HASSE INVARIANTS FOR THE CLAUSEN ELLIPTIC CURVES AHMAD EL-GUINDY AND KEN ONO Astract. Gauss s F x hyergeometric function gives eriods of ellitic curves in Legendre normal form. Certain truncations of this hyergeometric function give the Hasse invariants for these curves. Here we study another form, which we call the Clausen form, and we rove that certain truncations of F x and F x in F [x] are related to the characteristic Hasse invariants.. Introduction We egin y recalling three tyes of hyergeometric functions which give invariants of the Legendre normal form ellitic curves. E L λ : y = xx x λ, λ C \ {0, }. If n is a nonnegative integer, then define γ n y { if n = 0, γ n := γγ + γ + γ + n if n. The assical hyergeometric function in arameters α,..., α h, β,..., β j C is defined y hfj α, α,... α h α n α n α n α h n x := xn β,... β j β n β n β j n n!. n=0 By the theory of ellitic integrals, it is well known that Gauss s hyergeometric function F x := F x gives the eriods for examle, see age 8 of [7] of the Legendre normal form ellitic curves. In articular, if we denote the real eriod of E L λ y Ω L λ, then for 0 < λ < we have. Ω L λ = π F λ. Truncated hyergeometric functions also give invariants for these curves. Throughout let e an odd rime. Recall that an ellitic curve in characteristic is said to e suersingular if 000 Mathematics Suject Classification. Primary G, H. Key words and hrases. Hyergeometric functions, Hasse invariants. The second author thanks the suort of the NSF, the Hilldale Foundation and the Manasse family.
AHMAD EL-GUINDY AND KEN ONO it has no -torsion over F. We define the relevant truncated hyergeometric functions y. F tr x := n=0 n x n, n! and we define the characteristic Hasse invariant for the Legendre normal form ellitic curves y. H L x := x λ. λ F E L λ suersingular It turns out that H L x is in F [x], and it satisfies for examle, see age 6 of [7].5 H L x F tr x mod. There is a third kind of hyergeometric function, the finite field hyergeometric function. These functions also give information aout the Legendre normal form ellitic curves. We first recall their definition which is due to J. Greene []. If is a rime ower and A and B A are two Dirichlet characters on F extended so that A0 = B0 = 0, then let e the B normalized Jacoi sum A B := B JA, B = B AxB x. x F Here B is the comlex conjugate of B. If A 0,..., A n, and B,..., B n are characters on F, then the finite field hyergeometric function in these arameters is defined y n+fn ff A0, A,... A n x := A0 χ A χ An χ χx. B,... B n χ B χ B n χ Here χ denotes the sum over all characters χ of F. It has een oserved y many authors see [6], [], [8], [9], [], and [], to name a few that the Gaussian analog of a assical hyergeometric series with rational arameters is otained y relacing each with a character χ n n of order n and a with n χa n. Let ɛ e the trivial character on F and let φ e the character of order. Then the finite field analog of F x is More generally, we let F ff x :=.6 n+f ff n x := n+ F ff n χ φ φ ɛ x. φ, φ,... φ x. ɛ,... ɛ M. Koike roved [9] that if 5 is a rime for which E L λ has good reduction, and is a ower of then.7 F ff λ = φ a L λ;,
HASSE INVARIANTS FOR THE CLAUSEN ELLIPTIC CURVES where a L λ; is the trace of Froenius at for E L λ. We have now seen that the F x, F tr x and F ff x hyergeometric functions encode some of the most imortant invariants for the Legendre normal form ellitic curves. Loosely seaking, we have that Periods if =,.8 π F x = Hasse invariants if = tr, Traces of Froenius if = ff. Motivated y.7, the second author identified [, ] a second form, the Clausen form, which is similarly related to finite field hyergeometric functions. These curves are given y.9 E C λ : y = x x + λ. If λ {0, }, then E C λ is an ellitic curve with discriminant and j-invariant E C λ = 6λλ + and je C λ = 6λ λλ +. If E C λ has good reduction at a rime 5 and if is a ower of, then the second author roved see Theorem 5 of [] the following analog of.7: λ.0 + φ λ + F ff = a C λ;, λ + where a C λ; is the trace of the Froenius at for E C λ. It could e comuted y the formula. a C λ; = F φ + λ. λ λ+ Remark. This result imlies that the F ff sum which gives the trace of Froenius on E C λ. Greene and R. Evans [5] have otained a generalization of this henomenon for further F ff hyergeometric functions. is essentially the suare of the character Remark. A secial case of.0, which can e viewed as a finite field analog of the Clausen Theorem see Theorem., was roved first y Greene and Stanton [6]. Motivated y.8 and.0, D. McCarthy studied the relationshi etween F x := F x and the E C λ, and he roved that this assical hyergeometric function gives see Theorem. of [0] the suare of real eriods of these curves. Namely, if λ > 0, then. π λ + λ F = Ω C λ, λ + where Ω C λ is the real eriod of E C λ. This result may e interreted in terms of local zeta functions for a certain family of K surfaces [].
