Recent advances for Ramanujan type supercongruences

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1 Contemorary Mathematics Recent advances for Ramanujan tye suercongruences Sarah Chisholm, Alyson Deines, and Holly Swisher Abstract. In 9, Ramanujan listed 7 infinite series reresentations of /π of the form A(kx k = δ π, which were later used by J. Borwein and P. Borwein and D. Chudnovsky and G. Chudnovsky to find aroximations for π. Several of these formulas relate hyergeometric series to values of the gamma function. In 997, van Hamme develoed a -adic analogue of these series called Ramanujan tye suercongruences and conjectured 3 formulas relating truncated sums of hyergeometric series to values of the adic gamma function, three of which he roved. Since then, a handful more have been roved. In this survey, we discuss various methods to rove these suercongruences, including recent geometric interretations.. Introduction In a rather mysterious way Ramanujan [7 stated a number of reresentations for /π, including ( ( 3 k k! 3 (6k + k = π, where (a k denotes the rising factorial (a k = a(a + (a + k. Another examle due to Bauer [6 is ( ( 3 k k! 3 (k + ( k = π. Amazingly, the sums in ( and ( are of rational numbers, resulting in a transcendental number! Ramanujan s formulas gained oularity in the 980 s when they were discovered to rovide efficient means for calculating digits of π. In 987, J. & P. Borwein [9 roved all 7 of Ramanujan s identities, while D. & G. Chudnovsky [ derived additional series for /π. Digits of π were calculated in both aers resulting in a new world record (at the time of, 60, 33, 336 digits, by the Chudnovskys. Interesting adic analogues of Ramanujan s formulas for /π were develoed by van Hamme [3. In articular, he conjectured the following congruences, which corresond to ( and (, resectively. For rimes >, ( 3 k k! 3 (6k + k (mod and ( 3 k k! 3 (k + ( k (mod 3, 00 Mathematics Subject Classification. 33C0, A0, G07, F, F33. Key words and hrases. Ramanujan tye suercongruences, hyergeometric series, ellitic curves, modular forms. c 0000 (coyright holder

2 ( where is the Legendre symbol. Congruences of this form are called Ramanujan tye suercongruences. The term suercongruence originated with Coster [3. By suercongruence, we refer to the fact that the congruence holds modulo a ower of larger than exected by general theories which use formal grou laws. All told, van Hamme conjectured 3 Rananujan tye suercongruences. Among these conjectures, he rovided roofs for three of them. Since then, there have been several other roofs of a number of van Hamme s conjectures. In articular, in the work of McCarthy and Osburn [, Mortenson [5, Zudilin [3, Kilbourn [, and Long [3 each set of authors roved one or more of van Hamme s conjectures. Several conjectures remain oen. Ramanujan s formulas, and their associated suercongruences, are connected to certain hyergeometric series. For r a nonegative integer and α i, β i C, the hyergeometric series r+ F r is defined by r+f r [ α... α r+ β... β r ; x = (α k (α k... (α r+ k x k (β k... (β r k k!, which converges for x <, and for x = with aroriate conditions on α i, β i. One can immediately see a connection with the above equations ( (, since, for examle [ ( 3 3F ; k = k! 3. The remainder of this aer is organized as follows. In Section, we discuss the methods of the Borweins and Chudnovskys which rovide systematic ways to arrive at formulas for Ramanujan s reresentations for /π. In Section 3, we review van Hamme s -adic analogues to Ramanujan s formulas and state his conjectures. In Section we give a number of methods and recent roofs of van Hamme s conjectures. Lastly, in Section 6 we resent recent work of Long, Nebe, and the authors [, a geometric aroach to Ramanujan suercongruences via K3 surfaces.. Ramanujan tye formulas for /π Ramanujan s exansions of /π were even known rior to him by mathematicians including Bauer. Systematic ways to obtain such formulas have been studied by two airs of brothers, the Borweins and the Chudnovskys. Perhas rather surrisingly, the formulas are obtained using features of ellitic curves. In articular, one can use either the classical Legendre relation between the eriods and quasi-eriods or the Wronskian of the Picard-Fuchs equation associated to the curves. Here we outline a method using the Legendre relation. The motivated reader is encouraged to consult the standard reference of Silverman [9 for further details on this section. In addition, Baruah et al. [5 and Zudilin [33 rovide lovely surveys on formulas for /π and /π... Legendre relation. Let E/C be an ellitic curve and α and β be closed aths on the grou of comlex oints on E, E(C, which give a basis for the homology grou H (E, Z. Then dx ω = y, and ω = are R-linearly indeendent and are called the eriods of E. Let Λ = Zω + Zω be the corresonding lattice for E with I(ω /ω > 0. Recall that the Weierstrass -function, which is an ellitic function relative to a lattice Λ C, is given by (z : Λ = z + α β ω Λ,ω 0 dx y, ( (z ω ω In addition, the Weierstrass ζ-function is described as the equation ζ(z : Λ = z + ω Λ,ω 0 ( z ω + ω + z ω..

