THE PARTITION FUNCTION AND HECKE OPERATORS

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1 THE PARTITION FUNCTION AND HECKE OPERATORS KEN ONO Abstract. The theory of congruences for the partition function p(n depends heaviy on the properties of haf-integra weight Hecke operators. The subject has been compicated by the absence of cosed formuas for the Hecke images P (z T ( 2, where P (z is the reevant moduar generating function. We obtain such formuas using Euer s Pentagona Number Theorem and the denominator formua for the Monster Lie agebra. As a coroary, we obtain congruences for certain powers of Ramanujan s Deta-function.. Introduction and statement of resuts A partition of an integer n is a non-increasing sequence of positive integers that sum to n. Ramanujan investigated [7, 8] p(n, the number of partitions of n, and he proved that p(5n (mod 5, p(7n (mod 7, p(n (mod. These congruences have inspired many works (for exampe, see [, 2, 3, 5, 6, 7, 8, 0,, 2, 3, 4, 5, 9, 20, 2] to name a few. In particuar, Atkin [5] and Watson [9] proved Ramanujan s conjectures concerning congruences moduo powers of 5, 7 and. In the 60s, Atkin [6] surprisingy discovered congruences moduo some primes M 3 by making use of haf-integra weight Hecke operators. For exampe, he proved that (. p( n (mod 9. In the ate 90s, the author revisited Atkin s work using -adic Gaois representations and Shimura s theory of haf-integra weight moduar forms [5], and he proved that there are such congruences moduo every prime M 5. Ahgren and the author [, 2] ater extended this to incude a modui M coprime to 6. Other recent works by Weaver and Yang [20, 2] provide further resuts aong these ines. Despite these works, itte is known about the action of the Hecke operators on the partition generating function. To make this precise, we begin by recaing Dedekind s eta-function η(z := q n= ( qn (note. q := e 2πiz throughout. The moduar partition generating function is the weight /2 moduar form (.2 /η(z = P (z := p(nq n. The author thanks the support of the NSF, the Hidae Foundation, the Manasse famiy, and the Cander Fund for their generous support. n=0

2 2 KEN ONO For primes 5, we have the normaized Hecke action (for exampe, see 3. of [6] (.3 a(nq n T ( 2 := ( 3n 3 a(n 2 + a(n + a(n/ 2 q n. n n The genera theory of partition congruences depends on the properties of P (z T ( 2, and in the absence of a cosed formua, researchers have been required to design specia arguments which, under very specia circumstances, yied congruences such as (.. Here we consider the seemingy difficut probem of obtaining cosed formuas for P (z T ( 2. We obtain a simpe soution to this probem by making use of Euer s Pentagona Number Theorem and the denominator formua for the Monster Lie agebra. To state these formuas, et E 4 (z and E 6 (z be the usua Eisenstein series (.4 E 4 (z := + 0 σ 3 (nq n and E 6 (z := 504 σ 5 (nq n, n= where σ v (n := d n dv. Let (z be Ramanujan s weight 2 cusp form (.5 (z := η(z = q ( q n, and et j(z be Kein s moduar function (.6 j(z := E 4 (z 3 / (z = q q Finay, et (q; q be Euer s Pentagona Number generating function (.7 (q; q = q n=( n = ( k q (3k2 +k/2. k Z Using these q-series, we define poynomias A(m; x Z[x] as the coefficients of the series A(q = A(m; xq m := (q; q E4(z 2 E 6 (z (.8 (z j(z x m=0 n= n= = + (x 745q + (x 2 489x + 605q Remark. Each A(m; x is a monic degree m poynomia with integer coefficients. We show that P (z T ( 2 is obtained by mutipying P (z with A(( 2 /; j(z. Theorem.. If 5 is prime and δ := ( 2 /, then ( 3 P (z T ( 2 = P (z + A(δ ; j(z. N Remark. For fixed 5, this gives a method (see Exampe 3.3 for computing p 2 +. One needs A(δ ; x and short initia segments of j(z and P (z. It suffices to compute ( 3 P (z + A(δ ; j(z = q O(q N+.

