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Journal List > Proc Natl Acad Sci U S A > v.98(23); Nov 6, 2001

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Proc Natl Acad Sci U S A. 2001 November 6; 98(23): 12882–12884.
Published online 2001 October 23. doi: 10.1073/pnas.191488598.
PMCID: PMC60793
Copyright © 2001, The National Academy of Sciences
Mathematics
From the Cover
Congruence properties for the partition function
Scott Ahlgren* and Ken Ono†‡
*Department of Mathematics, University of Illinois, Urbana, IL 61801; and †Department of Mathematics, University of Wisconsin, Madison, WI 53706
‡To whom reprint requests should be addressed. E-mail: ono/at/math.wisc.edu.
Communicated by Richard A. Askey, University of Wisconsin, Madison, WI
Received August 15, 2000; Accepted September 17, 2001.
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Abstract
Eighty years ago, Ramanujan conjectured and proved some striking congruences for the partition function modulo powers of 5, 7, and 11. Until recently, only a handful of further such congruences were known. Here we report that such congruences are much more widespread than was previously known, and we describe the theoretical framework that appears to explain every known Ramanujan-type congruence.
1. Introduction and Statement of Results.
Let p(n) denote the usual partition function; p(n) is the number of ways to write a positive integer n as the sum of a nonincreasing sequence of positive integers. As usual, we agree that p(0) = 1 and that p(t) = 0 if t [negated set membership] Z≥0. Many of the most interesting arithmetic properties of this function were suggested (and often proved) by Ramanujan. Notice that if δ[ell] is defined by
equation M1
1.1
then the celebrated Ramanujan congruences may be written succinctly in the form
equation M2
Countless papers have been written on these three congruences and their extensions (already conjectured, and in some cases proved, by Ramanujan) to arbitrary powers of 5, 7, and 11 [see the fundamental works of Andrews, Atkin, Dyson, Garvan, Kim, Ramanujan, Stanton, and Swinnerton-Dyer (1–10)]. Each of these extensions lies within the class −δ[ell] (mod [ell]). The important role that this class plays in the theory is illustrated by the work of Kiming and Olsson (ref. 11, theorem 1), who proved that if [ell] ≥ 5 is prime and p([ell]n + β) [equivalent] 0 (mod [ell]) for all n, then β [equivalent] −δ[ell] (mod [ell]).
Work of Atkin, Newman, O'Brien, and Swinnerton-Dyer (12, 14, 15, 16) produced further congruences modulo [ell]m for primes [ell] ≤ 31 and small m. The examples discovered by Atkin and Newman in refs. 12 and 16 show that not every congruence lies within the progression −δ[ell] (mod [ell]). For example, we have
equation M3
1.2
We have shown (13, 17) that if [ell] ≥ 5 is prime and m is any positive integer, then there are infinitely many congruences of the form
equation M4
As in the case of Ramanujan's congruences, all of these arithmetic progressions lie within the class −δ[ell] (mod [ell]). To summarize, the current state of knowledge consists of a systematic theory of congruences within the progressions −δ[ell] (mod [ell]), as well as some sporadic examples of congruences that fall outside of this class. In view of this, it is natural to wonder what role the class −δ[ell] (mod [ell]) truly plays.
In this paper, we show that in general this class is not as distinguished as might have been expected. In fact, we prove that it is only one of ([ell] + 1)/2 classes modulo [ell] in which the partition function enjoys similar congruence properties. The results in this paper include the main results in refs. 13 and 17 as special cases and provide a theoretical framework that (to our knowledge) explains every known congruence for the partition function.
For each prime [ell] ≥ 5, define the integer [var epsilon][ell] [set membership] {±1} by
equation M5
1.3
and let S[ell] denote the set of ([ell] + 1)/2 integers
equation M6
1.4
Theorem 1. If [ell] ≥ 5 is prime, m is a positive integer, and β [set membership] S[ell], then a positive proportion of the primes Q [equivalent] −1 (mod 24[ell]) have the property that
equation M7
for all n [equivalent] 1 − 24β (mod 24[ell]) with gcd(Q, n) = 1.
Note that the case when β [equivalent] −δ[ell] (mod [ell]) already contains the main results in refs. 13 and 17.
