ELLIPTIC CURVES, MODULAR FORMS, AND SUMS OF HURWITZ CLASS NUMBERS (APPEARED IN JOURNAL OF NUMBER THEORY )

Transcription

1 ELLIPTIC CURVES, MODULAR FORMS, AND SUMS OF HURWITZ CLASS NUMBERS (APPEARED IN JOURNAL OF NUMBER THEORY BRITTANY BROWN, NEIL J. CALKIN, TIMOTHY B. FLOWERS, KEVIN JAMES, ETHAN SMITH, AND AMY STOUT Abstract. Let H(N denote the Hurwitz class number. It is known that if is a rime, then X H( r =. r < In this aer, we investigate the behavior of this sum with the additional condition r c (mod m. Three different methods will be exlored for determining the values of such sums. First, we will count isomorhism classes of ellitic curves over finite fields. Second, we will exress the sums as coefficients of modular forms. Third, we will maniulate the Eichler-Selberg trace formula for Hecke oerators to obtain Hurwitz class number relations. The cases m =, and are treated in full. Partial results, as well as several conjectures, are given for m = 5 and Introduction and Statement of Theorems We begin by recalling the definition of the Hurwitz class number. Definition 1. For an integer N 0, the Hurwitz class number H(N is defined as follows. H(0 = 1/1. If N 1 or (mod, then H(N = 0. Otherwise, H(N is the number of classes of not necessarily rimitive ositive definite quadratic forms of discriminant N, excet that those classes which have a reresentative which is a multile of the form x +y should be counted with weight 1/ and those which have a reresentative which is a multile of the form x + xy + y should be counted with weight 1/. Several nice identities are known for sums of Hurwitz class numbers. For examle, it is known that if is a rime, then H( r =, (1 r < where the sum is over integers r (both ositive, negative, and zero. See for examle [,. ] or [5,. 15]. In this aer, we investigate the behavior of this sum with additional condition r c (mod m. In articular, if we slit the sum according to the arity of r, then we have Theorem 1. If is an odd rime, then r <, r c (mod H( r =, if c = 0, +, if c = 1. Once we have the above result, we can use the ideas in its roof to quickly rove the next. Theorem. If is an odd rime, r <, r c (mod H( r = 5 7, c ±1 (mod, 6, c + 1 (mod,, c 1 (mod. We will also fully characterize the case m = by roving the following formulae. This work was artially funded by the NSF grant DMS:

2 Theorem. If is rime, then r <, r c (mod H( r =, if c 0 (mod, 1 (mod, 1, if c 0 (mod, (mod, 1, if c ±1 (mod, 1 (mod,, if c ±1 (mod, (mod. We also have a artial characterization for the sum slit according to the value of r modulo 5. Theorem. If is rime, then r <, r c (mod 5 H( r = 1, if c ±( + 1 (mod 5, ± (mod 5,, if c 0 (mod 5, (mod 5. All of the above theorems may be roven by exloiting the relationshi between Hurwitz class numbers and ellitic curves over finite fields. In Section, we will state this relationshi and show how it is used to rove Theorem 1. We will then briefly sketch how to use the same method for the roof of Theorem as well as several cases of Theorem 6 below. In Section, we will use a result about the modularity of certain artial generating functions for the Hurwitz class number to rove Theorem. The interesting thing about this method is that it leads to a far more general result than what is obtainable by the method of Section. Out of this result, it is ossible to extract a version of Theorem for not necessarily rime as well as the following. Theorem 5. If (n,6 = 1 and there exists a rime (mod such that ord (n 1 (mod, then H(n r = σ(n 1, r < n, r c n (mod where we take c n = 0 if n 1 (mod, and c n = 1 or if n (mod. A third method of roof will be discussed in Section, which uses the Eichler-Selberg trace formula. This method will allow us to rove the cases of the following result that remain unroven at the end of Section. Theorem 6. If is rime, then r <, r c (mod 7 H( r = 5, c 0 (mod 7,,5 (mod 7,, c 0 (mod 7, 6 (mod 7,, c ±( + 1 (mod 7,,,,5 (mod 7,, c ± (mod 7, 6 (mod 7. Finally, in Section 5, we list several conjectures, which are strongly suorted by comutational evidence. We also give a few artial results and discuss strategies for future work.. Ellitic Curves and Hurwitz Class Numbers The roofs we give in this section are combinatorial in nature and deend on the following, which is due to Deuring. Theorem 7 (See [] or [1]. If r is an integer such that r <, then the number of isomorhism classes of ellitic curves over F with exactly + 1 r oints is equal to the number of equivalence classes of binary quadratic forms with discriminant r. Corollary 1. For r <, the number of isomorhism classes of ellitic curves over F with exactly + 1 r oints is given by H( r + c r,, where 1/, if r = α for some α Z, c r, = /, if r = α for some α Z, ( 0, otherwise.

