OVERPARTITION M 2-RANK DIFFERENCES, CLASS NUMBER RELATIONS, AND VECTOR-VALUED MOCK EISENSTEIN SERIES

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1 OVERPARTITION M -RANK DIFFERENCES, CLASS NUMBER RELATIONS, AND VECTOR-VALUED MOCK EISENSTEIN SERIES BRANDON WILLIAMS Abstract. We prove that the generating function of overpartition M-rank differences is, up to coefficient signs, a component of the vector-valued mock Eisenstein series attached to a certain quadratic form. We use this to compute analogs of the class number relations for M-rank differences. As applications we split the Kronecker-Hurwitz relation into its even and odd parts and calculate sums over Hurwitz class numbers of the form Hn r. 1. Introduction and statement of results An overpartition of n N 0 is a partition in which the first occurance of a number may be overlined distinguished. Equivalently, an overpartition of n is a decomposition n = n 1 + n, with n 1, n 0, together with a partition of n 1 into distinct parts the overlined numbers and an arbitrary partition of n. For example, the 8 overpartitions of n = 3 are 3 = 3 = + 1 = + 1 = + 1 = + 1 = = In [7], Lovejoy introduced the M -rank statistic lλ M -rankλ = nλ + nλ 0 χλ, where lλ denotes the largest part of the overpartition λ, nλ is the number of parts, nλ o is the number of odd non-overlined parts, and 1 : the largest part of λ is odd and non-overlined; χλ = Let M e n and M o n denote the number of overpartitions of n with even M-rank resp. odd M-rank and define the M -rank difference α n = M e n M o n. Bringmann and Lovejoy proved [] that the generating function f q = α nq n = 1 + q + 4q q 4 + 8q 5 + 8q n=0 is a mock modular form of weight 3/, i.e. the holomorphic part of a harmonic weak Maass form. It was proved in [11] that the generating function for the difference counts between overpartitions with even and odd Dyson s ranks is, up to the signs of the coefficients, the e 0 -component of a vector valued mock Eisenstein series for the dual Weil representation attached to the quadratic form qx, y, z = x + y z on Z 3. It is also closely related to the mock Eisenstein series attached to the quadratic form qx = x. It was observed there that the function f seemed to be similarly related to the mock Eisenstein series attached to the quadratic form q x, y, z = x y + z or equivalently x + y z. The purpose of this note is to prove the assertion above. 010 Mathematics Subject Classification. 11E41,11F7,11F37. 1

2 Proposition 1. The series 1 α n q n = 1 q 4q q 4 8q 5 8q 6... n=1 is the e 0 -component of the mock Eisenstein series for the dual Weil representations attached to the quadratic form q x, y, z = x y + z and to the quadratic form q x, y, z = x y + z. As in the examples of [1], calculating the other components of the mock Eisenstein series leads to class number relations by comparing coefficients of certain vector-valued modular or quasimodular forms of weight. This is very similar to the arguments of [1] that use mixed mock modular forms to derive relations among class numbers. Let n N; and let σ 1 n = d n d, λ 1n = 1 Hurwitz class number. The identities we find in this case are: Proposition. Define αn = d n α n r + 4 H4n r + 4λ 1 n r odd Then αn satisfies the identity 8σ 1 n/ + 16σ 1 n/4 : n 0 mod 4; αn = 4σ 1 n/ : n mod 4; 4σ 1 n : n 1 mod. mind, n/d and let Hn denote the 4 : n = ; α n can be expressed in terms of Hurwitz class numbers and representations counts by sums of three squares, using theorem 1.1 of []. We can use this to isolate the term r odd H4n r in the identity above. This is a strengthening of theorem 1 of [3] which considers the case that n is prime. Corollary 3. For any n N, /3σ 1 n : n odd; H4n r = 4σ 1 n/ λ 1 n : n mod 4; r odd /3σ 1 n + σ 1 n/ 8/3σ 1 n/4 + 4λ 1 n/4 λ 1 n : n 0 mod 4. We can use similar arguments to give another class number relation for α of a different sort: Proposition 4. Write n N in the form n = ν m with m odd. Then α n r = σ 1 n, χ + χm ν + 4 b a Na=n + 4 : n = ; 0 : otherwise; where χ is the Dirichlet character modulo 8 given by χ1 = χ7 = 1 and χ3 = χ5 = 1; and σ 1 n, χ is the twisted divisor sum σ 1 n, χ = χn/dd; d n and a runs through all ideals of Z[ ] having norm n, and a + b a is a generator with minimal trace a > 0. As before, this translates into an identity for Hurwitz class numbers: Corollary 5. i For any n = ν m, m odd, H4n r = 3 σ 1n, χ ii If n is odd, then + χm ν+ + 1 Hn r = 4 + χn σ 1 n, χ Na=8n Na=4n