In general, there is no simple description of the set of primes Q occurring in Theorem 1. However, as Atkin (12) showed, when [ell] = 5, 7, or 13, the situation can be made quite explicit. For example, Atkin proved the following (see ref. 12 for analogous results when [ell] = 7 or 13).
Theorem 2 [Atkin (12)].
  • (1) Suppose [ell] [equivalent] 4 (mod 5) is prime and n is a positive integer with [ell][nmid]n. If n [equivalent] 23[ell] (mod 120) or n [equivalent] 47[ell] (mod 120), then
    equation M8
  • (2) Suppose [ell] [equivalent] 3 (mod 5) is a prime exceeding 3, and n is a positive integer with (−n/[ell]) = −1. If n [equivalent] 23 (mod 120) or n [equivalent] 47 (mod 120), then
    equation M9
    We should remark that Newman (16) discovered the simplest example of the congruences described in the first part of Theorem 2 (i.e., the case where [ell] = 19). Notice that in either part of Theorem 2, fixing n in an appropriate residue class modulo 120[ell] yields a Ramanujan-type congruence. For example, if [ell] = 13, then the second part of Theorem 2 implies, for every integer n, that
    equation M10
    1.5
    Arguing in this manner from Theorem 1, we obtain
Theorem 3. If [ell] ≥ 5 is prime, m is a positive integer, and β [set membership] S[ell], then there are infinitely many non-nested arithmetic progressions {An + B} [subset, dbl equals] {[ell]n + β}, such that for every integer n we have
equation M11
If M is an integer coprime to 6, then Theorem 3 and the Chinese Remainder Theorem guarantee the existence of congruences modulo x2133.
In Section 2, we construct half integral weight cusp forms whose coefficients capture the relevant values of the partition function, and in Section 3 we prove Theorem 1. The proof requires certain facts arising from the theory of Galois representations associated to modular forms and Shimura's theory of half integral weight modular forms. In Section 4, we consider those progressions [ell]n + β for β [negated set membership] S[ell]. We give heuristics that cast doubt on the existence of congruences within these progressions.
2. Half integral weight cusp forms and the partition function.
We assume familiarity with standard notation and facts from the theory of integral and half integral weight modular forms. Throughout, we agree that q := e2πiz, and we identify a modular form f(z) with its Fourier expansion f(z) = ∑equation M12a(n)qn. Recall Dedekind's eta-function
equation M13
2.1
Theorem 2.1. Suppose [ell] ≥ 5 is prime and m is a positive integer. If β [set membership] S[ell], then there is an integer λ[ell],mand a modular form F[ell],m,β(z) [set membership] S(2λ[ell],m+1)/2(Γ1(576[ell]5)) ∩ Z[[q]] such that
equation M14
Proof: If [ell] ≥ 5 is prime and t is a positive integer, then
equation M15
2.2
where χ[ell],t := ((−1)([ell]t−1)/2[ell]t/•). By using standard facts, it can be shown that if [ell] [nmid] a and 0 ≤ b < t, then orda/[ell]b(E[ell],t(z)) > 0. Hence, E[ell],t(z) vanishes at those cusps of Γ0([ell]t), which are not equivalent to ∞. Also, because (1 − X)[ell] [equivalent] (1 − X[ell]) (mod [ell]), for every m > 0 we have
equation M16
2.3
If [ell] ≥ 5 is prime, then define f[ell](z) = ∑equation M17a[ell](n)qn by
equation M18
2.4
Because ∑equation M19p(n)qn=[product]equation M20(1−qn)−1, 2.1 and 2.4 imply that
equation M21
2.5
Define f[ell](z) by
equation M22
2.6
By standard facts, we have f[ell](z) [set membership] M([ell]−1)/2(Γ0([ell]3), (equation M23)). By 2.3 and 2.6, if m′ is sufficiently large, then f[ell],m′(z) := Eequation M24(z)f[ell](z) is a cusp form on Γ0([ell]3) with character χ[ell],t[center dot](equation M25) for which
equation M26
2.7
and
equation M27
2.8
By 2.5 and 2.7, we have
equation M28
2.9
equation M29
Now 2.8 shows that (f[ell],m′(z)/η[ell]([ell]z))24 vanishes at ∞. Therefore, if m′ is sufficiently large, then this form vanishes at every cusp. It follows that f[ell],m′(24z)/η[ell](24[ell]z) is a cusp form on Γ0(576[ell]3). We have the general fact that if f(z) = ∑equation M30a(n)qn [set membership] Sλ+½(Γ1(N)), and r and t are positive integers, then ∑n[equivalent]r (mod t)a(n)qn [set membership] Sλ+½(Γ1(Nt2)). Theorem 2.1 follows by applying this fact to f[ell],m′(24z)/η[ell](24[ell]z). □
