

INCOMPLETE KLOOSTERMAN SUMS AND MULTIPLICATIVE INVERSES IN SHORT INTERVALS. xy 1 (mod p), (x, y) I (j)

INCOMPLETE KLOOSTERMAN SUMS AND MULTIPLICATIVE INVERSES IN SHORT INTERVALS T D BROWNING AND A HAYNES Abstract W invstigat th solubility of th congrunc xy (mod ), whr is a rim and x, y ar rstrictd to li