3 iii If n is odd, then Hn r = + χn σ 1 n, χ Na=n If there are no ideals of norm n then the error term 1 Na=n b a vanishes. For example, for primes p that remain inert in Z[ ] i.e. p 3 or p 5 modulo 8, we get the formulas H4p r = 7 6 p 1; Hp r = p 1 ; Hp r = p Proof of proposition 1 We first observe that α n is positive when n 1, mod 4 and α n is negative or 0 when n 0, 3 mod 4 with n = 0 as an exception. This is a consequence of equation 1.5 of [] which gives an exact expression for α n in terms of Hurwitz class numbers and representation counts by sums of three squares. It follows that fτ := 1 q 4q q 4 8q 5 8q 6... = 1 + i f τ + 1/4 + 1 i f τ 1/4. Theorem.1 of [] shows that the real-analytic correction of f τ is the harmonic weak Maass form 1 f τ N τ = f τ + i π i τ Θt iτ + t 3/ dt = f τ i π τ Θt dt τ + t 3/ of level 16, where Θt is the classical theta series Θτ = 1+q+q 4 +q It follows that the real-analytic correction of fτ is also fτ 1 + i N τ + 1/4 1 i N τ 1/4 i i = fτ π 1 [ 1 + i 1 i ] Θt 1/4 + τ τ + t 3/ Θt + 1/4 dt i i = fτ π Θt dt, τ + t 3/ τ using the fact that 1+i 1 i Θt 1/4 + Θt + 1/4 = Θt since the only nonzero Fourier coefficients of Θt occur with exponents n 0, 1 mod 4. Remark 6. This implies that f τ fτ = n 1, 4 α nq n = 4q + 8q + 16q q is a weight-3/ modular form of level 16. This is true; for example, we can write this as a difference of theta series, 4q + 8q + 16q q = ϑ A τ ϑ B τ for the matrices A = diag,, and B = diag4, 4,, which can be verified by computing finitely many coefficients. This gives another proof that every number n 0, 3 4 has the same number of representations in the form a + b + c as it does in the form a + b + c. We need to compare the weak Maass form above with the real-analytic correction of the mock Eisenstein series E 3/ τ. Following remark 17 of [11], this is i i E3/ τ, 0 = E ϑt 3/τ + dt, 16π τ t + τ 3/ where ϑt is the shadow ϑτ = γ Λ /Λ n Z q γ n 0 3 an, γq n e γ,

4 where Λ /Λ is the discriminant group of the quadratic form q x, y, z = x y + z and e γ are the corresponding basis elements of the group ring C[Λ /Λ] and the coefficients an, γ are given by 1 p s lim 1 + p 1 L 1 : n < 0; pn, γ, + s 1/ : n = 0; an, γ = 48 Λ /Λ s 0 bad p where the bad primes are p = and all primes dividing Λ /Λ or at which n has nonzero valuation, and L p is the series L p n, γ, s = ν=0 } p νs # v Z/p ν Z 3 : q v γ + n 0 mod p ν. We only need to compute a small number of these coefficients to identify ϑ among modular forms of weight 1/ for the Weil representation not its dual attached to the quadratic form q. A general result of Skoruppa [10] implies that the space of such forms is spanned by the theta series ϑ 1 τ = q + q 4 + q q 1/4 + q 9/4 + q 5/ e 0,0,0 + e 1/4,1/,1/4 + e 1/,0,1/ + e 3/4,1/,3/4 e 1/4,0,3/4 + e 1/,1/,0 + e 3/4,0,1/4 + e 0,1/,1/ and ϑ τ = 1 + q + q 4 + q e 0,0,0 + e 1/4,1/,3/4 + e 1/,0,1/ + e 3/4,1/,1/4 + q 1/4 + q 9/4 + q 5/ e 1/4,0,1/4 + e 1/,1/,0 + e 3/4,0,3/4 + e 0,1/,1/. Therefore, it is enough to compute the constant term of the shadow ϑτ. The only bad prime at n = 0 is p =, and the corresponding local L-functions are 4s s s 1 s 8 1 : γ = 0, 0, 0; L 0, γ, s = s 1 s 1 : γ = 1/, 1/, 0; 1 s 1 : otherwise; such that This implies that the shadow is lim 1 s 0 s L 0, γ, + s = : γ = 0, 0, 0 or γ = 1/, 1/, 0; 1 : otherwise. ϑτ = 8e 0,0,0 + e 1/,1/,0 4e 1/4,1/4,1/ + e 3/4,3/4,1/ + e 1/4,3/4,1/ + e 3/4,1/4,1/ + Oq 1/4 = 4ϑ 1 τ 4ϑ τ with e 0 -component 8Θτ, so the e 0 -component of E3/ τ, 0 is i i E3/ τ, 0 0 = E 3/ τ 0 π τ Θt dt t + τ 3/ has the same real-analytic part as fτ. It follows that E 3/ τ, 0 0 fτ is a modular form of level 16; we can easily verify that it is identically 0 by comparing Fourier coefficients up to the Sturm bound. The case of the quadratic form q x, y, z = x y +z is similar; here, the space of weight-1/ modular forms is spanned by the theta functions ϑ 1 τ = 1 + q + q 4 + q e 0,0,0 + e 1/4,1/4,0 + e 1/,1/,0 + e 3/4,3/4,0 + q 1/4 + q 9/4 + q 5/ e 0,0,1/ + e 1/4,1/4,1/ + e 1/,1/,1/ + e 3/4,3/4,1/ 4

