3-REGULAR PARTITIONS AND A MODULAR K3 SURFACEJEREMY LOVEJOY AND DAVID PENNISTON

1. Introduction
A k-regular partition of n (k > 1) is a non-increasing sequence of positive integers
whose sum is n, with the condition that no summand is divisible by k. We denotethe number of k-regular partitions of n by bk (n), and follow the convention thatbk (0) = 1. Elementary techniques in the theory of partitions [3] give the generating
functions¶∞∞ µX
Y
1 − q kn(1.1)
bk (n)q n =.
1 − qnn=0n=1
In classical representation theory, k-regular partitions of n label irreducible kmodular representations of the symmetric group Sn when k is prime [8]. Morerecently, such partitions have been studied for their arithmetic properties in connection with the theory of modular forms and Galois representations [1, 6, 10, 11, 12].
Although one may presumably use the ideas from [1, 10] to study the k-regular partitions modulo any prime, more focus has been placed on the most straightforward
case, the p-adic behavior of pj -regular partitions. For example, we have
Theorem 1 (Gordon-Ono [6]). If S(p, j, a) denotes the set of natural numbers nsuch that bpj (n) is not divisible by pa , then S(p, j, a) has arithmetic density 0.
In general there is no elementary characterization of the sets S(p, j, a), but inthe best cases we do have simple congruential formulas for bk (n). For example, theclassical expansions
∞∞
Y
X
(1.2)
(1 − q n ) =(−1)n q n(3n+1)/2n=−∞n=1and(1.3)

∞
Y
n=1

(1 − q n )3 =∞X(−1)n (2n + 1)q n(n+1)/2n=0

reveal that b2 (n) is even unless 24n + 1 is a square and b4 (n) is even unless 8n + 1is a square. The case of b2 (n) has a famous combinatorial proof by Franklin [3],while K. Ono and the second
author [11] have determined b2 (n) modulo 8 in terms√of the arithmetic of Z[ −6].
Here we undertake an investigation of the 3-adic behavior of b3 (n). Let
η(z) :=∞Y(1 − q n )n=1

1991 Mathematics Subject Classification. 11P83.
12JEREMY LOVEJOY AND DAVID PENNISTONdenote Dedekind’s eta funcion, where q := e2πiz . From (1.1) we have∞
Xb3 (n)q 12n+1 ≡ η 2 (12z) (mod 3),n=0

where η 2 (12z) is a weight 1 modular form which is the Mellin transform of anArtin L-function for Q(i). Modulo 9, it turns out that the generating function forb3 (n) is related to an eigenform which is essentially the Mellin transform of the”complicated factor” in the Hasse-Weil L-function for a certain K3 surface.
Theorem 2. Let X be the K3 surface defined by(1.4)

X : s2 = x(x + 1)y(y + 1)(x + 8y).

If p is a prime such that p ≡ 1 (mod 12), then¶µp−1(1.5)
b3≡ #X(Fp ) − (p + 1)212(mod 9).

Using the fact that the relevant eigenform has complex multiplication, we canuse Hecke theory and the arithmetic of the Gaussian integers to build a formula forthe number of 3-regular partitions modulo 9.
Theorem 3. Given a positive integer n, write
12n + 1 = N 2 M
with M squarefree. For every prime divisor p of 12n + 1, set
kp := ordp (12n + 1).
If p ≡ 1 (mod 12), let dp and ep be integers such that 3 | dp andp = d2p + e2p .
(1) If there is a prime p such that p | M and p ≡ 5, 7 or 11 (mod 12), then b3 (n) ≡ 0(mod 9).
(2) If every prime divisor p of M satisfies p ≡ 1 (mod 12), thenYYkp(−1)kp dp (kp + 1) ·(−1) 2(1.6) b3 (n) ≡ (3n + 1) ·p|(12n+1)p≡1 (mod 12)(mod 9).

p|(12n+1)p≡5 (mod 12)

For comparison with (1.2) and (1.3) we cite the following, which is a directconsequence of Theorem 3.
Corollary 4. b3 (n) is divisible by 3 unless both of the following hold:(i) All prime divisors p ≡ 5, 7, 11 (mod 12) of 12n + 1 divide 12n + 1 with evenorder.
(ii) All prime divisors p ≡ 1 (mod 12) of 12n + 1 divide 12n + 1 with order notcongruent to 2 modulo 3.
EXAMPLE. If n = 5, then 12n + 1 = 61 = 62 + 52 , so b3 (5) is not divisible by
3. More specifically, b3 (5) ≡ 16 · (−1)1·6 · 2 ≡ 5 (mod 9). Indeed, the 3-regularpartitions of 5 are 5, 4 + 1, 2 + 2 + 1, 2 + 1 + 1 + 1, and 1 + 1 + 1 + 1 + 1.

