PARTITION CONGRUENCES AND THE ANDREWS-GARVAN-DYSON CRANK

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1 PARTITION CONGRUENCES AND THE ANDREWS-GARVAN-DYSON CRANK KARL MAHLBURG Abstract. In 1944, Freeman Dyson conjectured the existence of a crank function for partitions that woud provide a combinatoria proof of Ramanujan s congruence moduo 11. Forty years ater, Andrews and Garvan successfuy found such a function, and proved the ceebrated resut that the crank simutaneousy expains the three Ramanujan congruences moduo 5, 7 and 11. This note announces the proof of a conjecture of Ono, which essentiay asserts that the eusive crank satisfies exacty the same types of genera congruences as the partition function. 1. Introduction and statement of resuts Whether these guesses are warranted by the evidence, I eave to the reader to decide. Whatever the fina verdict of posterity may be, I beieve the crank is unique among arithmetica functions in having been named before it was discovered. May it be preserved from the ignominious fate of the panet Vucan! Freeman Dyson [1] A partition of n is a non-increasing ist of positive integers λ 1 λ λ k that sum to n; we write λ = n. The partition function pn is defined to count the number of distinct partitions of a given integer n. Ramanujan s ceebrated congruences for the partition function state that 1.1 p5n mod 5, p7n mod 7, p11n mod 11. Foowing the spirit of Ramanujan s own work, Watson and Atkin extended these congruences to arbitrary powers of 5, 7 and 11 []. Sporadic progress was made in proving congruences for primes up to 31, unti Ono s semina paper from 000 achieved a surprising improvement [3]. He proved the existence of infinite famiies of partition congruences for every prime 5 by deveoping the p-adic theory of haf-integra weight moduar forms. This resut was expanded to incude congruences for every Date: February 8, Mathematics Subject Cassification. 11P83. The author thanks the generous support of an NSF Graduate Research Feowship. 1

2 KARL MAHLBURG moduus coprime to 6 by Ahgren and Ono [4, 5]. These resuts are typicay much more compicated than Ramanujan s origina congruences, as dispayed by the exampe 1. p n mod 17. However, there is another side to the story, which began when Freeman Dyson wondered whether a simpe statistic might group the partitions into natura casses and expain the Ramanujan congruences. The rank of the partition λ 1 + λ + + λ k, is defined by 1.3 rankλ := λ 1 k. Dyson observed empiricay that this function decomposes the Ramanujan congruences moduo 5 and 7 into casses of equa size [1]. For exampe, 1.4 N m, 5, 5n + 4 = 1 p5n m 4, 5 where N m, N, n is the number of partitions λ of n for which rankλ m mod N. His observations were proven 10 years ater by Atkin and Swinnerton-Dyer [6]. However, even the smaest exampes show that the rank does not equay dissect the Ramanujan congruence moduo 11. Instead, Dyson conjectured that there woud be a crank function for the fina Ramanujan congruence, athough it wasn t unti forty years had passed that Andrews and Garvan defined the function and showed that 1.5 Mm, 11, 11n + 6 = 1 p11n Here Mm, N, n is defined for the crank just as N m, N, n was for the rank [7, 8]. In these historic works, they aso showed that the crank dissects the Ramanujan congruences moduo 5 and 7 in a different way than the rank. If λ 1 + λ + + λ s has exacty r ones, then et oλ be the number of parts of λ that are stricty arger than r. The crank is given by { λ 1 if r = 0, 1.6 crankλ := oλ r if r 1. Ceary pn = M0, N, n + + MN 1, N, n, and for the Ramanujan congruences, a of these summands are equa. However, this behavior is atypica, and an unpubished conjecture of Ono asserted that a different approach woud show that congruences for the partition function are reated to the crank in a universa manner. Conjecture Ono. For every prime 5 and integer τ 1, there are infinitey many non-nested arithmetic progressions An + B for which for every 0 m 1. Mm,, An + B = 0 mod τ