3 Furthermore, the Weierstrass σ-function relative to Λ is defined as ( σ(z = σ(z; Λ = z z ex ((z/ω + ω (z/ω. ω Λ,ω 0 As er usual, when the lattice Λ is understood by context, we suress it from the notation. For a fixed lattice Λ, note the following relations between the Weierstrass, ζ and σ functions, d dz log σ(z = ζ(z and d ζ(z = (z. dz Also, for each vector ω Λ there are constants a, b, η(ω C, deending on ω such that for all z C, σ(z + ω = e az+b σ(z and ζ(z + ω = ζ(z + η(ω. The numbers η := η(ω, and η := η(ω are called the quasi-eriods of E. Integrating ζ(z along a fundamental arallelogram of Λ gives the Legendre relation on eriods and quasi-eriods: η ω η ω = πi. The comlete ellitic integral of first and second kind are defined as (3 K(k = and ( E(k = 0 dt ( t ( k t = π F [ 0 k t dt ( t = π F [ ; k ; k, resectively, roviding the connection between Ramanujan suercongruences and ellitic curves. Comlementary integrals E, K are defined as and K (k = K( k E (k = E( k. Together, the functions K(k and K (k, san the solution sace of a degree- hyergeometric ordinary differential equation (ODE. Similarly E(k and E (k K (k san the solution sace of another degree- hyergeometric ODE. Moreover, and de dk = E K k dk dk = E ( k K k( k. The Legendre relation can be interreted as the following. For any 0 < k <, (5 E(kK (k + E (kk(k K(kK (k = π. If x = k, then and F (x = π K(x = F [ F (x = F ( x = F [ ; x ; x Please note that this notation is standard in the literature, and does not indicate a derivative. 3

4 both satisfy the same order differential equation ( d dx + x d x( x dx u(x = 0. x( x The Wronskian of F (x is (6 df ( x df (x F (x dx dx F ( x = (F (xf x(x + F x (xf (x ( x = ex x( x dx c = x( x, where c = F ( F x( = π. It is straight forward to comute that F [ ; x = = ( k k! + k xk ( k k! k xk = x d dx [ x F [ ; x. What is imlied here is that the Legendre relation (5 is equivalent to the Wronskian described above. We note the following fact, which follows from hyergeometric transformation formulas [. Lemma.. For x R with x <, Letting a = / yields F [ a F [ a ; x ; [ a = F [ x = F a ; x. ; x. Clausen s formula, states that [ [ a b a b a + b (7 F a + b + ; x = 3 F a + b + a + b ; x. Letting a = /, b = /, we see that F [ [ ; x = 3 F ; x. Combining these results give us a connection between the corresonding 3 F and F hyergeometric series above, which leads to Ramanujan tye formulas for /π, of the form (8 ( 3 k k! 3 (ak + λn = δ π, where δ and a are algebraic numbers [9,. In articular, a Q(λ with λ <, for λ arametrizing a family of ellitic curves with comlex multilication (CM. When the endomorhism ring of an ellitic curve is larger than the rational integers, it is said to have comlex multilication.