5 and ϑ τ = 1 + q + q 4 + q e 0,0,0 + e 1/4,3/4,0 + e 1/,1/,0 + e 3/4,1/4,0 + q 1/4 + q 9/4 + q 5/ e 0,0,1/ + e 1/4,3/4,1/ + e 1/,1/,1/ + e 3/4,1/4,1/, and the local L-functions for n = 0 are exactly the same as those of equation although otherwise now stands for a different set of elements γ. It follows that e 0 component of E 3/ τ, 0 for q is the same as the e 0 -component of E 3/ τ, 0 for q, and therefore also equals fτ. 3. Proof of proposition The real-analytic Eisenstein series E3/ τ, 0 for q corresponds via the theta decomposition to a realanalytic Jacobi form E,1,0τ, z, 0 for the dual Weil representation ρ attached to the quadratic form x y, and we obtain a quasimodular form Q,1,0 the Poincaré square series of index 1 by projecting Eτ, 0, 0 orthogonally to the cusp space S ρ and then adding to it the quasimodular Eisenstein series E. Since S ρ = 0}, it follows that Q,1,0 τ = E τ. The e 0 -component of the series E τ for the quadratic form x y is E τ 0 = 1 4q 4q 16q 3 40q 4 4q 5... and using the structure of quasimodular forms of higher level proposition 1b of [6] this can be identified as 1 4q 4q 16q 3 40q 4... = 3 E τ + 3 E 4τ 1 3 E τ + 1/ 4 σn + 1q n+1, in which the coefficient of q n is 4σ 1 n : n 1 ; αn = 4σ 1 n/ : n 4; 8σ 1 n/ 16σ 1 n/4 : n 0 4. If Λ is an even e-dimensional lattice of signature σ 0 mod 4 with quadratic form q, then define the lattice Λ = Λ Z with quadratic form qv, λ = qv + λ. Suppose the weight-3/ mock Eisenstein series for Λ has coefficients denoted E 3/ τ = cn, γq n e γ with shadow ϑτ = γ Λ / Λ n Z q γ γ Λ / Λ n Z q γ n 0 The results of [1] imply that the coefficient formula for Q,1,0 τ = is bn, γ = cn r /4, γ, r/ γ Λ /Λ n Z qγ an, γq n e γ. bn, γq n e γ n=0 an r /4, γ, r/ r r 4n. In particular, the formula for bn, 0 involves knowing both the components of e 0,0,0 and e 0,0,1/ in the mock Eisenstein series E 3/ τ. This is the series E 3/ τ 0,0,1/ = 4q 7/4 8q 15/4 1q 3/4 1q 31/4... 5