3 PARTITION CONGRUENCES AND THE CRANK 3 In fact, the foowing theorem shows that the crank function actuay satisfies congruences beyond those predicted by Ono. Theorem 1.1. Suppose that 5 is prime and that τ and j are positive integers. Then there are infinitey many non-nested arithmetic progressions An + B such that simutaneousy for every 0 m j 1. Mm, j, An + B 0 mod τ Remark. The frequency of such congruences is quantified ater in this note by Theorem 4.1. An obvious impication of Theorem 1.1 is that pan + B 0 mod τ as we, corresponding to the congruences found by Ahgren and Ono. This announcement begins with a review of moduar forms in Section. Foowing that, Section 3 expains how to write the generating function of the crank in terms of Kein forms. A condensed proof of Theorem 1.1 is found in Section 4. Acknowedgments The author thanks Ken Ono for his great support throughout the course of this work. The author aso appreciates the comments of the anonymous referees, which improved the exposition and presentation of the resuts in this paper.. Haf-integra weight moduar forms This section contains the basic definitions and properties of moduar forms that wi be needed in Section 4 see [] for detais. Let Γ := SL Z denote the fu moduar group of -by- matrices, and for a given moduus N, et Γ 0 N and Γ 1 N denote the subsets of matrices that are congruent to 0 and mod N, respectivey. If k 1 Z, and Γ Γ is such a congruence subgroup, then M k! Γ denotes the vector space of neary hoomorphic moduar forms of weight k for the subgroup Γ these are hoomorphic on the upper haf-pane H, and meromorphic at the cusps of Γ. The forms that are hoomorphic at the cusps are denoted by M k Γ, and the forms that vanish at the cusps by S k Γ. If Γ = Γ 0 N, then M k! Γ 0N, χ, M k Γ 0 N, χ, or S k Γ 0 N, χ denotes the appropriate space of moduar forms of weight k on Γ 0 N with Nebentypus character χ. For a meromorphic function f on the upper haf pane H, and an integer k, the sash operator is defined by fz k a c d b := cz + d k f az+b cz+d, for any c d Γ. A key property is that this is a group action in the sense that for any M 1, M Γ, fz k M 1 k M = fz k M 1 M. Let q := e πiz. If f = n=0 anqn M k Γ 0 N, χ and ψ is a Dirichet character, then the twist of f by ψ is.1 fz ψ := ψnanq n.

4 4 KARL MAHLBURG Simpe facts about Gauss sums aow one to rewrite the twist of a moduar form using the sash operator in a manner that is independent of the weight. If p is a prime and g p := p 1 v=1 v p eπiv/p is the standard Gauss sum, then p 1. fz p = g p p v=1 v p fz 1 v/p 0 1 The haf-integra weight Hecke operators are important toos for finding congruences among the coefficients of moduar forms. If fz = anqn M k Γ 0 N, χ and k is not an integer, then for a prime p N the Hecke operator is defined by.3 fz T p := ap n + χ p 1 k 1/. n n p k 3/ an + χ p p k a q n, p p where χ n := χn. n A of these operators act on spaces of moduar forms in an easiy described manner. Proposition.1. Suppose that fz = anqn M k Γ 0 N, χ. 1 For a prime p N and a non-integra haf-integer k, the action of T p is space-preserving, i.e., fz T p M k Γ 0 N, χ. If ψ is a character with moduus M, then fz ψ M k Γ 0 NM, χψ. 3 If fz is a cusp form, t 1 and 0 r t 1, then anq n S k Γ 1 Nt. n r mod t In Section 4 we wi need to simutaneousy find congruences for two haf-integra weight moduar forms of different weights and eves. This is partiay addressed by Ono s Theorem. in [9], Ahgren and Ono s proof of Lemma 3.1 in [5], and Serre s arguments in [10]. The additiona ingredients needed to prove the next theorem are the integra weight Hecke operators, the famous Shimura correspondence, and the decomposition S k Γ 1 N = S k Γ 0 N, χ, where the sum is over a even characters χ. Theorem.. Suppose that k i and N i are positive integers for 1 i r, and et g 1 z,..., g r z be haf-integer weight cusp forms with agebraic integer coefficients such that g i z S ki +1/Γ 1 N i. If M 1, then a positive proportion of primes p 1 mod N 1 N r M have the property that for every i, g i z T p 0 mod M.