3 Thus, the number of isomorhism classes of ellitic curves E/F such that m #E(F is equal to ( H( r + c r,. ( r < r (mod m This is the main fact that we will exloit in this section. Another useful fact that we will exloit throughout the aer is the symmetry of our sums. In articular, H( r = H( r. ( r < r c (mod m r < r c (mod m Proof of Theorem 1. For =, the identities may be checked by direct calculation. For the remainder of the roof, we will assume is rime and strictly greater than. Let N, denote the number of isomorhism classes of ellitic curves over F ossessing -torsion, and recall that E has -torsion if and only if ( + 1 r. Thus, ( N, = H( r + c r,. (5 r < r (mod We will roceed by comuting N,, the number of isomorhism classes of ellitic curves ossessing - torsion over F. Then, we will comute the correction term, c r,. In light of (1 and (5, Theorem 1 will follow. We first recall the relevant background concerning ellitic curves with -torsion over F. The reader is referred to [10] or [15] for more details. If E is an ellitic curve with -torsion then, we can move a oint of order to the origin in order to obtain a model for E of the form The discriminant of such a curve is given by E b,c : y = x + bx + cx. (6 = 16c (b c. (7 We will omit from consideration those airs (b, c for which the resulting curve has zero discriminant since these curves are singular. Following [15,. 6-8], we take c = 16(b c and c 6 = b(9c b. Then since char(f,, E b,c is isomorhic to the curve The curves in this form that are isomorhic to (8 are E : y = x 7c x 5c 6. (8 y = x 7u c x 5u 6 c 6, u 0. (9 Thus, given any ellitic curve, the number of (A,B F for which the given curve is isomorhic to E : y = x + Ax + B is 1 6, if A = 0 and 1 (mod, 1, if B = 0 and 1 (mod, 1, otherwise. We are interested in how many curves E b,c give the same c and c 6 coefficients. Given an ellitic curve E : y = x +Ax+B with -torsion over F, each choice of an order oint to be moved to the origin yields a different model E b,c. Thus, the number of E b,c which have the same c and c 6 coefficients is equal to the number of order oints ossessed by the curves. This is either 1 or deending on whether the curves

4 have full or cyclic -torsion. Thus, the number of (b,c for which E b,c is isomorhic to a given curve is 1 6, c = 0, 1 (mod and -torsion is cyclic, 1, c 6 = 0, 1 (mod and -torsion is cyclic, 1, otherwise with cyclic -torsion, 1, c (10 = 0, 1 (mod and -torsion is full, ( 1, c 6 = 0, 1 (mod and -torsion is full, ( 1, otherwise with full -torsion. The roof of Theorem 1 will follow immediately from the following two roositions. Proosition 1. If > is rime, then the number of isomorhism classes of ellitic curves ossessing -torsion over F is given by +8, if 1 (mod 1, + N, =, if 5 (mod 1, +, if 7 (mod 1,, if 11 (mod 1. Proof. In view of (10, we want to count the number of curves E b,c that fall into each of six categories. Let A 1 denote the number of curves with cyclic -torsion and c = 0, A denote the number of curves with cyclic -torsion and c 6 = 0, A denote the number of curves with cyclic -torsion and c c 6 0, A denote the number of curves with full -torsion and c = 0, A 5 denote the number of curves with full -torsion and c 6 = 0 and A 6 denote the number of curves with c c 6 0. Then N, can be comuted by determining A i for i = 1,...,6 and alying (10. Now, an ellitic curve E b,c has full -torsion if and only if b c is a square modulo. Thus, the number of curves ossessing full -torsion over F is given by 1 1 b=0, c=1 b c 1 [( b c ] + 1 = ( 1(, (11 and the number of curves ossessing cyclic -torsion over F is given by [( b ] c ( 1 1 =. (1 b=0, c=1 b c ( ( Note that if c = 0, then b c c (mod and hence =. Thus, there are 1 nonsingular curves ( b (6 that give c = 0. If a nonsingular curve E b,c ossesses full -torsion and c = 0, then 1 = c = ( ( c = = (. Thus, when 1 (mod, all 1 nonsingular curves Eb,c with c = 0 will have full -torsion, and when (mod, all will have cyclic -torsion. Thus, 0, 1 (mod, Similar comutations lead to A 1 = A = 1, (mod, 1, 1 (mod, 0, (mod. A = 1, A 5 = ( 1.

5 Finally, using (11 and (1, we see that A = A 6 = ( 1 Combining these with (10, the result follows., 1 (mod, ( ( 1, (mod, ( 1( 7, 1 (mod, ( 1( 5, (mod. We now comute the correction term in (5. Proosition. The value of the correction term is given by 10/, 1 (mod 1,, 5 (mod 1, c r, = r < /, 7 (mod 1,, r 0 (mod 0, 11 (mod 1. Proof. By (, we see that each form roortional to x + xy + y contributes / to the sum while each form roortional to x + y contributes 1/. Forms roortional to x + xy + y arise for those r 0 (mod for which there exists α Z\0} such ( that r = α. Thus, = r+αi ( r αi. Recall that factors in Z [ 1+i ] if and only if 1 (mod. For each such, there are 6 solutions to the above, but only with r even. Thus, for 1 (mod, we must add / to the correction term, and for (mod, we add 0 to the correction term. Forms roortional to x + y arise for those r 0 (mod for which there exists α Z\0} such that r = α. Thus, = r +α = ( r + αi ( r αi. Recall that factors in Z[i] if and only if 1 (mod. Given a rime 1 (mod, there are choices for r/ and hence choices for r. So, we have forms and need to add to the correction term. When (mod we add 0 to the correction term. Combining the results in Proositions 1 and, we have H( r = N, r < r 0 (mod r < r 0 (mod c r, =. Theorem 1 now follows from (1. We now give a sketch of the roof of Theorem. The roof uses some of comutations from the roof of Theorem 1. Proof Sketch of Theorem. For c ±1 (mod, the identities follow directly from Theorem 1 and (. By (, r < r c (mod r < r (mod H( r = + 1 ( H( r + c r, is equal to the number of isomorhism classes of ellitic curves over E/F with #E(F. This is equal to the number of classes of curves having full -torsion lus the number of classes having cyclic -torsion over F. 5

6 As with the Proof of Theorem 1, the identities may be checked directly for =. So, we will assume that >. From the roof of Proosition 1, we see that the number of isomorhism classes of curves having full -torsion over F is given by +5, 1 (mod 1,, 5 (mod 1, +, 7 (mod 1,, 11 (mod 1. Following [10, ], we see that given any curve with -torsion over F, we can move the oint of order to the origin and lace the resulting curve into Tate normal to find a model for the curve of the form E b : y + xy by = x bx, (1 which has discriminant b = b (1 + 16b. Let P = (0,0 denote the oint of order on E b. Thus, as b runs over all of F, we see every class of ellitic curve ossessing -torsion over F. As before, we will omit b = 0,16 1 from consideration since these lead to singular curves. Given a curve of the form (1, we note that both P = (0,0 and