AHMAD EL-GUINDY AND KEN ONO To otain the full analogy with.8, we now show that the suares of Hasse invariants for the Clausen curves are given y truncated hyergeometric functions. If is an odd rime, then define the truncated hyergeometric function in arameters α,..., α h, β,..., β j C y hfj tr α, α,... α h x := β,... β j Following our earlier convention, we let. F tr x := F tr and we have the Hasse invariant. H C x := n=0 x α n α n α n α h n β n β n β j n xn n!. = λ F E C λ suersingular n=0 x λ. n x n, n! The following theorem, which nicely comlements.0 and., comletes the analogies of.8 for the Clausen ellitic curves and gives.5 π λ + λ Periods if =, F = Hasse invariants λ + if = tr, Traces of Froenius if = ff. Theorem.. If is an odd rime, then H C x is in F [x], and it satisfies x + x F tr H C x mod, x + where is the recirocal roduct of inomial coefficients :=. Remark. Since suersingular ellitic curves have models defined over F for examle, see. 69 of [7] or. 7 of [], it follows that the irreducile factors of F tr x in F [x] are linear or uadratic. The roof of Theorem. shows that.6 H C x F tr x mod, which in turn imlies that all of the roots of this truncated hyergeometric function are in F and are simle. Furthermore, we note that McCarthy see the roof of corollary. in [0] also otained the assical analog of.6, namely.7 Ω C λ = π F λ.
HASSE INVARIANTS FOR THE CLAUSEN ELLIPTIC CURVES 5 It is natural to wonder if a similar relation holds for Gaussian hyergeometric function with aroriate arameters. Indeed the following result is valid. Proosition.. Let e an odd rime and λ Q \ {0, } e such that ord λλ + = 0. If is a ower of such that mod and χ is a character of order defined on F, then we have.8 a C λ; = F ff χ χ ɛ λ. Proof. Following the roof of Theorem 5 in [] we see that if we define a function fx y fx := φ χ φ χ x χ, χ χ then for ord µ = 0 we have.9 f = φ µ x F \{ µ } χ φ x µ x + µ µx µ 5 µ. Dividing the right side y φ µ 6 =, setting λ := µ, and alying. we get µ Following Greene and Evans [5], we set Thus fλ + = φ a C ; λ φ λ +. F φ, ɛ ; x := fx + φ x..0 F φ, ɛ ; λ + = φ a C ; λ. Since χ = φ, we deduce using well-known roerties of Jacoi sums from Theorem. in [5] that. F φ, ɛ ; x = φ χ F ff χ χ x. ɛ However, for x 0,, Theorem.i of [] gives. F ff χ χ x = χ ɛ F ff χ χ ɛ x, and the result follows y setting x = λ + and noting that χ = φ = for mod. Remark. Note that Theorem.5 of [5], together with.0 imly that a C λ; = F ff χ χ λ. ɛ
6 AHMAD EL-GUINDY AND KEN ONO However, it seems one must go through an argument as in the roof aove in order to otain the more recise formula.8. Remark. A formula similar to.8 was stated, without an exlicit roof, in [9] for the family E K λ : y = x + x + λ x. Note that E K λ is the -uadratic twist of E Cλ. It follows that we have the following analog of.8 and.5, where the last line is valid only when an analog of exists; i.e. when mod.. π F λ = Periods Hasse invariants Traces of Froenius if =, if = tr, if = ff. Remark. It is well-known that the Gauss sum Gχ is the finite field analog of the gamma function see section.0 of [] for instance. Since Gφ = φ and Γ = π, we see that φ is indeed the Gaussian analog of π. On the other hand, the congruence for truncated hyergeometric series is one etween olynomials, rather than comlex numers. Hence its main content is that the zeros of the truncated hyergeometric series are the suersingular locus. The constant is merely resent to make the truncated hyergeometric series monic. It can t really e given an interretation as a truncated analog of the gamma function at the arameter since the corresonding constant in.8 is just, which simly means that the truncation in that case haens to e monic without the need for further normalization. Remark. As noted in [0] and confirmed y.7 and.8, the Gaussian analog of the real eriod is the negative of the trace of Froenius. Examle. The set of suersingular Clausen curves for = is and so it follows that H C x := {E C 5, E C 8, E C, E C, E C 7}, λ F E C λ suersingular x λ = x 5x 8x x x 7. One directly finds that = mod, and we have 508 x x + F tr = + 89 x + 8 x + 887 5 x + + 856856770755 x 57 Also, F tr x x 0 + 5x 9 + 9x 8 + + 9x + x + H C x mod. = 6 x + 05 0 x 55 68 x + 55 90 x 909907 670886 x5 x 5 + 9x + 8x + x + 7x + H C x mod.