5 PARTITION CONGRUENCES AND THE CRANK 5 3. The crank generating function and Kein forms Let Mm, n be the number of partitions λ of n such that crankλ = m. Using the generating function found by Andrews and Garvan [8], define 3.1 F x, z := x crankλ q λ m= Mm, nx m q n = λ = 1 q n 1 xq n 1 x 1 q n. Consider a positive integer N and set ζ := e πi/n. For any residue cass m mod N, eementary cacuations give the generating function for the crank, 3. Mm, N, nq n = 1 N = 1 N N 1 s=0 N 1 s=0 ζ ms F ζ s, zζ ms 1 q n. 1 ζ s q n 1 ζ s q n To prove congruences for this function, we need to show that it is a moduar form. Reca Dedekind s eta-function 3.3 ηz := q 1/4 1 q n, which is a moduar form of weight 1/. Perhaps ess famiiar are the Kein forms, which were studied extensivey by Kubert and Lang [11]. Definition 3.1. Let 1 s N 1. The 0, s Kein form is given by t 0,s z := ω s πi 1 ζ s q n 1 ζ s q n, 1 q n where ω s := ζ s/ 1 ζ s. Now write d for the east residue of d moduo N and set expz := e πiz. Understanding the action of Γ on the Kein forms is an important aspect of the proof of Theorem 1.1. The foowing formua comes from equation K on page 8 of [11]. Proposition 3.. If Γ c d 0 N, then t 0,s z = β t 1 c d 0,ds z, where β is a root of unity given by β := exp cs+ds ds cds. N N

6 6 KARL MAHLBURG A simpe cacuation shows that this mutipier system is aways trivia for a certain congruence subgroup. Coroary 3.3. If 1 s N 1, then t 0,s z M! 1Γ 1 N. Returning to the crank generating function, for 1 s N 1 equation 3.1 becomes 3.4 F ζ s, z = 1 ηzt 0,s z 1/4 ωsq. πi The generating function for the partition function is pnqn = 1, which 1 q n is the s = 0 term in 3.. Hence 3.5 Mm, N, nq n = 1 N 1 ω s ζ ms πin ηzt s=1 0,s z q1/4 + 1 N 4. The proof of Theorem 1.1 pnq n. For a prime 5, set δ := 1/4, and define ɛ := δ. Then define the set { } β + δ 4.1 S := 0 β 1 = 0 or ɛ. The foowing theorem is a more precise description of the congruences satisfied by the crank function, and it ceary impies Theorem 1.1. Theorem 4.1. Suppose that 5 is prime, τ and j are positive integers, and β S. Then a positive proportion of primes Q 1 mod 4 have the property that for every 0 m j 1, M m, j, Q3 n mod τ 4 for a n 1 4β mod 4 that are not divisibe by Q. For the rest of this section, et N := j be a fixed power of a fixed prime 5. Theorem. appies to moduar forms with agebraic integer coefficients, and thus 3.5 must be rescaed in defining g m z := N Mm, N, nq n+δ 4. 1 q n = 1 N 1 η ms z ωsζ πi ηz t s=1 0,s z + n z ηz. Let G m z and P z, respectivey, denote the two summands in the fina expression of 4..