5 .. Geometric connection. To see the connection with geometry, we first consider the Legendre family of ellitic curves, arametrized by λ and define E λ : y = x(x (x λ, ω λ = dx y and η λ = x dx y. Then E(k defined above is a differential of the second kind, and is a combination of ω λ and η λ, which corresonds to two different ways of writing the Legendre family of ellitic curves. The ellitic integrals (3 and ( are also related to one of the Jacobi theta functions, (9 K(k = π θ 3(q(k, where q(k = ex π K (k. K(k The eriod lattice of the corresonding ellitic curve is generated by K(k and ik (k. Assume that the ellitic curve has CM; this amounts to K /K being a quadratic number. In a secial case when K = K r with r Q, the following hold due to the Legendre relation (5 ( π E = rk + α(r (0 K, r ( E = π K + α(rk, where α(r = E /K π/k is the so-called singular value function and takes algebraic values when r Q +. Thus, when K /K = r, for r Q +, the Legendre relation (5 can be written in simly the terms of E and K. This aroach leads to formulas for /π related to hyergeometric series. To see the connection to the hyergeometric series 3 F, we consider the K3 surface X λ described by the equation X λ : z = x(x + y(y + (x + λy. When λ, this manifold is related to the one-arameter family of ellitic curves of the form Eλ : y = (x x +λ via the so-called Shioda-Inose structure [3, 3. In articular, for the imlication in terms of arithmetic, see the aer of Ahlgren, Ono, and Penniston [3. For d {, 3,, 6} let Ẽd(t denote the following families of ellitic curves arameterized by t. Ẽ (t : Ẽ 3 (t : y = x(x (x t, y + xy + t 7 y = x3, Ẽ (t : y = x(x + x + t, Ẽ 6 (t : y + xy = x 3 t 3. For t such that Ẽd(t has CM, let λ d = t(t, and write E d (λ d = Ẽd( λ d. Both the Borweins and Chudnovskys established the following theorem, which reiterates the identity (8 in more detail. Theorem.. Let d {, 3,, 6}. For λ d such that Q(λ d is totally real, and for any embedding λ d <, there exist algebraic numbers δ, a Q(λ d yielding the following Ramanujan tye formula for /π, ( ( k( d k( d d k (k! 3 (λ d k (ak + = δ π. 5

6 3. Ramanujan suercongrences of van Hamme Connecting artial sums of the related hyergeometric series to values of the -adic gamma function, van Hamme constructed analogues to 3 of Ramanujan s formulas. For instance, one of Ramanujan s formulas is given by ( 5 (k + ( k k k! 5 = Γ( 3, where Γ(x is the standard Gamma function (the value on the right hand side can be exressed in terms of /π. By considering the -adic gamma function Γ (x, van Hamme observed numerically that ( 5 (k + ( k k k! 5 = { Γ ( 3 if (mod 0 if 3 (mod. Baffled, van Hamme stated that he had no concrete exlanation for his observations, including the factor of / which distinguishes the right hand sides of the two equations. However, he roved 3 of the 3 congruences, (C., (H., and (I. in the notation below. For comleteness, we list all 3 of van Hamme s conjectures, together with their Ramanujan series counterart. Here the notation S( n, simly means take the left hand side of the corresonding Ramanujan series and truncate at ( /n. (A. (B. (C. (D. (E. (F. (G. (H. (I. (J. (K. (L. (M. Ramanujan Series Conjectures of van Hamme (k + ( ( k 5 (mod 3, if (mod k = ( k! 5 (A. S Γ 3 Γ 3 0 (mod 3, if 3 (mod (k + ( ( k 3 k = = k! 3 π (B. S ( Γ (mod 3, Γ (k + ( k = (C. S (mod 3, k! (6k + ( ( k = (D. S Γ k! (mod, if (mod 6 (6k + ( ( k 3 3 k = 3 3 = 3 (E. S (mod 3 k! 3 π Γ Γ (8k + ( ( k 3 k = = (F. S ( (mod k! 3 π Γ Γ 3 Γ Γ 3, if (mod 3 (8k + ( ( 3 k = k! (G. S Γ Γ (mod 3, if (mod πγ 3 Γ 3 { ( ( 3 k = ( π Γ k! 3 (H. S (mod, if (mod Γ 3 0 (mod, if 3 (mod ( k = = k+ k! π (I. S (mod 3, Γ 6k+ ( k ( 3 k k = = k! 3 π (J. S ( Γ (mod,, 3 Γ k+5 ( k ( 3 6 k k = 6 = 6 k! 3 π (K. S 5 Γ (mod, Γ 6k+ ( k ( 3 8 k k = = (L. S ( k! 3 π Γ Γ 3 Γ Γ 3 (mod 3, ( k : unknown (M. S a( (mod 3, k!. Proofs of van Hamme s suercongruences Three suercongruences (C., (H., and (I. were roved by van Hamme using various methods. For instance, to rove (C., he used a sequence of orthogonal olynomials k (x which satisfy a certain 6