HASSE INVARIANTS FOR THE CLAUSEN ELLIPTIC CURVES 7. Proof of Theorem. We refer to the ellitic curves E C λ as Clausen curves ecause they arise naturally in connection with an identity of Clausen relating F and F assical hyergeometric functions. First we recall this identity, along with other crucial oservations... Nuts and olts. We egin y recalling the following identity of Clausen. Throughout this section we view the assical hyergeometric series as a formal one. Theorem.. [Clausen] We have that F x = x F x. x Proof. A theorem of Clausen see. 86 of [] imlies that α β α + β α β F α + β α + β + x = F α + β + x. By the assical F transformation see. 0 of [], we have that a a c F c x = x a F x. c x The aim follows y letting α = β =, and y then letting a = = and c =. This theorem imlies a mod version for truncated hyergeometric functions. Corollary.. If is an odd rime, then x + x F tr F tr x + Proof. After relacing x y x x+ x in Theorem., use the fact that x + x + mod, x. To rove Theorem., we reuire the following descrition of F tr mod. x For the remainder of the aer, for an odd rime, set m := and m := m. Lemma.. If is an odd rime then m F tr x =0 m m Proof. It suffices to show, for 0 m, that m m! This early holds when = 0. x mod. mod. mod.
8 AHMAD EL-GUINDY AND KEN ONO The roof now follows y induction. Begin y noticing the following identities: m = m + + m, = +, + = + + Using these identities, it then suffices to show that m m m + + + This follows from the elementary congruence:. + + mod. 6 + 8m m + m m m + m + + + mod. Finally notice that m = 0 if > m... Proof of Theorem.. It is well known see Chater V of [] that E C λ is suersingular at a rime 5 if and only if the coefficient of xy is zero modulo in f λ x, y, where. f λ x, y := y x x + λ. The following lemma gives a formula for that articular coefficient. Lemma.. If is an odd rime, then the coefficient of xy modulo in f λ x, y is m m =0 m m λ. Proof. Oviously, we have that f λ x, y = c x c x + λ c y c. c c=0 Now oserve that xy occurs only in the middle of this sum, namely where c = m, and so it suffices to comute the coefficient of x in x m x + λ m. Now notice that m m x m x m + λ m = m a x a m x m λ a a=0 =0 m = m a m m λ x n. a n=0 a+m =n
HASSE INVARIANTS FOR THE CLAUSEN ELLIPTIC CURVES 9 One easily checks that the coefficient when n = is m m m m λ. =0 To comlete the roof, notice that m m > m. mod, and that m = 0 whenever Proof of Theorem.. By Corollary. and Lemma., we have that x + x F tr F tr m x + x m m To comlete the roof, it suffices to show that m 0 H C x m m m m =0 m =0 m x x mod. mod. By Lemma. and the receding discussion, oth olynomials have the same roots over F, and so they agree u to a multilicative constant. Since oth olynomials are monic y construction, they must e eual in F [x]. Remark. It follows from the roof aove that H C x has degree m. Acknowledgements The authors thank Marie Jameson for ointing out tyograhical errors in an earlier version of this aer. They are also grateful to the referee for helful suggestions. References [] S. Ahlgren, K. Ono, and D. Penniston, Zeta functions of an infinite family of K surfaces, Amer. J. Math. 00, ages 5-68. [] G. E. Andrews, R. Askey, R. Roy, Secial Functions,Camridge Univ. Press, Camridge, 998. [] W. Bailey, Generalized hyergeometric series, Camridge Univ. Press, Camridge, 95. [] J. Greene, Hyergeometric series over finite fields, Trans. Amer. Math. Soc. 0 987, ages 77-0. [5] J. Greene and R. Evans, Clausen s theorem and hyergeometric functions over finite fields, Finite Fields Al. 5 009, ages 97-09. [6] J. Greene and D. Stanton, A character sum evaluation and Gaussian hyergeometric series, J. Numer Theory 986, 6-8. [7] D. Husemöller, Ellitic Curves, Sringer Verlag, Graduate Texts in Mathematics, 00 [8] M. Ishiashi, H. Sato and K. Shiratani, On the Hasse invariants of ellitic curves, Kyushu J. Math.,8 99, no., ages 07-. [9] M. Koike, Orthogonal matrices otained from hyergeometric series over finite fields and ellitic curves over finite fields, Hiroshima Math. J. 5 995, ages -5. [0] D. McCarthy, F hyergeometric series and eriods of ellitic curves, Int. J. of Numer Th., 6 00, no., ages 6-70. [] K. Ono, Values of Gaussian hyergeometric series, Trans. Amer. Math. Soc. 50 998, ages 05-.
0 AHMAD EL-GUINDY AND KEN ONO [] K. Ono, We of Modularity: Arithmetic of the Coefficients of Modular Forms and -Series, Amer. Math. Soc., 00. [] J. Rouse, Hyergeometric functions and ellitic curves, Ramanujan J., 006, no., ages 97-05. [] J. Silverman, The arithmetic of ellitic curves, Sringer-Verlag, New York, 986. Deartment of Mathematics, Faculty of Science, Cairo University, Giza, Egyt 6 Deartment of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 570 E-mail address: a.elguindy@gmail.com, ono@math.wisc.edu