3. Proof of Theorems 1 and 2.
We begin with some general facts. Suppose that λ [set membership] Z, and that f(z) = ∑equation M31a(n)qn [set membership] Sλ+½(Γ1(N)) has algebraic coefficients. We have a decomposition
equation M32
3.1
further, we may write f(z) = ∑χ evenαfχfχ(z), where each αfχ is algebraic, and each form fχ(z) [set membership] Sλ+½(Γ0(N), χ) has algebraic integer coefficients. Suppose that the Fourier expansion of such a form is given by fχ(z) = ∑equation M33aχ(n)qn. If p is prime, then the action of the usual Hecke operator Tχ(p2) on fχ is described by
equation M34
3.2
Using 3.1 and 3.2, we define the operator T(p2) on Sλ+½(Γ1(N)) via linearity. In particular, if f(z) = ∑equation M35a(n)qn [set membership] Sλ+½(Γ1(N)) and p [equivalent] −1 (mod N) is prime, then
equation M36
3.3
Lemma 3.1. Suppose that f(z) = ∑equation M37a(n)qn [set membership] Sλ+½(Γ1(N)) has algebraic integer coefficients. If M is a positive integer, then a positive proportion of the primes p [equivalent] −1 (mod MN) have the property that f(z) | T(p2) [equivalent] 0 (mod M).
Proof: Write f(z) = ∑χ evenαfχfχ(z) as above and choose a positive integer D such that each Dαfχ is an algebraic integer. After replacing M by DM, we see that it will suffice to prove that a positive proportion of the primes p [equivalent] −1 (mod MN) have the property that fχ(z) | T(p2) [equivalent] 0 (mod M) for every character χ.
Fix a number field K such that the coefficients of each form fχ and the values of each character χ belong to the ring of integers OK. If t is an integer, then let χt denote the usual Kronecker character for Q(equation M38). For each form fχ and for every positive squarefree integer t, we have the Shimura lift (18)
equation M39
3.4
defined by St(fχ)(z) := ∑equation M40 Aχ, t(n)qn, where the Aχ,t(n) are given by
equation M41
3.5
If M, k, and N are positive integers, then let Sk(Γ1(N))OK/M [respectively (resp) Sk(Γ0(N), χ)OK/M] denote the reductions modulo M of those forms in Sk(Γ1(N)) [resp Sk(Γ0(N), χ)] with coefficients in OK, and let T(p) [resp Tχ(p)] denote the usual integral-weight Hecke operator. Serre (ref. 19, 6.4) proved that a positive proportion of the primes p [equivalent] −1 (mod MN) have
equation M42
By using a straightforward modification of the same argument, one can show that a positive proportion of the primes p [equivalent] −1 (mod MN) have
equation M43
After 3.4, we conclude that a positive proportion of the primes p [equivalent] −1 (mod MN) have
equation M44
Because the Shimura correspondence commutes with the action of the Hecke algebra, it follows that if p is such a prime, then
equation M45
3.6
Lemma 3.1 follows from 3.5 and 3.6. □
Proof of Theorem 1: We apply Lemma 3.1 to the forms F[ell],m,β(z) given in Theorem 2.1. Fix a prime [ell] and an integer β [set membership] S[ell] and write
equation M46
equation M47
By Lemma 3.1, a positive proportion of the primes Q [equivalent] −1 (mod 24[ell]) have the property that F[ell],m,β(z) | T(Q2) [equivalent] 0 (mod [ell]m). After replacing n by Qn in definition 3.3, we see that if n [equivalent] 1 − 24β (mod 24[ell]) and gcd(Q, n) = 1, then
