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Material Type: Paper; Class: Intermediate Algebra; Subject: MATHEMATICS; University: University of Wisconsin - Madison; Term: Unknown 1989;
Typology: Papers
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Scott Ahlgren Ken Ono Department of Mathematics Department of Mathematics University of Illinois University of Wisconsin Urbana, Illinois 61801 Madison, WI 53706
email: [email protected] email: [email protected]
Major Classification: Physical Sciences; Minor Classification: Mathematics.
∼ 17,000 Characters in the paper.
Abstract. Eighty years ago, Ramanujan conjectured and proved some striking con- gruences 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 which appears to explain every known Ramanujan-type congruence.
(1.1) δ:=
(^2) − 1 24 ,
then the celebrated Ramanujan congruences may be written succinctly in the form
p(n − δ) ≡ 0 (mod `).
There have been countless papers 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 [2, 3, 6, 9, 10, 11, 15, 16, 17, 18]). Each of these extensions lies within the class −δ(mod). The important role which this class plays in the theory is illustrated by the work of Kiming and Olsson [12, Thm. 1], who proved that if ≥ 5 is prime and p(n + β) ≡ 0 (mod ) for all n, then β ≡ −δ (mod ). Work of Atkin, Newman, O’Brien, and Swinnerton-Dyer [4, 5, 7, 13] produced further congruences modulom^ for primes ≤ 31 and small m. The examples discov- ered by Atkin and Newman in [4, 13] show that not every congruence lies within the progression −δ (mod `). For example, we have
(1.2) p(17303n + 237) ≡ 0 (mod 13).
The current authors [1, 14] have shown that if ` ≥ 5 is prime and m is any positive integer, then there are infinitely many congruences of the form
p(An + B) ≡ 0 (mod `m).
As in the case of Ramanujan’s congruences, all of these arithmetic progressions lie within the class −δ(mod). To summarize, the current state of knowledge consists of a systematic theory of congruences within the progressions −δ(mod), as well as some sporadic examples of congruences which fall outside of this class. In view of this, it is natural to wonder what role the class −δ(mod) truly plays.
Theorem 3. If ≥ 5 is prime, m is a positive integer, and β ∈ S, then there are infinitely many non-nested arithmetic progressions {An + B} ⊆ {n + β} such that for every integer n we have p(An + B) ≡ 0 (modm).
If M is an integer coprime to 6, then Theorem 3 and the Chinese Remainder The- orem guarantee the existence of congruences modulo M. Although Theorem 3 shows that congruences modulo often lie outside the progressionn − δ, we speculate that all congruences of the form p(An + B) ≡ 0 (mod M ) have the property that An + B is a subprogression ofn − δfor some prime. This is true, for example, of the congruences (1.2) and (1.5). In §2 we construct half integral weight cusp forms whose coefficients capture the relevant values of the partition function, and in §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 §4 we consider those progressions n + β for β 6 ∈ S. We give heuristics which cast doubt on the existence of congruences within these progressions.
n=0 a(n)q
n. Recall Dedekind’s
eta-function
(2.1) η(z) := q^1 /^24
n=
(1 − qn).
Theorem 2.1. Suppose that ≥ 5 is prime and that m is a positive integer. If β ∈ S, then there is an integer λ,m and a modular form F,m,β (z) ∈ S(2λ,m+1)/ 2 (Γ 1 (576^5 ))∩ Z[[q]] such that
F`,m,β (z) ≡
n=
p(n + β)q^24n+24β−^1 (mod `m).
Proof. If ` ≥ 5 is prime and t is a positive integer, then
(2.2) E`,t(z) :=
η`
t (z) η(`tz)
∈ M(`t−1)/ 2
Γ 0 (t), χ,t
where χ`,t :=
((−1)(t−1)/ (^2)t
. Using standard facts, it can be shown that if - a and 0 ≤ b < t, then orda/b (E,t(z)) > 0. Hence, E,t(z) vanishes at those cusps of Γ 0 (`t)
which are not equivalent to ∞. Also, since (1 − X)^ ≡ (1 − X) (mod `), for every m > 0 we have
(2.3) E`
m− 1 ,t (z)^ ≡^1 (mod^
m).
If ≥ 5 is prime, then define f(z) =
n=1 a`(n)q
n (^) by
(2.4) f`(z) =
n=
a`(n)qn^ :=
η(z) η(z)
`
Since
n=0 p(n)q
n (^) = ∏∞ n=1(1^ −^ q
n)− (^1) , (2.1) and (2.4) imply that
n=
a`(n)qn^ =
n=
p(n)qn+δ`
n=
(1 − qn).
