ON SUPERSINGULAR ELLIPTIC CURVES AND HYPERGEOMETRIC FUNCTIONS

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1 ON SUPERSINGULAR ELLIPTIC CURVES AND HYPERGEOMETRIC FUNCTIONS KEENAN MONKS Abstract The Legedre Family of ellitic curves has the remarkable roerty that both its eriods ad its suersigular locus have descritios i terms of the F z hyergeometric fuctio Here we study ellitic curves ad ellitic itegrals with resect to the F z ad F z hyer- geometric fuctios, ad rove that the suersigular -ivariat locus of certai families of ellitic curves are give by these fuctios Itroductio ad statemet of results Let be a rime ad F a field of characteristic A ellitic curve E/F is called suersigular if the grou EF has o -torsio Oe well-kow ad widely studied family of ellitic curves is the Legedre Family, which we deote by E : y = xx x for 0, We defie its suersigular locus by S, := 0 F suersigular E 0 0 The locus S, ad the eriods of E have beautiful ad simle descritios i terms of the hyergeometric fuctio F a b c z = a b z c! Here a, b, z C, c C \ Z 0, x 0 =, ad x = xx + x + is the Pochhammer symbol For ay rime, defie a b F c z a b z mod c! 000 Mathematics Subject Classificatio H, C60

2 KEENAN MONKS It is atural to study hyergeometric fuctios related to ellitic itegrals ellitic itegral of the first kid is writte as Kk = π 0 dθ k si θ From [] we have the followig idetities for aroriate rages of k: K π k = F k, K 8 9 k = h F π kk h, K π k = kk F kk, kk K π k = kk F J Here k = k, J = kk 7kk A ad h is the smaller of the two solutios of 9 8h = J 6h 6 h For the locus S,, it is a classical result see [] ad [6] that S, F mod El-Guidy ad Oo studied i [] the family of curves defied by E : y = x x + They roved a result aalogous to the classical case, amely 0 F 0 F suersigular E 0 Here we rove two other cases of this heomeo that cover the other hyergeometric fuctios related to ellitic itegrals listed i We defie the followig families of ellitic curves: E : y + yx + y = x, E : y = x 7x 7 We ote that if {0, 7} res {0, }, the E res E is sigular

3 ON SUPERSINGULAR ELLIPTIC CURVES AND HYPERGEOMETRIC FUNCTIONS We also defie, for each i {,, } ad all rimes, S,i := 0 0 F suersigular E i 0 Geeralizig the results above, we rove the followig for E ad E Theorem For ay rime, we have S, F 7 mod Theorem For ay rime, we have the followig: If, mod, the S, c F if 7, mod, the where c = 6 S, c 7 F + d mod, mod,, ad d = 0,,, for,, 7, mod resectively Remark The j-ivariat of E Notice that E its j-ivariat is 0 ad udefied whe j = 78 is 7 ad the j-ivariat of E is sigular whe = 0 ad j = 0 Also, E Throughout, let be rime Prelimiaries is 78 is sigular whe Defiitio The Hasse ivariat of a ellitic curve defied by fw, x, y = 0 is the coefficiet of wxy i fw, x, y Likewise, the Hasse ivariat of a curve defied by y = fx is the coefficiet of x i fx Remark The rojective comletios of E ad E are ad wy + wxy + x = 0 wy x + 7w x + 7w = 0 We have the followig well-kow characterizatio of suersigular ellitic curves see [], [6], [7]

