ON THE SUPERCONGRUENCE CONJECTURES OF VAN HAMME

Transcription

1 ON THE SUPERCONGRUENCE CONJECTURES OF VAN HAMME HOLLY SWISHER Abstract In 997, van Hamme develoed adic analogs, for rimes, of several series which relate hyergeometric series to values of the gamma function, originally studied by Ramanujan These analogs relate truncated sums of hyergeometric series to values of the adic gamma function, and are called Ramanujan tye suercongruences In all, van Hamme conjectured such formulas, three of which were roved by van Hamme himself, and five others have been roved recently using a wide range of methods Here, we exlore four of the remaining five van Hamme suercongruences, revisit some of the roved ones, and rovide some extensions Introduction In 9, Ramanujan listed 7 infinite series reresentations of /π, including for examle + π Γ Several of Ramanujan s formulas relate hyergeometric series to values of the gamma function In the 980 s it was discovered that Ramanujan s formulas rovided efficient means for calculating digits of π In 987, J and P Borwein [ roved all 7 of Ramanujan s identities, while D and G Chudnovsy [5 derived additional series for /π Digits of π were calculated in both aers resulting in a new world record at the time by the Chudnovsys of, 0,, digits All of these Ramanujan tye formulas for /π are related to ellitic curves with comlex multilication CM In 997, van Hamme [8 develoed adic analogs, for rimes, of several Ramanujan tye series Analogs of this tye are called Ramanujan tye suercongruences, and relate truncated sums of hyergeometric series to values of the adic gamma function In a recent aer [, the author along with S Chisholm, A Deines, L Long, and G Nebe rove a general adic analog of Ramanujan tye suercongruences modulo for suitable truncated hyergeometric series arising from CM ellitic curves Zudilin conjectured that the generic otimal strength in this setting should be modulo In all, van Hamme conjectured Ramanujan tye suercongruences, which we list below in Table We note that in the right column of Table, Sm denotes the corresonding sum from the left column truncated at m Furthermore, in the last suercongruence M, an denotes the nth Fourier coefficient of the eta-roduct fz : ηz ηz q n q n q n n anq n, where q e πiz The value of the corresonding Ramanujan series M is given in terms of the L-value Lf, This was not originally nown by van Hamme, rather it was recently shown by Rogers, Wan, and Zucer [7 Date: July, Mathematics Subject Classification C0, A0 Key words and hrases Ramanujan tye suercongruences, hyergeometric series

2 Table The van Hamme Conjectures A B C D E F G H I J K L M Ramanujan Series Conjectures of van Hamme Γ A S Γ mod, if mod 0 mod, if mod + π Γ B S Γ mod, + C S + 0 D S + π Γ Γ 8 + π Γ Γ E S 8 + πγ G S π Γ H S + π Γ mod, Γ 9 mod, if mod mod, if mod F S Γ Γ mod, if mod Γ Γ Γ mod, if mod { Γ mod, if mod 0 mod, if mod I S + π Γ J S +5 π Γ K S + 8 π Γ Γ L S π Lf, M S mod, Γ mod,, 5 Γ mod, Γ Γ mod, a mod, Proofs of the suercongruences labeled C, H, and I were given by van Hamme Kilbourn [0 roved M via a connection to Calabi-Yau threefolds over finite fields, maing use of the fact that the Calabi-Yau threefold in question is modular, which was roved by Ahlgren and Ono [, van Geemen and Nygaard [9, and Verrill [0 The conjectures, A, B and J, have been roved using a variety of techniques involving hyergeometric series McCarthy and Osburn [ roved A using Gaussian hyergeometric series The suercongruence B has been roved in three ways, by Mortenson [5 using a technical evaluation of a quotient of Gamma functions, by Zudilin [ using the W-Z method, and by Long [ using hyergeometric series identities and evaluations Long also uses a similar but more general method in [ to rove J The congruence D has now been roved by Long and Ramarishna in a recent rerint [ In addition, they rove that H holds modulo when mod, and rovide extensions for D and H to additional rimes This leaves five congruences left to rove: E, F, G, K, and L In this aer, we first observe that Long s method to rove B in [ can be used to rove C, E, F, G, and L as well Furthermore we extend E, F, and G to results for additional rimes, and show that C and G holds in fact modulo We also revisit A to show it holds in fact modulo 5 when mod Using similar methods, He [8, 7 has indeendently verified the results for cases E, F, and G