Define f˜`(z) by
(2.6) f˜`(z) :=
n=
( (^) n `
a`(n)qn.
By standard facts, we have f˜(z) ∈ M(−1)/ 2
`
. By (2.3) and (2.6), if m′^ is
sufficiently large, then f,m′ (z) := E
m′ , 3 (z) f˜(z) is a cusp form on Γ 0 (`^3 ) with character
χ`,t ·
`
for which
(2.7) f,m′^ (z) ≡ f˜(z) (mod `m),
and
(2.8) ord∞(f,m′^ (z)) ≥ δ + 1.
By (2.5) and (2.7), we have
f,m′ (z) η(`z)
n≡0 (mod `)
p(n − δ`)qn−^
` 242
(n)=−
p(n − δ`)qn−^
242 (modm).
Now (2.8) shows that
( (^) f ,m′^ (z) η(`z)
vanishes at ∞. Therefore if m′^ is sufficiently large,
then this form vanishes at every cusp. It follows that f,m′ (24z)/η(24z) is a cusp form on Γ 0 (576^3 ). We have the general fact that if f (z) =
n=1 a(n)q
n (^) ∈ S λ+ 12 (Γ^1 (N^ )), and r and t are positive integers, then
n≡r (mod t) a(n)q
n (^) ∈ S λ+ 12 (Γ^1 (N t (^2) )). Theo-
rem 2.1 follows by applying this fact to f,m′ (24z)/η(24`z).
n=1 a(n)q
n (^) ∈ S λ+ 12 (Γ^1 (N^ )) has algebraic coefficients. We have a decomposition
(3.1) Sλ+ 12 (Γ 1 (N )) =
χ even
Sλ+ 12 (Γ 0 (N ), χ);
Using a straightforward modification of the same argument, one can show that a positive proportion of the primes p ≡ −1 (mod M N ) have
F (z) | T (p) ≡ 0 (mod M ) for all F ∈ Sk(Γ 1 (N ))OK /M.
After (3.4), we conclude that a positive proportion of the primes p ≡ −1 (mod M N ) have St(fχ) | T (p) ≡ 0 (mod M ) for all χ and t.
Since the Shimura correspondence commutes with the action of the Hecke algebra, it follows that if p is such a prime, then
(3.6) St(fχ | T (p^2 )) ≡ 0 (mod M ) for all χ and t.
Lemma 3.1 follows from (3.5) and (3.6).
Proof of Theorem 1. We apply Lemma 3.1 to the forms F,m,β (z) given in Theorem 2.1. Fix a prime and an integer β ∈ S`, and write
F`,m,β (z) =
n=
a`,m,β (n)qn^ ≡
n≡ 24 β−1 (mod 24`)
p
( (^) n+ 24
qn^ (mod `m).
By Lemma 3.1, a positive proportion of the primes Q ≡ −1 (mod 24) have the property that F,m,β (z) | T (Q^2 ) ≡ 0 (mod m). After replacing n by Qn in the definition (3.3), we see that if n ≡ 1 − 24 β (mod 24) and gcd(Q, n) = 1, then
0 ≡ a`,m,β (Q^3 n) ≡ p
Q^3 n+ 24
(mod `m),
since Q^3 n ≡ 24 β − 1 (mod 24`). Theorem 1 follows.
. Suppose that β ∈ { 0 ,... , − 1 }. If we could produce a cusp form F,m,β (z) as in Theorem 2.1, then we would obtain the statement of Theorem 1 for β. Since F,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 6 = 0. For every prime ≥ 5 and every m, it is straightforward to show that there exists an integral weight modular form H,m(z) such that∑
n 6 ≡δ(mod)
p(n − δ)q^24 n−
2 ≡ H,m(24z)/η(24z) (modm).
However, to construct H,m(z) requires the twists of f(z) by all of the Dirichlet characters modulo ; it results that H,m(z) is a form on Γ 1 (^3 ). By contrast, the form which we constructed in (2.6) required only a single quadratic twist, and so remained on Γ 0 (^3 ).
It is clear that if we had the analog for Γ 1 (t) of the form E,t(z) used in the proof of Theorem 2.1, then we could prove Theorem 1 for all non-zero β. Using the work of Hecke, it is possible to construct an Eisenstein series on Γ 1 (t) which has the proper cusp conditions (in fact, up to scalar multiplication, exactly one such series exists). It remains to determine if this series can be defined over the algebraic numbers, and, if so, to determine the-adic nature of its coefficients. Unfortunately, the answers to both of these problems seems 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 [8]. 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 [19, Thm. 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.
Acknowledgements. The first author is supported by a National Science Foundation grant. The second author 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