P have order. We see that moving P to origin and lacing the resulting curve in Tate normal form gives us exactly the same normal form as before. Thus, there is exactly one way to reresent each cyclic -torsion curve in the form (1. We are only interested in counting the classes which have cyclic -torsion and not full -torsion (since these have already been counted above. Thus, given a curve (1, we move P to the origin and lace the ( resulting curve in the form (6. Thus, we see that the curve has full -torsion if and only if 16b+1 = 1. Hence, we conclude that there are ( 1/ isomorhism classes of curves ossessing cyclic -torsion but not ossessing full -torsion over F. Finally, in a manner similar to the roof of Proosition, we check that 7/, 1 (mod 1, 1, 5 (mod 1, c r, = r < /, 7 (mod 1,, r (mod 0, 11 (mod 1. Combining all the ieces, the result follows. For the remainder of this section, we will need the following result, which allows us to avoid the roblem of detecting full m-torsion by requiring that our rimes satisfy 1 (mod m. Proosition. If E is an ellitic curve ossessing full m-torsion over F, then 1 (mod m. Proof. Let G be the Galois grou of F (E[m]/F. Then G = φ, where φ : F (E[m] F (E[m] is the Frobenius automorhism. We have the reresentation ρ m : G Aut(E[m] = Aut (Z/mZ Z/mZ = GL (Z/mZ. See [15, ]. Now, suose that E has full m-torsion. Then F (E[m]/F is a trivial extension. Whence, G is trivial and ρ m (φ = I GL (Z/mZ. Therefore, alying [15, Pro. V...], we have det(ρ m (φ 1 (mod m. We omit the roof of Theorem since it is similar to, but less involved than the following cases of Theorem 6. Proof Sketch of Theorem 6 (Cases: 0,1 (mod 7;c ±( + 1 (mod 7. If =, the identities may be checked directly. We will assume that,7 and rime. Since we also assume that 1 (mod 7, we know that no curve may have full 7-torsion over F. Thus, if P is a oint of order 7, E[7](F = P = Z/7Z. Now, suose that E ossesses 7-torsion, and let P be a oint of order 7. In a manner similar to [10, ], we see that we can move P to the origin and ut the resulting equation into Tate normal form to 6

7 obtain a model for E of the form E s : y + (1 s + sxy (s s y = x (s s x, (1 which has discriminant s = s 7 (s 1 7 (s 8s + 5s + 1. First, we examine the discriminant. We note that s = 0,1 both result in singular curves and so we omit these values from consideration. The cubic s 8s +5s+1 has discriminant 7 and hence has Galois grou isomorhic to Z/Z (See [9, Cor. V..7]. Thus, the slitting field for the cubic is a degree extension over Q; and we see that the cubic will either be irreducible or slit comletely over F. One can then check that the cubic slits over the cyclotomic field Q(ζ 7, where ζ 7 is a rimitive 7th root of unity. Q(ζ 7 has a unique subfield which is cubic over Q, namely Q(ζ 7 +ζ7. 6 Thus, Q(ζ 7 +ζ7 6 is the slitting field for the cubic s 8s + 5s + 1. By examining the way that rational rimes slit in Q(ζ 7, one can deduce that rational rimes are inert in Q(ζ 7 +ζ7 6 unless ±1 (mod 7, in which case they slit comletely. Thus, we see that the cubic s 8s +5s+1 has exactly roots over F if ±1 (mod 7 and is irreducible otherwise. Hence, as s ranges over all of F, we see 5 nonsingular curves (1 if ±1 (mod 7 and nonsingular curves (1 otherwise. Second, we check that the maing s (1 s + s, (s s is a one to one maing of F \0,1} into