3 As mentioned earlier, since the K conjecture corresonds to a CM ellitic curve, we now that it holds modulo by [, however our method did not yield the full strength modulo However, K has been recently roven by Osburn and Zudilin using the W-Z method [ We rove the following theorems Theorem Let a {,, }, and an odd rime we require 5 when a / Let b when mod a, and let b a when mod a Then ab a + a b ab b mod Γ aγ a We observe that when a,, and mod a, Theorem gives B, E, and F, resectively Furthermore, for rimes mod, Theorem yields the following generalization of E + mod Similarly, for rimes mod, Theorem yields the following generalization of F 8 + Γ Γ mod Theorem Let a {, }, and an odd rime we require 5 when a / Let b when mod a, and let b a when mod a Then ab a a + ab bδ Γ aγ a mod, where δ δ ab when a, b {,,, }, and δ / when a, b, We observe that when a, Theorem gives C modulo the stronger ower When a and mod, Theorem gives G modulo the stronger ower Moreover, when a and mod for 5, Theorem gives the following generalization of G modulo, Theorem For any odd rime, 8 + Γ Γ mod + 8 Γ Γ mod Theorem yields the following corollary, which is difficult to rove otherwise see Remar of [ Corollary For any odd rime, + j j j 8 0 mod

4 We also have the following theorem which strengthens the A congruence when mod to a congruence modulo 5 Theorem 5 For any rime mod, with > Γ mod 5 In Section we discuss the gamma and -adic gamma functions, as well as some useful lemmas In Sections - we rove our results In Section 7, we conclude with some more general conjectures, which are suorted by comutational evidence from wor done in Sage This leaves only K from the original van Hamme conjectures, which doesn t seem to yield to this method As mentioned earlier, since this case corresonds to a CM ellitic curve, we now that K holds modulo by [; however it remains to be roved modulo as conjectured Preliminaries In this section we review hyergeometric series notation, some facts about the gamma function Γz, and the adic gamma function Γ z First, we give a lemma that will be imortant for us later Recall the definition of the rising factorial for a ositive integer, a : aa + a + For r a nonnegative integer and α i, β i C, the hyergeometric series r+ F r is defined by [ α α r+ α α α r+ r+f r ; λ λ β β r β β r, which converges for λ < We write [ α α r+ r+f r ; λ β β r n n to denote the truncation of the series after the λ n term α α α r+ β β r λ, Lemma Let be rime and ζ a rimitive nth root of unity for some ositive integer n If a, b Q Z and is a ositive integer such that a + j Z for each 0 j, then a b a bζ a bζ n a n mod n, and does not vanish modulo Moreover for an indeterminate x, and is invertible in Z [[x n a bx a bζx a bζ n x Z [[x n, Proof Exanding each term as a rising factorial, we can write a b a bζ a bζ n n a + j bζ i j0 i0 Let σ i x 0,, x n denote the ith elementary symmetric olynomial in n variables Then we have σ i, ζ,, ζ n 0, for i n and σ n, ζ,, ζ n ± For a fixed 0 j, we thus have n a + j bζ i a + j n ± b n n a + j n mod n i0

5 Together with we see that the result holds, and is nontrivial recisely when a + j Z for each 0 j Relacing by x in gives the additional result for series Since the constant term for a bx a bζx a bζ n x is j0 a + jn Z, the series is invertible in Z [[x n The gamma function The gamma function Γz is a meromorhic function on C with simle oles recisely at the nonositive integers, which extends the factorial function on ositive integers, namely 5 Γn n! for ositive integers n It also satisfies the functional equation Γz + zγz, which immediately yields that for comlex z and ositive integers, 7 Γz + Γz z Also imortant is the following reflection formula due to Euler For comlex z, π 8 ΓzΓ z sinπz Furthermore, Γ satisfies the dulication formula 9 ΓzΓ z + z πγz The following lemma will be useful in the next section Lemma Let a, b Q and rime such that ab is a ositive integer Then Γ abγ + ab Γ aγ + a ab b Proof By 7, we have that Γ + ab + a Γ + a ab and Γ a Γ ab ab ab ab a ab, which gives the desired result The adic gamma function We note that many of the facts we state in this section can be found in Morita [ Let be an odd rime Set Γ 0, and for ositive integers n, define 0 Γ n n j 0<j<n j The adic gamma function is the extension to Z defined by Γ α lim n α Γ n, where n are ositive integers adically aroaching α With this definition, Γ α is a uniquely defined continuous function on Z 5