3 THE PARTITION FUNCTION AND HECKE OPERATORS 3 Theorem. aso gives the foowing congruences for powers of the Deta-function. Coroary.2. If 5 is prime, then we have that (z δ A(δ ; j(z (mod. In Section 2 we reca the denominator formua for the Monster Lie agebra, and we then use a cassica emma due to Atkin on P (z to then prove Theorem.. In Section 3 we give some exampes of Theorem. and Coroary Proofs Here we prove Theorem. and Coroary.2. We begin by recaing Faber poynomias, a sequence of poynomias whose generating function is essentiay equivaent to the denominator formua for the Monster Lie agebra. 2.. Faber poynomias. If p := e 2πiτ, then the denominator formua for the Monster Lie agebra is j(τ j(z = p ( p m q n c(mn. m>0 and n Z Here the exponents c(n are the coefficients of j(z. This identity may be reformuated in terms of a sequence of moduar functions j m (z. We et j 0 (z := and j (z := j(z 744. For m 2 we et j m (z be the unique moduar function on SL 2 (Z with an expansion of the form (2. j m (z = q m + c m (nq n. It is not difficut to show that the denominator formua is equivaent to ( j(τ j(z = p exp j n (z pn. n The moduar functions j m (z are speciaizations of poynomias J m (x which were previousy defined by Faber [9] (aso see [4]. These poynomias are defined by the generating function (2.2 J m (xq m := E 4(z 2 E 6 (z (z j(z x = +(x 744q +(x2 488x+59768q m=0 Here we reca some of the main properties of these poynomias (see [4, 9, 22]. Theorem 2.. Assuming the notation above, the foowing are true. n= n= ( If m 0, then j m (z = J m (j(z. (2 If m 2, then j m (z = J (j(z T 0 (m, where T 0 (m is the normaized mth weight 0 Hecke operator.

4 4 KEN ONO 2.2. Proof of Theorem.. If 5 is prime, then define F (z by (2.3 F (z := η(z (P (z T ( 2. The nonzero coefficients are supported on exponents which are mutipes of. After etting z z/, a standard argument invoving the definition of T ( 2 and the transformation aw for Dedekind s eta-function impies that F (z is a moduar function on SL 2 (Z. (note. This fact was previousy observed by Atkin (see Lemma 2 of [6]. Since F (z is hoomorphic on the upper haf of the compex pane, it is a poynomia in j(z. By direct cacuation, we have that 3 P (z T ( 2 = q 2 + q + O(q 23. Euer s Pentagona Number Theorem then gives 3 ( F (z = q ( k q 2 +ω(k + q 2 +ω( k k= where ω(k := (3k 2 + k/2. By etting z z/, we obtain F (z = q δ ( ( k q δ+ω(k + q δ +ω( k + O(q. k= + O(q 23, We now show that this poynomia in j(z is A(δ ; j(z. By Euer s Pentagona Number Theorem, Theorem 2., (.8, (2. and (2.2, it foows that A(δ ; j(z is a moduar function on SL 2 (Z with the property that 3 + A(δ ; j(z F (z = O(q. This moduar function must then be a poynomia in j(z. Since every nonconstant moduar function on SL 2 (Z has a poe, and since this function does not have a poe at infinity, we have 3 + A(δ ; j(z = F (z. After etting z z, the theorem foows from (2.3 by dividing A(δ ; j(z by η(z = q(q ; q. The proof is compete because of the fact that P (z = η(z Proof of Coroary.2. If 5 is prime, then Theorem. impies that By (.3, we have that P (z T ( 2 P (z A(δ ; j(z P (z T ( 2 Putting these together, we find that p(nq (n 2 P ( 2 z n=0 A(δ ; j(z P (z P ( 2 z (mod. (mod. (mod.

5 THE PARTITION FUNCTION AND HECKE OPERATORS 5 By direct cacuation we have P (z P ( 2 z = η(2 z η(z and so the coroary foows by etting z z/. η(z 2 = (z δ (mod, 3. Exampes Here we iustrate the resuts described in the introduction. Exampe 3.. Here we iustrate Theorem. for = 7. Then we have that A(δ 7 ; x = A(2; x = x 2 489x Therefore, we find that A(2; j(z = q 48 q q q , 7 which in turn gives ( 3 P (z A(2; j(z = q 49 7q q q This iustrates Theorem. since one directy finds that P (z T (7 2 = q 49 7q q q Exampe 3.2. Here we iustrate Coroary.2 for = 3. We have that A(δ 3 ; x = A(7; x = x x x x x x x , which in turn gives A(7; j(z = q7 + q 8 + 2q q 5... q 7 + q 8 + 2q 9 + 3q 0 + 5q + 7q 2 + q 3 + 2q 4 + 9q 5 + 4q (mod 3. This iustrates Coroary.2 because (z δ 3 = (z 7 = q 7 68q q q q 7 + q 8 + 2q 9 + 3q 0 + 5q + 7q 2 + q 3 + 2q 4 + 9q 5 + 4q (mod 3. Exampe ( 3.3. Here we iustrate how one may efficienty compute partition numbers of the form p. We consider the simpe case where N = 7 and = 5, and so our aim is to N 2 + cacuate p(74. We compute p(74 using p(0,..., p(4, the first five coefficients of j(z, and the poynomia A(δ 5 ; x = x 745. By (.2 and (.3, if we et n a 5 (nq n := P (z T (5 2,