F. Hence, as s ranges over all of F \0,1}, we see distinct equations of the form (1. Next, we check that if we choose to move P to the origin instead of P, we will obtain exactly the same Tate normal form for E. Moving P or P to the origin each result in different normal forms unless s = 0,1 or is a nontrivial cube root of 1, in which case both give exactly the same normal form as moving P to the origin. Note that by the above argument, moving P to the origin will give the same normal form as P and moving P to the origin will give the same normal form as P. Now, s = 0,1 both give singular curves; and nontrivial cube roots of 1 exists in F if and only if 1 (mod, in which case there are exactly. Thus, the number of isomorhism classes of curves ossessing 7-torsion over F is given by +, ±1 (mod 7, 1 (mod, 1, 6 (mod 7, 1 (mod,, ±1 (mod 7, (mod, 5 Finally, we check that, for 1 (mod 7,, 6 (mod 7, (mod. r <, r (mod 7 c r, = /, 1 (mod, 0, otherwise. The result now follows for c ±( + 1 (mod 7 by ( and (. The remaining cases of Theorem 6 will be treated in Section.. Modular Forms and Hurwitz Class Numbers We do not give an exhaustive account of modular forms. Instead we refer the reader to Miyake s book [1] and Shimura s aer [1] for the details. Recall the definition of a modular form. Definition. Let f : h C be holomorhic, let k 1 Z, and let χ be a Dirichlet character modulo N. Then f is said to be a modular form of weight k, level N, and character χ if χ(d(cz + d k f(z, if k Z, (1 f(γz = χ(d ( ( c k d d (cz + d k f(z, if k 1/ + Z ( a b for all γ = Γ c d 0 (N; ( f is holomorhic at the cuss of h/γ 0 (N. We denote this sace by M k (N,χ. In the case that χ is trivial, we will omit the character. We also recall that the sace is a finite dimensional vector sace over C and decomoses as M k (N,χ = E k (N,χ S k (N,χ, 7

8 where S k (N,χ is the subsace of cus forms and E k (N,χ is the Eisenstein subsace. For the remainder, we ut q := q(z = e πiz. It is well-known that modular forms have natural reresentations as Fourier series. Here we will develo some notation and show how to construct weight Eisenstein series. Given two Dirichlet characters ψ 1,ψ with conductors M 1 and M resectively, ut M = M 1 M ; and ut E (z;ψ 1,ψ := a n q n, where and, for n 1, n=0 0, if ψ 1 is non-trivial, M 1 a 0 =, if ψ 1 and ψ are both trivial, B k,(ψ1ψ /, otherwise, a n = d n ψ 1 (n/dψ (dd. Here B k,χ denotes the k th generalized Bernoulli number associated to χ, whose generating function is F χ (t = m χ(ate at a=1 e mt 1, where m is the conductor of χ. Then, subject to a coule of technical conditions on ψ 1 and ψ, one can show E (z;ψ 1,ψ E (M,ψ 1 ψ. See [1, ]. When comuting with modular forms, the following result allows us to work with only a finite number of coefficients. Theorem 8 (See Pro. 1.1 in [6]. Suose that k Z and f(z = a n q n M k (N,χ. Put m = k 1 N ( N Then f(z is uniquely determined by its Fourier coefficients a 0,a 1,...,a [m]. In [8,. 