7 recurrence relation that enabled him to deduce that ( k k ( / = k!, k+ ( / = 0. Further analysis of these olynomials yields (C.... Aéry numbers. Let A(n denote the Aéry numbers defined by n n + j n A(n =, j j j=0 which were used to rove the irrationality of the values of the Riemann zeta function ζ( and ζ(3. In addition, define a(n to be the nth Fourier coefficient of the modular form (3 η(z η(z = a(nq n, where q = e πiz, and η(z is the usual Dedekind eta-function. Beukers [7, 8 roved the congruence A a( (mod, for rimes >. Furthermore, defining the sum B(n = j=0 he roved that for rimes >, ( B b( n= n n + j n, j j (mod, where b(n are similarly defined as coefficients of the modular form η(z 6 = n= b(nqn. Furthermore, he conjectured that the following statements hold modulo (5 A a( (mod (6 B b( (mod. Interestingly, equation (6 is actually van Hamme s (H. in disguise, which he later showed in his work with Stienstra [3... Suercongruences arising from Calabi-Yau threefolds. Rodriguez-Villegas uses Calabi-Yau threefolds over finite fields to numerically discover Beukers-like suercongruences modulo 3 [8. For instance, for odd rimes, (7 a( ( k k! (mod 3. Notice that equation (7, incredibly, is van Hamme s (M.. More secifically, for odd rimes, the coefficients a( are connected to a Calabi-Yau threefold defined by (8 x + x + y + y + z + z + w + w = 0, by the relation (9 a( = 3 7 N(, 7

8 where N( counts the number of solutions to the manifold (8 over F. The relation (9 was roved by Ahlgren and Ono [, and they used it to reresent a( in terms of Gaussian hyergeometric series. Additionally, they roved conjecture (5 of Beukers [. Kilbourn roved observation (7, making use of the fact that the Calabi-Yau threefold (8 is modular [. His method involves writing a( in terms of Gauss sums, and then using the Gross-Koblitz formula [6 to relate the Gauss sums to the -adic gamma function. 5. Hyergeometric methods for van Hamme s conjectures Three of van Hamme s conjectures, (A., (B. and (J., have been roved using a variety of techniques involving hyergeometric series. We include details below. 5.. Proof of Conjecture (A.. One aroach to roving conjecture (A. is due to McCarthy and Osburn [. Their roof also invloves Gaussian hyergeometric series, which we now introduce. If A, B are characters of F, the normalized Jacobi sum is given as ( A B = B( x F A(xB( x. We extend multilicative characters χ of F to F by defining χ(0 = 0. For characters A 0,..., A n and B,..., B n of F, the Gaussian hyergeometric series is defined for x F by [ A0 A A n n+f n B 0 ; x = A0 χ A χ An χ χ(x. χ B χ B n χ When each of the A i are Legendre symbols modulo, and all of the B j are trivial characters, we simly write n+ F n (x, following suit of McCarthy and Osburn. The Gross-Koblitz formula rovides a wonderful connection between Gauss sums and the -adic Gamma function. To state this formula, we first observe that we may view a character χ for F as taking values in Z. Let π C be a fixed root of x + = 0, and let ζ be the unique th root of unity in C for which χ ζ + π (mod π. We then define the Gauss sum for a character χ : F C by g(χ = χ(xζ x. x=0 If ω is the Teichmüller character, a rimitive character defined by the roerty that ω(x x (mod for x = 0,...,, then the Gross-Koblitz formula states that for 0 j, j (0 cg(ω j = π j Γ. In the roof of (A. by McCarthy and Osburn, they use the following theorem of Osburn and Schneider [6, which states that for an odd rime, an integer n, and an element λ F, n+ n n+f n (λ ( n+ [ X(, λ, n + Y (, λ, n + Z(, λ, n (mod 3. Here, is the Legendre symbol modulo, and X(, λ, n, Y (, λ, n, and Z(, λ, n are truncated sums involving differences of generalized harmonic sums, n H n (i = j i. For exlicit formulas see the work of McCarthy and Osburn [. 8 j=