6 For α Z, the following fact is found in [: Γ α + Γ α From, the following lemma follows immediately { α if α Z, if α Z Lemma For a ositive integer, and α Z, if α, α +,, α + Z, then More generally, We also have for x Z that Γ α + Γ α Γ α + Γ α α α + j j0 α+j Z Γ xγ x a 0x, where a 0 x is the least ositive residue of x modulo Let G a Γ a/γ a, where Γ denotes the th derivative of Γ, and G 0 a Long and Ramarishna [ show that for a Z, G 0 a, G a G a, G a + G a G a Furthermore, they rove the following useful theorem Theorem Long, Ramarishna [ Let 5 be rime, r a ositive integer, a, b Q Z, and t {0,, } Then Γ a + b r Γ a t G a b r mod t+r Remar 5 Fixing r, we can extend this result modulo when > 5 Let a Q Z, b Z, and let v x denote the -order of x From the roof of Theorem in [ using also their Proosition we have that Γ a + b Γ a G a b We can show that v G a b when > 5 This is because as in [, v G a 0 for all <, and in general, v G a + So we have that v G a b when < For, we have that and so the inequality we need is G a v b +, + The inequality holds for all 5 when, and holds for all when 7 When 5, we see that holds for all 8, which leaves the cases 5,, 7 When, 7 a

7 calculation shows that v G a b, however the 5 case remains elusive We thus obtain that for rimes > 5, 5 Γ a + b Γ a G a b mod Proof of B, E and F with generalizations For this section we will mae use of the following identities of While see 5 and in [ F + a c d a + a c + a d ; Γ + a cγ + a d Γ + aγ + a c d, and 7 F + a c d e f 5 a + a c + a d + a e + a f ; Γ + a eγ + a f Γ + aγ + a e f F [ + a c d e f + a c + a d ; Observe that the left hand side targets for B, E, and F can all be exressed by the truncated hyergeometric series 8 a [ a a + a F + a a a a ;, a where a {,, }, and is a rime such that mod a We will also consider rimes for which mod a to obtain additional van Hamme tye suercongruences related to E and F We now rove Theorem Proof of Theorem For fixed a {,, } and an odd rime, let b Q be defined by { if mod b ba, a, a if mod a, so that a b is a negative integer, and b Z we require 5 when a / Consider the hyergeometric series F F + a a b a + b a + ab ab ; Then F naturally truncates at ab since a b is a negative integer Letting c a b, d a + b in, we get using Lemma 9 F Γ + abγ ab Γ + aγ a ab b When mod a, shows that F is the right hand side target for B, E, and F Switching to a variable x and truncating at ab, we define F x F + a a bx a + bx a + abx abx ; 7 ab

8 By Lemma, F x Z [[x, and so F x C 0 + C x + C x +, for C i Z Notice that C 0 is recisely our left hand side target 8 Thus if C, then letting x gives the desired congruence F C 0 mod To see that C, let c a bx, d a + bx, e, and f a b in 7 We then have by, 0 F 5 + a a bx a + bx a b a + abx abx a + ab ; ΓaΓ + ab Γ + aγab F [ a a b + abx abx ; b F [ a a b + abx abx ; The hyergeometric series on both sides of 0 naturally truncate at ab since a b is a negative integer Also, modulo, the left hand side is congruent to F x Thus, as a series in x, we have [ a a b F x b F + abx abx ; mod Thus C and we have roven Theorem Proof of C and G modulo with a generalization For this section we use the following identities of While see 5 and 77 in [ 5F + a c d e a + a c + a d + a e ; Γ + a cγ + a dγ + a eγ + a c d e Γ + aγ + a d eγ + a c dγ + a c e, and 7F + a c d e f g a + a c + a d + a e + a f + a g ; Γ + a eγ + a fγ + a gγ + a e f g Γ + aγ + a f gγ + a e fγ + a e g F [ + a c d e f g e + f + g a + a c + a d ; rovided the F series terminates As in Section, we first observe that the left hand side targets for C, and G can be exressed by the truncated hyergeometric series a a a + 5F + a a a a a ;, a where a {, }, and is a rime such that mod a We will also consider rimes for which mod a when a / to obtain an additional van Hamme tye suercongruence related to G We now rove Theorem 8,