4 KEENAN MONKS Lemma A ellitic curve E is suersigular if ad oly if its Hasse ivariat is 0 It is well-kow that two ellitic curves defied over F are isomorhic if ad oly if they have the same j-ivariat Recall the followig formula for the umber of isomorhism classes of suersigular ellitic curves over F see [7] We write = m + 6ɛ + δ, where ɛ, δ {0, } Lemma U to isomorhism, there are exactly m + ɛ + δ suersigular ellitic curves i characteristic Remark It is kow that δ = oly whe mod ie whe 0 is a suersigular j-ivariat ad ɛ = oly whe mod whe 78 is a suersigular j-ivariat Also, i all cases m = Proof of Mai Results We first rove several relimiary lemmas Lemma There are exactly distict values of for which E is suersigular over F Proof To calculate the degree of S,, we must cosider how may differet values for yield a curve E with a give suersigular j-ivariat From [] we have that je = 7 ad that the discrimiat E = 8 7 Hece there are usually four -ivariats for a give j-ivariat, but there are certai excetios Sice the oly roots of i this case are 0 ad 7, we kow that these ad 78 are the oly ossible j-ivariats for which there are less tha four corresodig -ivariats However, there are four distict values of for which je = 7 Also, oly = 8 ± 6 gives a value of 78 for j, so the corresodece is -to- i this case As metioed reviously, the curve is sigular for = 0, so the oly value of that will give a j-ivariat of 0 is = The corresodece is thus oe-to-oe for j = 0 Usig the ideas of Lemma, we have that each of the m suersigular j- ivariats is obtaied from four suersigular -ivariats, δ ca come from at most oe -ivariat, ad ɛ comes from two, if ay, -ivariats Thus the total umber of -ivariats, ad the degree of S, easily verified that this equals, is m + δ + ɛ = + δ + ɛ It is for every rime, ad so we are doe

5 ON SUPERSINGULAR ELLIPTIC CURVES AND HYPERGEOMETRIC FUNCTIONS Lemma There are exactly distict values of for which E is suersigular over F Proof The j-ivariat of E is je = 78 This is a oe-to-oe corresodece from -ivariats to j-ivariats for j 78 Also, the secial cases j = 0 ad j = 78 do ot aly here, for the curve is sigular for these resective j-ivariats Thus by Lemma there are exactly values of for which E is suersigular Proof of Theorem The curve E ca be defied as fw, x, y = wy + wxy + w y x = 0 We first comute its Hasse ivariat A geeral term i the exasio of wy + wxy + w y x is of the form wy a wxy b w y c x d, where a + b + c + d = I order for this to be a costat multile of a ower of wxy, we must have a = c = d Thus the terms that we are cocered with are of the form wy w y x wxy = wxy For a give, there are ways to choose which of the fw, x, y factors we obtai each of the wy, w y, ad x terms from Summig over all ossible values of, we determie the Hasse ivariat to be!!!!! mod mod

6 6 KEENAN MONKS By defiitio, we have F 7 7! x However, if >, the will aear i the umerator of either or makig those terms cogruet to 0 modulo, so F Thus F mod! 7 So by Lemma, is a root of F E mod!!! mod is cogruet modulo to the Hasse ivariat of E 7, 0 mod if ad oly if is suersigular, ie if ad oly if is a root of S, x Sice the least ower of i F 7 is, / F has the same roots as F 7 7, with the excetio of 0, which is ot a -ivariat as show i Lemma, ad thus is ot a root of S, The degree of F 7 is exactly Sice the degree of S, is also by Lemma, it follows that F 7 c S, mod However, c is sice F 7 is moic, so we are doe Proof of Theorem Assume, mod The fuctio fz = F z satisfies the

7 ON SUPERSINGULAR ELLIPTIC CURVES AND HYPERGEOMETRIC FUNCTIONS 7 secod order differetial equatio z z d f dz + z df dz f = 0 Substitutig z = x, we see that gx = F Hece, h = F x x d g dx + x x satisfies d h d x dg dx g = 0 dh d h = 0 satisfies The fuctio h is a Lauret series i with -itegral ratioal cofficiets However, its reductio modulo yields a olyomial i This olyomial must satisfy the reductio of modulo, so F = F satisfies d F d + df d F 0 6 A similar calculatio shows that F = F 7 mod also satisfies the same differetial equatio whe 7, mod Now, to comute the Hasse ivariat, we cosider a geeral x term i the exasio of x 7x 7 This is of the form x 7x 7, where 6 For a give i this rage, there are exactly ways to choose which of the x 7x 7 factors the x terms ad 7x terms came from Summig over all yields the Hasse ivariat to be = 6 7, ito which we ca substitute = k, ad usig the fact that mod, we obtai 6 7 k k= k k k