9 After careful analysis of the X(, λ, n and Y (, λ, n terms, they reduced the roblem to showing that ( 5 (k + ( k k k! 5 k= Z(,, (mod 3. From here, they used the following secial case of While s 7 F 6 hyergeometric transformation [ a + a ( c d e f 6F 5 a + a c + a d + a e + a f ; = which enabled them to rove the result. Γ( + a eγ( + a f Γ( + aγ( + a e f 3F [ + a c d e f + a c + a d ; 5.. Proofs of Conjecture (B.. Conjecture (B. of van Hamme has been roved in two different ways. Mortenson [5 observed that the suercongruence (A. shared the right hand side with a suercongrence of Beukers. Namely, for odd rimes ( 5 (k + ( k k k! 5 ( 3 k k! 3 (mod 3. Armed with this idea, Mortenson roved (B. from van Hamme s list. More secifically, ( 3 (k + ( k k k! 3 (mod 3. To do so, he first demonstrated a technical lemma to evaluate a quotient of gamma functions. Then, following McCarthy and Osburn, he utilized the secial case of While s transformation (. Zudilin roved conjecture (B. by using a method known as the WZ method [3. This method, designed by Wilf and Zeilberger [3, is a owerful tool for roving identities for hyergeometric series. In fact, equied with the WZ method, Zudilin was able to rove a -adic analog for a formula for /π discovered by Guillera [7. More secifically, Guillera s series ( 3 ( ( 3 k k k k! 5 (0k + 3k = k π, has the -adic analog for odd rimes ( ( 3 ( 3 k k k k! 5 (0k + 3k + 3 k 3 (mod Proof of Conjecture (J.. The aroach by Long [3 is more general for conjecture (J.. Here, motivated by the techniques of McCarthy and Osburn, Mortenson, and Zudilin, she utilized articular hyergeometric series identities and evaluations in order to rove the following more general theorem. Theorem 5.. Let > 3 be rime and r a ositive integer. Then r In addition, she roved conjecture (J. of van Hamme, (k + ( k k! r (mod 3+r. (6k + ( 3 k k! 3 k ( (mod. 9

10 Her method differs with the use of articularly nice hyergeometric evaluation identities. For instance, the strange valuation of Goser [5, [ a b b + a 3 n 5F a + b a + b + a 3 + a + n ; (a + = n(a + n (a + b +. n(a b + n 6. Ramanujan suercongruences arising from K3 surfaces We now turn our attention to recent work of Long, Nebe, and the authors on Ramanujan suercongruences arising from K3 surfaces. First, we describe some work of Atkin and Swinnerton-Dyer on noncongruence subgrous which will be involved in our discussion. Atkin and Swinnerton-Dyer [ studied the arithmetic roerties of coefficients of modular forms of noncongruence subgrous of SL (Z, i.e. those finite index subgrou of SL (Z not containing any rincial congruence subgrous. Due to the lack of efficient Hecke theory, these coefficients are not as desirable as the coefficients of Hecke eigenforms; for examle, the discriminant function, to name one, describes some of the coefficients. Desite this difficulty, Atkin and Swinnerton-Dyer discovered interesting adic analogues of classical Hecke recursions. To illustrate, we state their result in a secial case. Let E : y = x 3 Bx C be a nonsingular ellitic curve over Z such that E has good reduction modulo. Let ξ be any local uniformizer of E at infinity over Z that is a formal ower series of x y with coefficients in Z. Then the holomorhic differential ( dx y = ( + a(nξ n dξ n has coefficients in Z. Then, taking A = + #(E/F, and the coefficient a(n from the holomorhic differential (, the following congruence holds for all n, a(n Aa(n + a(n/ 0 (mod +ordn. Atkin and Swinnerton-Dyer s results insired the later work of Cartier [0 and Katz [0. Here we shall use Katz s aroach. In the sirit of their results, a sequence {a(n} is said to satisfy a weight k Atkin and Swinnerton-Dyer (ASD congruence at a fixed rime if there are -adic integers A,, A s such that for all n, a(n + A a(n + A a(n/ + + A s a(n/ s 0 (mod (k (ordn+. We require that the congruence is written in the way which is comatible with the congruences satisfied by the logarithms of dimensional commutative formal grous [30, Aendix A.8. The convention that a(n/ e = 0 if e n is taken in the receding congruence. Moreover, the characteristic olynomial T s + A T s A s customarily has imortant arithmetic meaning. In the case of the above mentioned ellitic curves, when E has a model defined over Q, then T AT + is the Hecke olynomial of a weight Hecke eigenform. This is the result of the Taniyama-Shimura-Weil conjecture, since roved by Wiles, Taylor-Wiles et al. A common technique in roving weight ASD congruences is the theory of formal grous [9. As a result, any ASD congruence of weight greater than is also referred to as a suercongruence. 0