9 Proof of Theorem For fixed a {, } and an odd rime, define b Q by { if mod b ba, a, a if mod a, so that a b is a negative integer, and b Z requiring that 5 when a / Let ω be a rimitive third root of unity and consider the hyergeometric series G 5 F + a a b a bω a bω a + ab + abω + abω ;, which naturally truncates at ab Moreover by Lemma, 5 G 5 F + a a a a a ; mod ab Letting c a b, d a bω, and e a bω in, gives that G Γ + abγ + abωγ + abω Γ a Γ + aγ a abγ a abω Γ a abω We will show in that this gives the right hand side target from Theorem In the meantime, as in Section, we consider the series obtained from G by switching with an indeterminate x and truncating at ab, Gx 5 F + a a bx a bωx a bω x a + abx + abωx + abω x ; ab By Lemma, Gx Z [[x, and so we have Gx C 0 + C x + C x +, for C i Z where C 0 is our left hand side target Thus if C, then letting x gives the desired congruence G C 0 mod Let c a bωx, d a bω x, e a bx, f a b, and g in to get 7 7F + a a bωx a bω x a bx a b a + abωx + abω x + abx + ab a ; Γ + abxγ + abγaγabx + b Γ + aγabγabxγ + abx + b F [ a abx a bx a b abx + b + abωx + abω x ; where since a b is a negative integer, both sides of 7 terminate at ab By, we have that Γ + abxγ + abγaγabx + b Γ + aγabγabxγ + abx + b b x bx + b Z [[x, since the integers b and b are in Z As series in x, 7F + a a bωx a bω x a bx a b a + abωx + abω x + abx + ab a ; Gx mod, thus C 0, as desired To finish the roof of Theorem, it remains to show that G gives the aroriate right hand side target 9,

10 Fix ab From and 7 we can rewrite G as G + a a ab + ab + abω + abω + ab We first observe that since + is a ositive integer, Lemma, 0, and show that the denominator satisfies the congruence 8 + abω + abω + ab Γ + /Γ mod To next evaluate the numerator in terms of adic gamma functions, we emloy Lemma for α { + a, + ab, a ab} When α + a, the factors α, α +,, α + are not in Z, but α + ab Thus using, 9 + a ab Γ + a + Γ + a b Γ + ab Γ a b Γ aγ + ab When α + ab, one sees that none of the factors α, α +,, α + are in Z, so that 0 + ab Γ a + ab Γ + ab Finally, when α a ab, we first note that for a /, each factor α + j is an integer in the range < α + j so none are in Z When a /, we see that each factor α + j is in Z When mod, b so each α + j satisfies / < α + j / and so none are in Z Thus using we have when a, b {,,, }, a ab Γ a Γ a ab Γ aγ a + ab When mod, b, the integer j / < yields α + j / Since each α + j in this case satisfies < α + j /, this is the only factor in Z Thus in this case, Putting this together, we have a ab Γ aγ a + ab a ab δ ab Γ aγ a + ab, where δ ab is defined to be when a, b {,,, }, and / when a, b, Combining 8, 9, 0,, and using, the factor of at least one gives the following congruence modulo, G bδ ab Γ aγ aγ a ab Γ a + abγ a + ab mod By Theorem, we have that Γ a ab Γ a [ abg a + a b G a Γ a + ab Γ a [ + abg a + a b G a mod mod Γ a + ab Γ a [ + abg a + a b G a mod Using, we see that and so Γ a ab Γ a + abγ a + ab Γ a mod, G bδ ab Γ aγ a mod 0

11 as desired 5 Proof of L For this section we use the following identity from [9 see 8 which gives that a + b b F a+b+ a b+ ; a+b+ Γ Γ a b+ a 8 Γ a+ Γ a+ Proof of Theorem Let be an odd rime Observe that using, the right hand side target for L can be written as 8 + Γ Γ The left hand side target for L can be exressed by the truncated hyergeometric series [ F Consider the hyergeometric series L F [ 7 ; Then L naturally truncates at since is a negative integer As shown in [ Lemma, L, our right hand side target for L Switching to a variable x and truncating at, we define Lx F [ 7 x +x x + x By Lemma, Lx Z [[x, and so Lx C 0 + C x + C x +, for C i Z Notice that C 0 is recisely our left hand side target Thus if C, then letting x gives our desired congruence L C 0 mod To see that C, observe that as a series in x, [ 7 F x +x x + x ; 8 ; 8 ; 8 Lx mod, where since the left hand side is naturally truncating at, we actually have that it is a rational function in x Moreover, letting a and b x in, we see that this rational function is actually 0, since [ 7 F x +x x + x ; 8 Γ x Γ + x Γ Γ 5 and one of or 5 is a nonositive integer, yielding a ole for Γ