6 6 KEN ONO then we find that 7 5 a 5 (7 = p 5p = 5 3 p(74 5p(3. By Theorem., we have that P (z T (5 2 = P (z (j(z 750 = ( q + q q q 7 + 5q (q q +... = q 25 5q q q q Since a 5 (7 = and p(3 = 3, we then find that p(74 = References [] S. Ahgren, The partition function moduo composite integers M, Math. Ann. 38 (2000, pages [2] S. Ahgren and K. Ono, Congruence properties for the partition function, Proc. Nat. Acad. Sci., USA 98 (200, pages [3] G. E. Andrews, The theory of partitions, Cambridge Univ. Press, Cambridge, 984. [4] T. Asai, M. Kaneko, and H. Ninomiya, Zeros of certain moduar functions and an appication, Comm. Math. Univ. Sancti Paui 46 (997, pages [5] A. O. L. Atkin, Proof of a conjecture of Ramanujan, Gasgow Math. J. 8 (967, pages [6] A. O. L. Atkin, Mutipicative congruence properties and density probems for p(n, Proc. London Math. Soc. (3 8 (968, pages [7] A. O. L. Atkin and J. N. O Brien, Some properties of p(n and c(n moduo powers of 3, Trans. Amer. Math. Soc. 26 (967, pages [8] A. O. L. Atkin and H P. F. Swinnerton-Dyer, Some properties of partitions, Proc. London Math. Soc. 66 No. 4 (954, pages [9] G. Faber, Über poynomische Entwickeungen, Math. Ann. 57 (903, pages [0] M. Hirschhorn and D.C. Hunt, A simpe proof of the Ramanujan conjecture for powers of 5, J. Reine Angew. Math. 326 (98, pages 7. [] T. Hjee and T. Kove, Congruence properties and density probems for coefficients of moduar forms, Math. Scand. 23 (968, pages [2] J. Lovejoy and K. Ono, Extension of Ramanujan s congruences for the partition function moduo powers of 5, J. reine Ange. Math. 542 (2002, pages [3] M. Newman, Periodicity moduo m and divisibiity properties of the partition function, Trans. Amer. Math. Soc. 97 (960, pages [4] M. Newman, Congruences for the coefficients of moduar forms and some new congruences for the partition function, Canadian J. Math. 9 (957, pages [5] K. Ono, The partition function moduo m, Ann. of Math. 5 (2000, pages [6] K. Ono, The web of moduarity: arithmetic of the coefficients of moduar forms and q-series, CBMS 02, Amer. Math. Soc., Providence, [7] S. Ramanujan, Congruence properties of partitions, Proc. London Math. Soc. (2 9, pages [8] S. Ramanujan, Ramanujan s unpubished manuscript on the partition and tau-functions (Commentary by B. C. Berndt and K. Ono The Andrews Festschrift (Ed. D. Foata and G.-N. Han, Springer-Verag, Berin, 200, pp [9] G. N. Watson, Ramanujan s vermutung über zerfäungsanzahen, J. reine Angew. Math. 79 (938, pages [20] R. Weaver, New congruences for the partition function, Ramanujan J. 5 (200, pages [2] Y. Yang, Congruences of the partition function, Int. Math. Res. Notices, to appear. [22] D. Zagier, Traces of singuar modui, Motives, poyogarithms, and Hodge theory I (Ed. F. Bogomoov and L. Katzarkov, Int. Press, Somervie, (2003, pages 2-4.

7 THE PARTITION FUNCTION AND HECKE OPERATORS 7 Department of Mathematics and Computer Science, Emory University, Atanta, Georgia E-mai address: ono@mathcs.emory.edu Department of Mathematics, University of Wisconsin, Madison, Wisconsin E-mai address: ono@math.wisc.edu