11 6.. Results of Chisholm, Deines, Long, Nebe, and Swisher. Alternatively, for d {, 3,, 6} we give a geometric roof of Ramanujan tye suercongruences related to hyergeometric series of the form [ d d 3F ; λ truncated modulo. We outline the secific case for d = and note that the results for the other cases follow similarly. Consider the K3 surface X λ described by the equation X λ : z = x(x + y(y + (x + λy. This manifold is a natural dimensional analogue of the one-arameter family of ellitic curves of the form ( E λ : y = (x x. + λ Each ellitic curve E λ has a model over the field F λ = Q(j(E λ, where the j invariant of E λ is given by (λ + 3 j(e λ = 6 λ. In fact, E λ is isomorhic to E (λ as defined in Section. Let Q(τ denote an imaginary quadratic number field. When λ Q is such that E λ admits comlex multilication by an order of Q(τ, following Ramanujan s idea, there exist numbers a and λ in K λ = Q(τ, j(τ, λ, and an algebraic number δ, giving an instance of the Ramanujan tye formula [9, (3 ( 3 k k! 3 (ak + λk = δ π. Now we will state the main result in full generality for d {, 3,, 6}. Take E d (λ d as in Section.. Theorem 6.. Let λ d Q such that Q(λ d is totally real, the ellitic curve E d (λ d has comlex multilication, and λ d < for an embedding of λ d to C. For each rime that is unramified in Q( λ d and corime to the discriminant of E d (λ d such that a, λ d can be embedded in Z (and we fix such embeddings, then ( where ( k( d k( d d k k! 3 (λ d k (ak + sgn ( λd (mod, λd is the Legendre symbol, and sgn = ±, equaling if and only if E d (λ d is ordinary modulo. The roof involves ASD congruences, discussed above, which was insired by results due to Katz. Also required are ellitic curves with comlex multilication with a quartic twist. As well, we view the curve with a model over the ring of Witt vectors, and consider its de Rham cohomology. The theory of modular forms lays an integral role as well as essential identities due to Clausen. 6.. Ramanujan tye suercongruences for λ in Q. In this case the the coefficients of the Ramanujan formula (3 are obtained from the Hilbert class field K λ = Q. In fact, the congruence (5 ( 3 k k! 3 (k + ( k ( (mod 3, is an examle of such a formula with a =, λ =, taken modulo 3. Corresonding to this suercongruence is the conjectural ASD congruence of the form ( 3 k k! 3 ( k α (mod 3.