12 Proof of Corollary By Lemma, we have directly that L is congruent to our left hand side target modulo Considering the difference, L , we observe that due to cancelation, + + j j +, where the remaining terms all have a factor of n for n Also we observe that the denominator + contains no factors of Thus as a corollary to Theorem we obtain the desired result, 5 + j j 0 mod 8 j j Proof of A modulo 5 for mod For this section we use the following identity from [ see Thm 55 ii, which gives that [ a b c F e f ; πγeγf when a + b and e + f c + c Γ a+e Γ a+f b+e b+f Γ Γ, Proof of Theorem 5 Let mod be rime, with > 5 Observe that the left hand side target for A can be exressed by 7 Consider the hyergeometric series 8 A F 5 [ 5 + [ F 5 i which naturally truncates at By Lemma, 9 A F 5 [ 5 +i ; + + i i + ; ; mod,

13 Letting c i, d +i, e +, and f A Γ + Γ [ Γ F in 7, gives that Γ + i i ; Note that by and 8, Γ Γ π Thus, letting a, b +, c, e + i, and f i in, we see that a + b and e + f c, so we obtain Γ + Γ Γ + i Γ i 0 A Γ Γ Γ Γ i +i + i + ++i We will show in that this gives the right hand side target from Theorem 5 modulo 5 In the meantime, as in Sections and, we consider the series obtained from A by changing to an indeterminate x and truncating at Ax F 5 [ 5 ix, +ix x +x + ix ix + x x By Lemma, Ax Z [[x, and so we have Ax C 0 + C x + C 8 x 8 +, for C i Z where C 0 is our left hand side target 7 Thus if C, then letting x gives the desired congruence A C 0 mod 5 Considering instead [ 5 A x F 5 ix +ix x ; +x + ix ix + x x we see that A x naturally truncates at, so is actually a rational function in x in Z [[x Modulo, Ax and A z have the same coefficients in Z [[x However, by 7, we see that A x Γ + x Γ x Γ Γ F [ x +x + ix ix where the F series naturally truncates and is a rational function in x and in Z [[x The Γ-factor in front however gives that A x 0, since, are negative integers Thus since modulo, Ax and A z have the same coefficients in Z [[x we must have that C in fact all of the C i, and so [ 5 A F 5 ; mod 5 Fix We now show that A is congruent to the right hand target modulo 5 Starting from 0, we first observe that 7 together with 8 yields that Γ + Γ + π Furthermore, by 9 we have that Γ + i Γ i π Γ + i Γ i ; ; Γ + i Γ i,,

14 Using 7, we can thus rewrite A from 0 as A +i i + i i We now use Lemma for α {,, + i, i, +i, i } to analyze the factors in terms of Γ Note that since mod we have i Z When α, the factors α, α+,, α+ are not in Z, but α + When α, none of the factors α + j are in Z Thus with and we have, Γ + Γ Γ Γ + Γ Γ When α ± i, we have that none of the factors α + j are in Z, and so + i i Γ + i Γ i Γ + +i Γ + i Similarly when α ±i none of the factors α + j are in Z, so + i i Γ + i Γ i Γ + +i Γ + i Together,, and give that A Γ + Γ Γ + i Γ i Γ + i Γ i Γ + +i Γ + i Γ + +i Γ + i Using the discussion in Remar 5 we can analyze this quotient modulo First, note that Γ + Γ Γ [ + G G mod, using that G G by Similarly, Γ + i Γ i [ Γ G + G Also, since Γ / by we have Γ + i Γ i [ Γ G + G mod mod Using the same technique on the denominator we see that [ i + i + Γ + Γ Γ i G G 8 [ i i Γ + Γ Γ + i G G 8 mod mod and so Γ + i i + i i + Γ + Γ + Γ + Γ mod

15 Putting this together, the factor of in front gives the desired congruence modulo 5 A Γ Γ mod 5 7 Conjectures The following more general van Hamme tye congruence conjectures are suorted by comutational evidence comuted with Ling Long and Hao Chen using Sage Note that some of these conjectures extend van Hamme s conjectures in the r case, which motivated several of the theorems in this aer A B C D S r Γ S r mod 5r mod, r S 0 mod mod S r S r mod 5r mod, r S r Γ S r mod r mod S Γ mod mod S r S r mod r mod, r r r S S mod r S r Γ 9 S r mod r mod E S 0 mod mod S r S r mod r+ mod, r even S r 5 S r mod r mod, r odd S r S r mod r mod S r S r mod r mod, r even S 0 mod mod S r S r mod r mod, r odd 5