8 8 KEENAN MONKS We show the Hasse ivariat satisfies the differetial equatio by showig that for ay t, the t term i the resultig exasio is cogruet to 0 mod Let ck = 7 k k k k The the t term has coefficiet d d ct dt dt ct + d dt ct d dt ct + 6 ct + 6 ct, which we exad to obtai t t t t t t t t This is cogruet to 0 modulo if ad oly if t 7 t 6 tt t + 6 t t + t + 6 t t t + t is also cogruet to 0 We ow exad the first biomials to obtai t t + t t t! t + t t + t! 7 t 6 t t + 6 t t + 6 which is cogruet to 0 modulo if ad oly if t t 7 t t t t + 6 t t t + 6 is as well A similar cacellatio method o the remaiig biomials shows that it is sufficiet to rove t + t t + t 7 t + t t 6 t t + 0 6, mod, which is easily verified Thus the Hasse ivariat satisfies the same secod order differetial equatio as 7 both F ad F For >, otice that both the Hasse ivariat ad the trucated hyergeometric fuctios have

9 ON SUPERSINGULAR ELLIPTIC CURVES AND HYPERGEOMETRIC FUNCTIONS 9 o term with a degree less tha For each case, this imlies that the trucated olyomials are cogruet modulo to the Hasse ivariat u to multilicatio by a costat For the case =, it is easy to comute that F =, ad the Hasse ivariat is, so this roerty still holds Therefore, we kow that the two trucated hyergeometric fuctios have the same roots modulo as the Hasse ivariat, so by Lemma, is a root of the hyergeometric fuctios if ad oly if E is suersigular Notice that F, res F 7 has the same roots as multilied by the resective trucated fuctios with the excetio of 0, which is as desired sice E 0 is sigular Also, whe, mod the degree of F c such that S, c F is, so by Lemma, there exists a costat mod Similarly for rimes 7, mod, S, c F 7 mod Fially, we exlicitly comute the costat c Notice that S, is moic, so c is the coefficiet of the leadig term i F as the costat term i F For > will be cogruet to 0 modulo Hece, the costat term of, which is the same, oe of or F =!, is!

10 0 KEENAN MONKS For mod, we have! +! mod Also,! mod Therefore, c = 6! mod For mod, c =! 6 + mod 6 + A similar method ca be used to comute c mod whe mod ad whe mod, which comletes the roof Examles Examle Theorem Cosider = 9 The suersigular j-ivariats mod 9 are kow to be 8 ad 7 From formula we fid that the values of where j = 8 are ± i 6 oly this j corresods to j = 78 The values of for which j = 7 are 6 ± ad ± Thus S 9, = + i 6 i mod mod 9

11 ON SUPERSINGULAR ELLIPTIC CURVES AND HYPERGEOMETRIC FUNCTIONS The Hasse ivariat is the coefficiet of wxy 8 i wy + wxy + w y x 8 This is H S 9, mod 9 I additio, F S 6 9, mod 9 Examle Theorem Cosider = 9, which is modulo The suersigular j-ivariats mod 9 are kow to be 0, 7 corresodig to 78, 8, 7, 8, ad From formula, we fid the -ivariats corresodig to 8, 7, 8, ad are,,, ad, resectively Note that we do ot iclude the cases j = 0 or j = 78 for sigularity reasos Thus S 9, = mod 9 The Hasse ivariat is the coefficiet of x 9 i x 7x 7 8 This is H S 9, mod 9 I additio, F 7 9 Also, c 9 8 mod S 9, mod 9 mod 9 Refereces [] D Husemöller, Ellitic Curves, Graduate Texts i Mathematics, Sriger-Verlag 00 [] C Leo, A Trace Formula for Certai Hecke Oerators ad Gaussia Hyergeometric Fuctios, rerit [] D McCarthy, F Hyergeometric Fuctios ad Periods of Ellitic Curves, Iteratioal Joural of Number Theory, Volume: 6, Issue: 00, ages 6-70 [] A El-Guidy, K Oo, Hasse Ivariats for the Clause Ellitic Curves, rerit [] J M Borwei, P B Borwei, Pi ad the AGM, Caadia Mathematical Society Series of Moograhs ad Advaced Texts, Wiley-Itersciece, 987 [6] J H Silverma, The Arithmetic of Ellitic Curves, Graduate Texts i Mathematics, Sriger- Verlag 986 [7] L C Washigto, Ellitic Curves: Number Theory ad Crytograhy, Chama & Hall, 00 7 N James St, Hazleto, PA 80 address: keeaeek@gmailcom