90] ([16] also, Zagier defines a non-holomorhic q-series whose holomorhic art is the generating function for the Hurwitz class number, H(N. He then shows that the series transforms like a modular form of weight / on Γ 0 (. In [1, Cor.], Cohen oints out that a holomorhic form can be obtained by only summing over those N that fall into certain arithmetic rogressions. He states without roof that the resulting series should be a form on Γ 0 (A, where he secifies A. However, using techniques similar to those found in [11, ], we were only able to rove the following version of this result. Theorem 9. If b is a quadratic non-residue modulo a, then H 1 (z;a,b := N b (mod a H(Nq N M / (G a, where ( α β G a = γ δ } Γ 0 (A : α 1 (mod a, and we take A = a if a and A = a otherwise. We now turn to the roof of Theorem. Proof of Theorem. It is well-known that the classical theta series θ(z := s= qs M 1/ (. Alying Theorem 9, we see that H 1 (z;,1 = H(Nq N M / (6. Note that in this case, G = Γ 0 (6. Thus, we can N 1 (mod check that roduct H 1 (z;,1θ(z M (6. Observe that the coefficients of the roduct bear a striking 8

9 resemblance to the sums of interest. Indeed, H 1 (z;,1θ(z = = s= N 1 (mod n 1 (mod + n (mod H(Nq N+s s < n, s 0 (mod s < n, s ±1 (mod H(n s qn H(n s qn. We will rove Theorem by exressing H 1 (z;,1θ(z as a linear combination of basis forms with nice Fourier coefficients. Note that Theorem 8 says that we will only need to consider the first 1 coefficients in order to do this. Let χ 0 denote the rincial character of conductor 1, and let χ 0, and χ 0, denote the trivial characters modulo and resectively. Finally let ( denote the Legendre symbol modulo. Then one can show that E (6 has dimension 11 over C and is sanned by E (z;χ 0,χ 0,, E (z;χ 0,χ 0,, E (z; ( (,, E (z;χ 0,χ 0,, E (z;χ 0,χ 0,, E (9z;χ 0,χ 0,, E (6z;χ 0,χ 0,, E (18z;χ 0,χ 0,, E (z;χ 0,χ 0,, E (z; (, (, E (z; (, ( The cus sace S (6 is 1 dimensional and is sanned by the cus form associated to the ellitic curve which is the inverse Mellin transform of the L-series E : y = x + 1, L(E,s = (1 a( s + 1 s 1 = 6 ( = (n,6=1 n l ord(n m k ord (n (n,6=1 a(n n s k ord (n k. a( k ord(n ( ord(n k 1 n s, where a( := + 1 #E(F, and we take a( 0 = 1 even if a( = 0. We will denote this cus form by f E (z. One can verify comutationally that H 1 (z;,1θ(z = 1 16 E (z;χ 0,χ 0, + 16 E (z;χ 0,χ 0, + 1 E (z; (, ( + 1 E (z;χ 0,χ 0, + 1 E (z;χ 0,χ 0, + 16 E (9z;χ 0,χ 0, + E (6z;χ 0,χ 0, + E (18z;χ 0,χ 0, + 16 E (z;χ 0,χ 0, + 1 ( ( 8 E (z;, + 1 ( ( E (z;, f E(z. 9

10 Let σ(n := σ 1 (n = d n d, and define the arithmetic functions µ 1 (n := d, µ (n := µ (n := d n, d 0 (mod d n, d 0 (mod ( n σ(n. Extend these to Q by setting µ i (r = 0 for r Q\Z (i = 1,,. Comaring n-th coefficients, we have the following roosition. Proosition. s < n H(n s = 1 16 µ 1(n + 16 µ (n 1 µ (n 1 µ 1(n/ + 1 µ 1(n/ 16 µ 1(n/9 + µ 1 (n/6 µ 1(n/18 16 µ (n/ 1 8 µ (n/ 1 µ (n/ 1 1 a(n, where the * denotes the fact that if n 1 (mod, we take the sum over all s 0 (mod ; if n (mod, we take the sum over all s ±1 (mod. Using ( and the fact that a(n = 0 if (n,6 > 1, we are able to extract the identities r <, r c (mod H( r = d,, if 1 (mod, c = 0,, if (mod, c = 1 or. So, using (1, we are able to obtain Theorem as a corollary. At this oint, we also note that there is a more general identity than (1, which is due to Hurwitz. See [,. 6]. In articular, if we let λ(n := 1 d n min(d,n/d, then H(N r = σ(n λ(n. r < N So, this identity together with Proosition makes it ossible to generalize Theorem for not necessarily rime. In addition, if we study the cus form in our basis carefully, we can extract other nice formulae as well. For examle, we can use the fact that a( = 0 if (mod to obtain Theorem 5.. The Eichler-Selberg Trace Formula and Hurwitz Class Numbers The Eichler-Selberg trace formula gives the value of the trace of the n-th Hecke oerator acting on S k (N,ψ. The following gives a formula for comuting the trace in a secific setting, which is useful for our uroses. The trace formula is, in fact, much more general and we refer the reader to [7,. 