equation M48
because Q3n [equivalent] 24β − 1 (mod 24[ell]). Theorem 1 follows. □
4. Final Remarks.
One naturally questions whether Theorem 1 can be extended to the remaining residue classes modulo [ell]. Suppose that β [set membership] {0, … , [ell] − 1}. If we could produce a cusp form F[ell],m,β(z) as in Theorem 2.1, then we would obtain the statment of Theorem 1 for β. Because F[ell],m,0(z) would necessarily have a pole at infinity, this approach seems hopeless when β = 0. The situation, however, is less clear when β ≠ 0. For every prime [ell] ≥ 5 and every m, it is straightforward to show that there exists an integral weight modular form H[ell],m(z) such that
equation M49
However, to construct H[ell],m(z) requires the twists of f[ell](z) by all of the Dirichlet characters modulo [ell]; it results that H[ell],m(z) is a form on Γ1([ell]3). By contrast, the form that we constructed in 2.6 required only a single quadratic twist and so remained on Γ0([ell]3).
It is clear that if we had the analog for Γ1([ell]t) of the form E[ell],t(z) used in the proof of Theorem 2.1, then we could prove Theorem 1 for all nonzero β. By using the work of Hecke, it is possible to construct an Eisenstein series on Γ1([ell]t) that has the proper cusp conditions (in fact, up to scalar multiplication, exactly one such series exists). It remains to determine whether this series can be defined over the algebraic numbers, and, if so, to determine the [ell]-adic nature of its coefficients. Unfortunately, the answers to both of these problems seem to depend on the arithmetic of certain unknown values of Dirichlet L-functions at positive integral arguments. Although many of these values can be described in terms of generalized Bernoulli numbers, the remaining values are (up to unknown algebraic factors) values of certain regulators defined via canonical maps from higher K-groups into Minkowski-type spaces (20).
We conclude by remarking that computer calculations seem to cast some doubt on whether such forms exist in general. If they did, this evidence suggests a contradiction to Serre's famous result (ref. 19, theorem 4.7) that if M is any given integer, then almost all of the coefficients of an integral weight modular form with integer coefficients are multiples of M.
Acknowledgments
S.A. is supported by a National Science Foundation grant. K.O. is supported by a National Science Foundation Presidential Early Career Award, an Alfred P. Sloan Foundation Research Fellowship, and a David and Lucile Packard Research Fellowship.
References
1. Andrews G E, Garvan F. Bull Am Math Soc (NS). 1988;18:167–171.
2. Atkin A O L. Glasgow Math J. 1967;8:14–32.
3. Atkin A O L, Swinnerton-Dyer H P F. Proc London Math Soc. 1954;4:84–106.
4. Berndt B C, Ono K. Sem Lothar Comb. 1999;42:B42.
5. Dyson F J. Eureka (Cambridge). 1944;8:10–15.
6. Garvan F, Kim D, Stanton D. Invent Math. 1990;101:1–17.
7. Ramanujan S. Trans Cambridge Philos Soc. 1916;22:159–184.
8. Ramanujan S. Proc Cambridge Philos Soc. 1919;19:207–210.
9. Ramanujan S. Proc London Math Soc. 1920;18:xix.
10. Ramanujan S. Math Z. 1921;9:147–153.
11. Kiming I, Olsson J. Arch Math (Basel). 1992;59:348–360.
12. Atkin A O L. Proc London Math Soc. 1968;18:563–576.
13. Ahlgren S. Math Ann. 2000;318:795–803.
14. Atkin A O L, O'Brien J N. Trans Am Math Soc. 1967;126:442–459.
15. Atkin A O L, Swinnerton-Dyer H P F. Proc Sympos Pure Math. 1971;19:1–25.
16. Newman M. Math Comp. 1967;21:481–482.
17. Ono K. Ann Math. 2000;151:293–307.
18. Shimura G. Ann Math. 1973;97:440–481.
19. Serre J-P. L'Enseign Math. 1976;22:227–260.
20. Beilinson A A. Current Problems in Mathematics, Itogi Nauki i Tekhniki Akad Nauk SSSR, Vsesoyuz Inst Nauchn i Tekhn Inform, Moscow. 1984;24:181–238.
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