12 The congruence ertains to ordinary rimes, i.e., 3 (mod 8 and α is the adic unit root of H (X, λ = X a X + ψ λ (. The quadratic function H (X, λ is the th Hecke olynomial of a weight 3 Hecke cusidal eigenform g(z = b(nq n. Exlicitly, the first few coefficients of H (X, λ are a 5 = a 3 = a 3 = 0, a 7 =, a 9 = 3, a 67 = 6, a 73 =, which can be identified from the L function of X λ. For the corresonding ellitic curve at λ =, E, whenever the curve has ordinary reduction, for examle at the rimes = 7, 9, 67, the ASD congruence becomes (6 ( 3 k k! 3 ( k b( (mod 3. When E is suersingular at, for examle at the rimes = 3, 3, 3, then the ASD congruence is given by (7 ( 3 k k! 3 ( k b( = 0 (mod. It is also ossible to write the coefficient b( in terms of an exlicit value of a -adic Gamma function as related to Gauss or Jacobi sums, using the Gross-Koblitz formula (0. These congruences can be roved modulo, by counting oints on the K3 surface X. Also, congruence (6 can be roved modulo, using Coster and van-hamme s result [. The keen reader is encourage to consult the work of Kibelbek et al. for further details [. Numerical results similar to those in equations (6 and (7 are observed when λ = /, /8, / Ramanujan tye suercongruences for λ not in Q. For larger fields K λ, less has been formulated, and here we rovide some numerical evidence comuted using Sage. Secific values for a, λ and δ are considered below. We claim that the infinite sums, when reduced modulo, take the articular form stated in their resective conjecture. We note that in rivate communication with Long, Zudilin conjectured that Theorem 6. should in fact hold modulo 3 for each λ for which E λ has comlex multilication. For further conjectured examles, see also the work of Guillera [8. Examle. [9, Exercise 5 ( [ 3 k k! k + (3 3( 3(( 3 k = 3 / π. Here a = /(3 3, λ = ( 3, and the Hilbert class field is K λ = Q( 3,. This sum in condensed form is ( 3 k k! 3 [(3 3 + k( 3 k+ = 3 / π. Truncating the sum modulo we make the following claim. Conjecture 6.. For any rime ± (mod, ( 3 [ k k! 3 (3 3 + k ( ( 3 3 k+ (3 3( 3 (mod. This congruence holds for rimes at least u to 0, 000. Examle. [9, (5.5.5 ( 3 k k! 3 [ 5 ( 5 k + (( 5 5 k = + 5 π.

13 Here a = 5/( 5, λ = ( 5, and the Hilbert class field is K λ = Q( 5,. To see how this follows from equation (5.5.5 [9, let N = 5, k 5 =, k5 = 5, α(5 = 5 5, G 5 = 5, and The condensed form is We conjecture the following. a n (5 = (α(5 5k 5 + n 5(k 5 k 5 = ( 3 k k! 3 [( 5 / + k 5( 5 k = 5 [ ( 5 / + n π Conjecture 6.3. For any rime ± (mod 0, ( 3 [ k k! 3 ( 5 / + k 5 ( ( ( 5 ( 5 5 k (mod. This congruence holds for rimes at least u to 0, 000. Examle 3. [9, Exercise 6 ( [ 3 k 8 k! k + (7 6( 3 ( ( 3 8 k = 3 π. Here a = 8/(7 6, λ = ( 3 8, and the Hilbert class field is K λ = Q(, 3. In condensed form, this is ( 3 k k! 3 ( k [( k( 3 8k+ = 3π. Modulo 3 we claim the following. Conjecture 6.. For any rime ± (mod, ( 3 [ k k! 3 ( k ( k ( 3 8k+ (7 6( 3 (mod 3. This congruence holds for rimes at least u to 0, 000. It is our desire that the numerous techniques described in this aer will rove fruitful in further investigation of these conjectures. 7. Acknowledgements The authors would like to thank the Banff International Research Station and the Women in Numbers (WIN worksho for the oortunity to engage in this collaboration. We would esecially like to extend our gratitude to Ling Long and Gabriele Nebe, our WIN grou leaders, for their insightful comments and suort while rearing this aer. The second author thanks National Science Foundation Grant No. DMS for the comuter on which the comutations where run. The third author thanks Robert Osburn, for many encouraging and engaging conversations about this roject, and the Association for Women in Mathematics for a travel grant to attend WIN. References [ Scott Ahlgren and Ken Ono. A Gaussian hyergeometric series evaluation and Aéry number congruences. J. Reine Angew. Math., 58:87, 000. [ Scott Ahlgren and Ken Ono. Modularity of a certain Calabi-Yau threefold. Monatsh. Math., 9(3:77 90, 000. [3 Scott Ahlgren, Ken Ono, and David Penniston. Zeta functions of an infinite family of K3 surfaces. Amer. J. Math., (: , 00. [ Aurthur O. L. Atkin and Henry P. F. Swinnerton-Dyer. Modular forms on noncongruence subgrous. In Combinatorics (Proc. Symos. Pure Math., Vol. XIX, Univ. California, Los Angeles, Calif., 968, ages 5. Amer. Math. Soc., Providence, R.I., 97. 3

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