7 PARTITION CONGRUENCES AND THE CRANK 7 As expained in [5], if t is a positive integer, then there is a Dirichet character χ,t such that z 4.3 E t z := ηt η t z M t 1 Γ 0 t, χ,t. This form vanishes at every cusp a with c t not dividing c, and aso satisfies E t z τ 1 mod τ+1 for any τ 0. Simiar facts about eta-quotients, aong with Coroary 3.3, show that P z = η z ηz M 1 Γ 0,, and G m z M! +1 Γ 1 N. For any function fz with a q-expansion, define fz := fz ɛ fz. The ine of argument in [5] impies that if τ is sufficienty arge, then there is some integer λ 1 and some character χ such that 4.6 P 4z η 4z E j+14z τ S λ +1/Γ max{3,j+1}, χ. We now show that a simiar cusp form exists for G m z. Lemma 4.. If τ is sufficienty arge, then there is some λ 1 such that 4.7 G m 4z η 4z E j+14z τ S λ+1/ Γ N. Proof. Recaing equation 4.3, if τ sufficienty arge, then it ony needs to be shown that G m z/η z vanishes at each cusp a with N c. If c c d Γ 0N, then the expansion of 1/η z at that cusp is q / up to a root of unity. Thus it must be proven that the expansion of G m z at a is c qh +... for some h > /4. Using Proposition 3. and equation 4.4, cacuate 4.8 G m z +1 = 1 d η N 1 z ω s ζ ms c d πi ηz β s t 0,ds z, where the β s are the roots of unity described by Proposition 3.. To find the expansion of G m z, first observe that for any v 1 v/ a b = 1 v / c d c d, where 0 1 a b a cv/ b cvv c d := / + av dv/ c d + cv. / s=1

8 8 KARL MAHLBURG Pick v d v mod for the subsequent arguments, and et g be the Gauss sum as in.. Proposition 3. and equations 4.8 and 4.9 show that 4.10 G m z +1 c d = g d N 1 1 v η z πi ηz ω s ζ ms 1 v / β s t 0,d s z. 0 1 s=1 v=1 Since d d mod N and N c, a short computation verifies that β s = β s. An additiona cacuation then shows that the first term of 4.8 is ɛ times the first term of Thus the expansion of G m z has the form q δ , and δ +1 > /4. The foowing theorem is now proved, as F z is defined by 4.6, and F m z is described by Lemma 4.. Theorem 4.3. For any τ 0 and 0 m N 1, there is a character χ, positive integers λ and λ, and moduar forms F m z S λ+1/ Γ N and F z S λ +1/Γ max{3,j+1}, χ such that g m 4z η 4z F mz + F z mod τ. Deduction of Theorem 4.1 from Theorem 4.3. Suppose that β S. Starting from the definition of g m z in 4., restrict g m 4z/η 4z to those indices n β + δ mod, which then gives a new series scaed by a factor of 1/ when β δ mod h m,β z := NMm, N, n δ q 4n 4.11 n β+δ mod = But Theorem 4.3 impies that n 4β 1 mod 4 NM m, N, n + 1 q n h m,β z F m,β z + F β z mod τ, where F m,β z and F β z are defined by restricting F mz and F z to ony those indices with n β + δ mod. Proposition.1 and Theorem. then show that a positive proportion of primes Q 1 mod 4 have the property that F m,β z T Q F β z T Q 0 mod τ for a m. Repace n by Qn in.3 to see that 4.13 NM m, N, Q3 n mod τ 4

9 PARTITION CONGRUENCES AND THE CRANK 9 for a n 1 4β mod 4 that are not divisibe by Q. Dividing by N then competes the proof. References [1] Dyson, F. 1944, Eureka Cambridge 8, [] Ono, K. 004 The Web of Moduarity, CBMS Regiona Conference Series in Mathematics 10 American Mathematica Society. [3] Ono, K. 000 Ann. of Math. 151, [4] Ahgren, S. 000 Math. Ann. 318, [5] Ahgren, S., and Ono, K. 001 Proc. of Nat. Acad. of Sci. 98, [6] Atkin, A. L., and Swinnerton-Dyer, P Proc. London Math. Soc. 3 4, [7] Garvan, F Trans. Amer. Math. Soc. 305, [8] Andrews, G., and Garvan, F Bu. Amer. Math. Soc. 18, [9] Ono, K. 001 J. Reine Angew. Math. 533, [10] Serre, J.-P L Ens. Math., [11] Kubert, D., and Lang, S Moduar Units, Grundehren der mathematischen Wissenschaften 44, Springer-Verag, New York. University of Wisconsin-Madison E-mai address: mahburg@math.wisc.edu