1-1] for more details. Theorem 10. Let be a rime. The trace of the -th Hecke oerator acting on S (N is given by tr,n (T = φ((s 1/ /f c(s,f,l s H f t l N 1 h((s /f ω((s /f c(s,f,l, s E 1 E where H = s : s = t }, E 1 = s : s = t m, 0 > m 1 (mod }, and E = s : s = t m, 0 > m, (mod }. 10 f t l N

11 Here, φ is the Euler φ-function. For d < 0, h(d is the class number of the order of Q( d of discriminant d, and ω(d is 1/ the cardinality of its unit grou. For a rime l N, c(s,f,l essentially counts the number of solutions to a certain system of congruences. It is well-known that if s E 1 E, then H( s = f t h((s /f ω((s /f. In fact, one may even define the Hurwitz class number in this way. See [,. 18]. So, if it is ossible to control the c(s,f,l in articular, if it is ossible to make them constant with resect to f, then there is hoe that Hurwitz class number relations may be extracted from the trace formula. Indeed, if is quadratic nonresidue modulo l for all rimes l dividing N, then the comutation of c(s, f, l is quite simle and does not deend on f. For examle, if we aly the trace formula to T acting on S (7 = 0} for,5,6 (mod 7,, (mod 7, s 0, ±,, 5 (mod 7, s 0, ±1, c(s,f,l =, 6 (mod 7, s 0, ±, 0, otherwise. The resulting Hurwitz class number relation is the following. Proosition 5. s < H( s = 1, where the * denotes the fact that if (mod 7, the sum is over all s 0, ± (mod 7; if 5 (mod 7, the sum is over all s 0, ±1 (mod 7; and if 6 (mod 7, the sum is over all s 0, ± (mod 7. Combining the above roosition with the cases of Theorem 6 that were roven in Section, we are able to obtain the remaining formulae in the theorem. 5. Conjectures For all rimes sufficiently large, the tables in this section give conjectured values for the sum H( r. r <, r c (mod m These values have been checked for rimes < 1,000,000. Where an entry is bold and marked by asterisks, the formula is handled by a theorem somewhere in this aer; where an entry is blank, we were not able to recognize any simle attern from the comutations. We note that for m = 5 and 7, neither the curve counting method of Section alone nor the basis of modular forms aroach of Section alone will be sufficient for a comlete characterization of these sums. Rather a combination of the two methods should work. c = 0 c = ±1 c = ± ( ( (5 7 1 (mod 5 1 ( ( (mod 5 ( 1 ( (mod 5 ( 1 ( (mod 5 ( (5+5 ( 1 Table 1. m = 5 For making further rogress on the above tables, it aears that the method of Section will be most fruitful. The difficulty in using this method is that, in each case, the grou G m (defined in Theorem 9 is strictly contained in Γ 0 (m. So, we will need a much larger basis of modular forms. Certainly a basis for 11