3

2. Proof of Theorem 2Letη 6 (4z) :=

(2.1)

∞
Xa(n)q n ,n=1a¡ weight
3 cusp form for the congruence subgroup Γ0 (16) with character χ−1 (d) :=¢−1.
Wedenote the space of such forms by S3 (Γ0 (16), χ−1 ) (see [9] for definitions
drelated to modular forms). It is well-known [5] that η 6 (4z) has complex multiplication by K = Q(i). Specifically, let OK denote the ring of integers of K, and let∗χ be the character on (OK /(2)) defined by χ(i) = −1. Extending χ to the setof all elements of K ∗ prime to (2), we find that for d + ei ∈ OK with d + e odd,
χ(d + ei) = (−1)e . Denote by c the Hecke character on K with conductor (2) andexponent 2 given by(2.2)

c((d + ei)) = χ(d + ei)(d + ei)2 .

Thenη 6 (4z) =(2.3)

Xc(I)q N (I) ,where the sum is over ideals I of OK prime to (2).
This form is the fundamental object in our work, as it relates 3-regular partitions,
the K3 surface (1.4), and the arithmetic of the Gaussian integers.
Proof of Theorem 2. Letη 8 (12z)
F (z) := 2,
η (36z)which is easily seen to be a modular form in S3 (Γ0 (1296), χ−1 ) (see [10], for example). From (1.1) and the fact that
η 9 (z)≡1η 3 (3z)
we have∞X(mod 9),

b3 (n)q 12n+1 ≡ F (z) (mod 9).

n=0

By definition, a(n) = 0 unless n ≡ 1 (mod 4), and therefore∞³n´ ³n´´X
1 X ³³ n ´(2.4)a(n) +a(n) q n =2 n=1333n≡1a(n)q n .

(mod 12)From [9], p. 127, (2.4) is a modular form in S3 (Γ0 (1296), χ−1 ). By computation,
the first 648 coefficients of F (z) and (2.4) are equivalent modulo 9, and hence by atheorem of Sturm [13] we have for every n,(2.5)

b3 (n) ≡ a(12n + 1)

(mod 9).

To complete the proof, we recall the modularity of the surface (1.4) [2]. For everyprime p ≥ 5, we have(2.6)#X(Fp ) = 1 + p2 + 20p + a(p).
¤4JEREMY LOVEJOY AND DAVID PENNISTONREMARK. Since the L-series for X is the symmetric square of the L-series forthe congruent number elliptic curve given by the equation E : y 2 = x3 − x [2], thecongruence (2.5) is dictated by Galois actions on certain points on E. Specifically,let g(n) denote the Fourier coefficients of the associated eigenform:η 2 (4z)η 2 (8z) :=∞Xg(n)q n .

n=1

Then for every prime p ≥ 5, a(p) = g(p)2 − 2p. Denote by GQ the absolute Galois
group of Q, and by E[n] the group of n-division points of E for any n ≥ 1 (as agroup, E[n] ∼= (Z/nZ)2 ). If ` is prime, GQ acts on the Tate module
T` (E) = lim
E[`m ] ∼= Z` × Z` ,←−mand therefore we obtain a representationρ` : GQ → GL2 (Z` ).
If frobp denotes a Frobenius element for p (p 6= `), then trace(ρ` (frobp )) = g(p).
With (2.5), this shows that the behavior of b3 (n) modulo 9 is determined by theGalois action on the 3-division points of E.
3. Proof of Theorem 3Since η 6 (4z) =that(3.1)

P∞

n=1 a(n)qn∈ S3 (Γ0 (16), χ−1 ) is a Hecke eigenform, we havea(mn) = a(m)a(n)if (m, n) = 1and(3.2)

a(pk+1 ) = a(p)a(pk ) − χ−1 (p)a(pk−1 )p2

if p ≥ 5 is prime and k ≥ 0.

In light of (2.5), (3.1), and (3.2), we begin by studying the a(p) for p prime.
Proposition 5. Let p be an odd prime.
(1) If p ≡ 3 (mod 4), then a(p) = 0.
(2) If p ≡ 5 (mod 12), then 3 | a(p).
(3) If p ≡ 1 (mod 12) and we write p = d2p +e2p with 3 | dp , then a(p) ≡ (−1)dp ·2p(mod 9).
Proof. For (1), see (2.1), or recall (2.3) and note that since (p) is prime in OK ,there are no ideals of norm p in OK .
Now suppose p ≡ 1 (mod 4). Then there are integers dp and ep with p = d2p + e2p ,and hence the prime ideals of OK of norm p are (dp ± ep i). Since χ(dp ± ep i) =(−1)ep , (2.2) and (2.3) give us that(3.3)a(p) = (−1)ep (2d2p − 2e2p ) = (−1)ep (4d2p − 2p).