16 F S r 8 S r mod r mod S r S r mod r mod, r even S r S r mod r mod, r odd G S r 8 Γ S r mod r mod S r S r mod r mod, r even S r S r mod r+ mod, r odd H I J K For > 5, L r S S r Γ S r mod r mod S 0 mod mod S r S r mod r mod, r r S r mod r+ r S r S S mod r 5 mod S r S r 8 S r S mod r r mod r 8 Acnowledgements The author would lie to than Ling Long for numerous helful conversations which rovided the ideas behind this wor, and Robert Osburn for many insiring conversations and for introducing her to the van Hamme conjectures The author also thans Tulane University for hosting her while woring on this roject, and Sage Days 5, where she ran comutations related to this roject with Ling Long and Hao Chen

17 References [ Scott Ahlgren and Ken Ono Modularity of a certain calabi-yau threefold Monatshefte für Mathemati, 9:77 90, 000 [ George E Andrews, Richard Asey, and Ranjan Roy Secial functions, volume 7 of Encycloedia of Mathematics and its Alications Cambridge University Press, Cambridge, 999 [ Jonathan M Borwein and Peter B Borwein Pi and the AGM Canadian Mathematical Society Series of Monograhs and Advanced Texts John Wiley & Sons Inc, New Yor, 987 A study in analytic number theory and comutational comlexity, A Wiley-Interscience Publication [ Sarah Chisholm, Alyson Deines, Ling Long, Gabriele Nebe, and Holly Swisher adic analogues of Ramanujan tye formulas for /π Mathematics, :9 0, 0 [5 David V Chudnovsy and Gregory V Chudnovsy Aroximations and comlex multilication according to Ramanujan In Ramanujan revisited Urbana-Chamaign, Ill, 987, ages 75 7 Academic Press, Boston, MA, 988 [ Bernard Dwor A note on the -adic gamma function In Study grou on ultrametric analysis, 9th year: 98/8, No Marseille, 98, ages Ex No J5, 0 Inst Henri Poincaré, Paris, 98 [7 Bing He On a -adic suercongruence conjecture of l van hamme 05 [8 Bing He Some congruences on truncated hyergeometric series 05 [9 Per W Karlsson Clausen s hyergeometric function with variable /8 or 8 Math Sci Res Hot-Line, 7:5, 000 [0 Timothy Kilbourn An extension of the Aéry number suercongruence Acta Arith, :5 8, 00 [ Ling Long Hyergeometric evaluation identities and suercongruences Pacific J Math, 9:05 8, 0 [ Ling Long and Ravi Ramarishna Some suercongruences occurring in truncated hyergeometric series rerint arxiv:05 [ Dermot McCarthy and Robert Osburn A -adic analogue of a formula of Ramanujan Arch Math Basel, 9:9 50, 008 [ Yasuo Morita A -adic analogue of the-function J Fac Sci Univ Toyo, :55, 975 [5 Eric Mortenson A -adic suercongruence conjecture of van Hamme Proc Amer Math Soc, : 8, 008 [ Robert Osburn and Wadim Zudilin On the suercongruence of van hamme 05 [7 M Rogers, J G Wan, and I J Zucer Moments of ellitic integrals and critical l-values The Ramanujan Journal, 7: 0, 05 [8 Lucien van Hamme Some conjectures concerning artial sums of generalized hyergeometric series Lecture Notes in Pure and Al Math, 9:, 997 [9 Bert Vangeemen and Niels O Nygaard On the geometry and arithmetic of some siegel modular threefolds Journal of Number Theory, 5:5 87, 995 [0 H A Verrill Arithmetic of a certain Calabi-Yau threefold In Number theory Ottawa, ON, 99, volume 9 of CRM Proc Lecture Notes, ages 0 Amer Math Soc, Providence, RI, 999 [ F J W While On well-oised series, generalized hyergeometric series having arameters in airs, each air with the same sum Proc London Math Soc, s-:7, 9 [ Wadim Zudilin Ramanujan-tye suercongruences J Number Theory, 98:88 857, 009 Deartment of Mathematics, Oregon State University, 8 Kidder Hall, Corvallis, OR 97, USA address: swisherh@mathoregonstateedu 7