12 1 (mod 7 (mod 7 (mod 7 (mod 7 5 (mod 7 6 (mod 7 c = 0 c = ±1 c = ± c = ± ( ( ( ( ( ( 5 Table. m = 7 ( ( ( ( ( ( ( ( ( M (Γ 1 (m would be sufficient. However, to comletely fill in the table, another method will be needed. Perhas an adatation of the curve counting method of Section for 1 (mod m would be best. The main obstacle to overcome is that one must deal with the resence of full m-torsion curves when 1 (mod m. In general, for a rime m, we note that the curve counting method will give roofs for c ±( + 1 (mod m. We also note that, for rimes m 1 (mod, the basis of forms method will give roofs for the cases when c is not a square; and for rimes m (mod, the basis of forms method will give roofs for the cases when c is a square. Thus, for a rime m, we will obtain half the cases from the basis of forms aroach and we will obtain (m more from the curve counting aroach assuming that the case 1 (mod m can be adequately handled. So, at least for rimes greater than 7, a third method will be necessary to fully characterize how the sum slits. 6. Acknowledgments The authors would like to thank Ken Ono for suggesting that we consider the Eichler-Selberg trace formula. The authors would also like to thank Robert Osburn for helful conversation. References [1] Henri Cohen. Sums involving the values at negative integers of L-functions of quadratic characters. Math. Ann., 17(:71 85, [] Henri Cohen. A Course in Comutational Algebraic Number Theory. Sringer-Verlag, New York, [] David A. Cox. Primes of the Form x + ny. Wiley-Interscience, New York, [] Max Deuring. Die Tyen der Multilikatorenringe ellitischer Funktionenkörer. Abh. Math. Sem. Hansischen Univ., 1:197 7, 191. [5] Martin Eichler. On the class of imaginary quadratic fields and the sums of divisors of natural numbers. J. Indian Math. Soc. (N.S., 19: (1956, [6] Gerhard Frey. Construction and arithmetical alications of modular forms of low weight. In Ellitic curves and related toics, volume of CRM Proc. Lecture Notes, ages 1 1. Amer. Math. Soc., Providence, RI, 199. [7] Hiroaki Hijikata, Arnold K. Pizer, and Thomas R. Shemanske. The basis roblem for modular forms on Γ 0 (N. Mem. Amer. Math. Soc., 8(18:vi+159, [8] F. Hirzebruch and D. Zagier. Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentyus. Invent. Math., 6:57 11, [9] Thomas W. Hungerford. Algebra. Sringer-Verlag, New York, 197. [10] Anthony W. Kna. Ellitic Curves. Princeton University Press, Princeton, 199. [11] Neal Koblitz. Introduction to Ellitic Curves and Modular Forms. Sringer-Verlag, New York, 199. [1] H. W. Lenstra, Jr. Factoring integers with ellitic curves. Ann. of Math. (, 16(:69 67, [1] Toshitsune Miyake. Modular Forms. Sringer-Verlag, New York, [1] Goro Shimura. On modular forms of half integral weight. Ann. of Math. (, 97:0 81, 197. [15] Joseh H. Silverman. The Arithmetic of Ellitic Curves. Sringer-Verlag, New York, [16] Don Zagier. Nombres de classes et formes modulaires de oids /. C. R. Acad. Sci. Paris Sér. A-B, 81(1:Ai, A88 A886,

13 Brittany Brown, Deartment of Mathematics and Actuarial Science, Butler University, 600 Sunset Ave., Indianaolis, IN 608 Neil J. Calkin, Deartment of Mathematical Sciences, Clemson University, Box 0975 Clemson, SC address: Timothy B. Flowers, Deartment of Mathematical Sciences, Clemson University, Box 0975 Clemson, SC address: Kevin James, Deartment of Mathematical Sciences, Clemson University, Box 0975 Clemson, SC address: Ethan Smith, Deartment of Mathematical Sciences, Clemson University, Box 0975 Clemson, SC address: Amy Stout, Deartment of Mathematics, LeConte College, 15 Greene Street, University of South Carolina, Columbia, SC 998 1