If p ≡ 5 (mod 12), then since p ≡ 2 (mod 3), it follows that 3 - dp ep . Henced2p ≡ e2p ≡ 1 (mod 3), and the proof of (2) is complete.
To finish the proof of (3), if p ≡ 1 (mod 12), then 3 | dp ep . We assume without
loss that 3 | dp . Then by (3.3),

5a(p) ≡ (−1)ep +1 · 2p = (−1)dp · 2p (mod 9).
¤Combining Proposition 5 with (3.2), it is straightforward induction to show
Proposition 6. Let p be an odd prime, k a positive integer.
(1) If p ≡ 3 (mod 4), then a(p2k−1 ) = 0 and a(p2k ) ≡ p2k (mod 9).
(2) If p ≡ 5 (mod 12), then 3 | a(p2k−1 ) and a(p2k ) ≡ (−p2 )k (mod 9).
(3) If p ≡ 1 (mod 12) and p = d2p + e2p with 3 | dp , then a(pk ) ≡ (−1)kdp (k + 1)pk
(mod 9).
Theorem 3 follows now from (2.5), (3.1), and Proposition 6.
We have not observed any simple congruence condition which determines theparity of dp as a function of p, which is tantamount to distinguishing between
primes of the form x2 + 36y 2 and those of the form 4x2 + 9y 2 . In this direction it isknown [4] that for all but finitely many primes p ≡ 1 √(mod 4), p is represented by
x2 + 36y 2 if and only if the minimal polynomial for j( −36) has a root modulo p.
4. Concluding remarks
Since the generating functions for partition theoretic objects are typically products and quotients of the η function, connections to objects in arithmetic geometry
such as that given by Theorem 2 are not unexpected. A striking example of this is inrecent work of L. Guo and K. Ono [7], where it is shown that values of the ordinary
partition function reveal structure of Tate-Shafarevich groups of motives of modular forms. In our case, an examination of, for instance, the five 3-regular partitionsof 5 and the 4920 F61 -points on our K3 surface gives one little reason to expectthat there is something in the combinatorics of 3-regular partitions or irreducible
3-modular representations of Sn that is related to the structure of modular surfaces
or the arithmetic of Q(i). We must for now be content that the theory of modularforms is a meeting place for diverse mathematical objects whose connections often
cannot be otherwise explained.
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Mathematika, to appear.
[2] S. Ahlgren, K. Ono, and D. Penniston, Zeta functions of an infinite family of K3 surfaces,
Amer. J. Math., to appear.
[3] G. E. Andrews, “The Theory of Partitions,” Cambridge Univ. Press, Cambridge, 1998.
[4] D. A. Cox, “Primes of the form x2 + ny 2 ”, John Wiley and Sons, Inc., 1989.
[5] D. Dummit, H. Kisilevsky, and J. McKay, Multiplicative products of η-functions, Contemp.
Math. 45 (Finite groups - coming of age), 89-98.
[6] B. Gordon and K. Ono, The divisibility of certain partition functions, Ramanujan J. 1 (1997),
25-34.
[7] L. Guo and K. Ono, The partition function and the arithmetic of certain modular L-functions,Internat. Math. Res. Notices 21 (1999), 1179-1197.
[8] G. James and A. Kerber, “The Representation of the Symmetric Group,” Addison-Wesley,Reading, 1979.
[9] N. Koblitz, “Introduction to Elliptic Curves and Modular Forms,” Springer-Verlag, New York,1984.
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6JEREMY LOVEJOY AND DAVID PENNISTON[11] K. Ono and D. Penniston, The 2-adic behavior of the number of partitions into distinct parts,
J. Combin. Theory Ser. A 92 (2000), 138-157.
[12] D. Penniston, The pa -regular partition function modulo pj , preprint.
[13] J. Sturm, On the congruence of modular forms, Lect. Notes in Math. 1240, Springer-Verlag,
Ner York, 1984, 275-280.
Department of Mathematics, University of Wisconsin, Madison, WI 53706
E-mail address: lovejoy@math.wisc.edu
Department of Mathematics, Furman University, Greenville, SC 29613
E-mail address: david.penniston@furman.edu