6 Lemma 7. Let Hn denote the Hurwitz class number of n N 0 ; then E 3/ τ 0,0,1/ = 4 Hnq n/4. n 7 8 Proof. In general, the formula of [4] for the weight-3/ mock Eisenstein series for an even lattice of signature b +, b and dimension e = b + b under the assumption 3 + b + b 0 4 implies that the coefficient cn, γ of E 3/ τ is b+ b /4 L1, χ D 1 n Λ /Λ π p Λ /Λ [ [ lim s 0 d f ] µdχ D dd 1 σ 1 f/d 1 p e 3/ s ] 1 p L p n, γ, 1/ + e/ + s, where D is a discriminant defined in theorem 4.5 of [4], and χ D d = D d is the Kronecker symbol, and σ 1 n = d n d 1, and f is the largest square dividing n but coprime to Λ /Λ. Finally, recall that L p n, γ, s denotes the series } L p n, γ, s = p νs # v Z/p ν Z e : qv γ + n 0 mod p ν. ν=0 For a fixed lattice, the series L p n, γ, s can always be evaluated in closed form in both n and s for example, using the p-adic generating functions of [5] although the result tends to be messy particularly for p =. In our case, most of these terms are the same between the lattice Z with quadratic form x in which case, E 3/ is essentially the Zagier Eisenstein series and cn, γ is 1H4n and the lattice Z 3 with quadratic form x y + z. The only differences are the order of the discriminant group and the local L-functions. For qx = x, the result is that 8n + 7 lim 1 s 0 s L, 1 4, 1 + s 1 s 1+s + 1 = lim s 0 1+s = 3 1, while for qx, y, z = x y + z the result is that and 8n + 7 lim 1 s 0 s L, 1 4, + s 1 s +s + 4 = lim s 0 +s = 4 8n + 3 lim 1 s 0 s L, 1 4, + s = 0. }} =1 Since the discriminant group of x y + z is 16 times the size of that of x, we conclude that the coefficient of q 4n+7/4 in the series E 3/ τ 0,0,1/ is 1 4 3/ 1H4n + 7 : n 0 ; 0 : n 1 ; which implies the claim. Finally, the coefficients an r /4, 0, r/ of the shadow ϑτ are 8 : n r /4 = 0; an r /4, 0, r/ = 16 : r 4n is a square; Comparing coefficients in Q,1,0 = E τ 0 gives the formula: 6

7 Lemma 8. If n 0 mod 4, then 8σ 1 n/+16σ 1 n/4 = If n mod 4, then If n 1 mod, then Since 4σ 1 n/ = 4σ 1 n = r 4n= this implies proposition. α n r +4 H4n r + r odd r 4n= α n r + 4 H4n r + α n r + r odd r r 4 n + 1 : n = ; 4n r 4n= r r 4n. r r 4 n + 1 : n = ; 4n 0 : otherwise; r r 4n = 4 n : n = ; mind, n/d + 0 : otherwise; d n Corollary 3 follows from this and the identity r 3 n/3 : n 1, 4; 4 α n = 8Hn r 3 n/3 : n 0, 3 4; where r 3 n is the representation count of n as a sum of three squares, using the identities r 3 n r = 8σ 1 n 3σ 1 n/4 Jacobi s formula, and r 1 r r 3 n r = 1 n 1/ 4σ 1 n, n odd, r which follows from Jacobi s formula and the class number relations H4n r = σ 1 n λ 1 n and Hn r = 1 3 σ 1n λ 1 n, n odd, the latter of which is due to Eichler. See also equation 1.3 of [9]. We leave the details to the reader. 4. Proof of proposition 4 This identity is derived analogously to proposition, but it arises from the equality Q,,0 = E for the dual Weil representation attached to the quadratic form x y. The discriminant group of this quadratic form has nonsquare order 8, so E τ is a true modular form; its e 0 -component is E τ 0 = 1 18q 34q 8q 3 66q 4 56q 5 60q 6... M Γ 1 8 in which the coefficient of q n is 8σ 1 n, χ + χm ν+, n = ν m, m odd, where χ is the Dirichlet character modulo 8 given by χ1 = χ7 = 1, χ3 = χ5 = 1 and where σ 1 n, χ = d n χn/dd. This can be calculated using the coefficient formula of [4], for example. 7

8 The relevant components of the weight-3/ Eisenstein series for x y + z are E 3/ τ 0,0,0 = 1 q 4q 4q 4 8q 5... E 3/ τ 0,0,1/ = q 1/ 4q 3/ 4q 5/ 8q 7/ 6q 9/... E 3/ τ 0,0,1/4 = E 3/ τ 0,0,3/4 = 4q 7/8 4q 15/8 4q 3/8 8q 31/8 4q 39/8... Lemma 9. i The coefficient of q n/ in E 3/ τ 0,0,1/ is 1/ times the number of representations of n by the quadratic form 4a + b + c. ii The coefficient of q n/8 in E 3/ τ 0,0,1/4 is 1/ times the number of representations of n by the quadratic form 4a + b + c. Proof. The components E 3/ τ 0,0,1/ and E 3/ τ 0,0,1/4 are modular forms because the components e 0,0,1/ and e 0,0,1/4 do not appear in the shadow ϑτ. Once an equality between two modular forms has been conjectured here, the components of E 3/ τ and two theta series, it can always be proved by comparing a finite number of coefficients. In principle this could also be proven directly via the same argument as lemma 7. The coefficient formula for the e 0 -component of the index- series Q,,0 τ 0 = bnq n is now bn = cn r /8, 0, 0, r/ n=0 an r /8, 0, 0, r/4 r r 8n, where cn, γ is the coefficient of q n e γ in the mock Eisenstein series above and an, γ is the coefficient of q n e γ in its shadow. Here, cn r /8, 0, 0, r/4 = α n r 1 r A 4n r + r B 8n r 4 : n = ; + r odd Here, r A n is the representation count of n by 4a + b + c and r B n is the representation count of n by 4a + b + c. Remark 10. The generating function of the coefficients r odd r An r is the difference of theta functions for the quadratic forms 4a +b +c +d and 4a +b +c +4d. In particular, n=0 r odd r A4n r q n is a modular form of weight ; we can identify it as the eta product r A 4n r q n = 8q + 16q + 16q 3 + 3q = 8ητ3 η4τη8τ ητ. Similarly, n=0 r odd n=0 r odd r B 8n r q n = 16q + 3q + 3q q = 16ητ3 η4τη8τ ητ. The eta product ητ3 η4τη8τ ητ is one of the few such products with multiplicative coefficients, as classified by Martin [8], and its coefficient of q n is the twisted divisor sum σ 1 n, χ = d n χn/dd for the character χ1 = χ7 = 1, χ3 = χ5 = 1 mod 8 that we consider throughout. Therefore, we can simplify the above sum to cn r /8, 0, 0, r/4 = α n r 1σ 1 n, χ : n = ;

9 The correction term 1 8 an r /8, 0, 0, r/4 r r 8n = r 8n= r r 1 : r 8n; 8n 1/ : r = 8n; is more difficult to calculate than the corresponding term in the proof of proposition because the discriminant order Λ /Λ = 8 is not square. Following section 7 of [1], this term can be calculated by finding minimal solutions to the Pell-type equation a 8b = 64n. The true Pell equation a 8b = 1 has fundamental solution a = 3, b = 1. We let µ i = a i + b i 8, i 1,..., N} denote the representatives of orbits of elements in Z[ ] up to conjugation having norm n and minimal positive trace; then 1 r 8n= [ r r 8n 1 : r 8n 0; ] 1/ : r 8n = 0; = 4 N i=1 b i a i : µ i /µ i / Z[ ]; 1 : µ i /µ i Z[ ]. Since Q has class number one, these orbits correspond to the ideals of Z[ ] with ideal norm n and we can write [ r r 1 : r 8n 0; ] 8n 1/ : r = 4 b a, 8n = 0; r 8n= Na=n where a runs through the ideals of Z[ ] of norm n and a + b a is a generator with minimal a > 0. Comparing coefficients between Q,,0 τ 0 and E τ 0 results in the identity α n r = σ 1 n, χ + χm ν + 4 Na=n as claimed, where a + b a is a generator with minimal a > 0. b a + 4 : n = ; 0 : otherwise; Example 11. Let n = 7. The ideals of Z[ ] of norm 14 are 4 ± and the trace 8 is minimal within both ideals. The left side of lemma 11 is α 7 r = α 7 + α 5 = 4, while the right side is σ 1 7, χ + χ = = 4. Remark 1. When n = p is a prime that remains inert in Z[ ] i.e. χp = 1 this identity simplifies to α p r = p 1. To prove the corollary, we again use equation 4. In the first case, α 4n r = 8 H4n r 1 1 r r 3 4n r. 3 r The generating function 1 r r 3 4n r q n = 1 18q 34q 8q 3 66q 4 56q 5 60q 6... n=0 is a difference of theta functions and therefore a modular form of level 8; and we identify it as E τ 0, giving the identity 1 r r 3 4n r = 8σ 1 n, χ + χm ν+, n = ν m, m odd. 9

10 Therefore, H4n r = 1 3 σ 1n, χ + χm ν σ 14n, χ + χm ν+ + 1 = 3 σ 1n, χ + χm ν+ + 1 Na=8n In the second case, for odd n, α n r = 8 Hn r r r 3 n r. 3 r Here, so 1 r r 3 n r = 8 + χn σ 1 n, χ, n odd, Hn r = 8 + χn σ 1 n, χ σ 1n, χ = 4 + χn σ 1 n, χ Na=4n In the third case, for odd n, α n r = 8 Hn r χn 3 r where and therefore + χn + 1 Na=8n Na=4n 1 r r 3 n r, 6σ 1 n, χ : n 1 8; 1 r r 3 n r σ 1 n, χ : n 3 8; = σ 1 n, χ : n 5 8; 6σ 1 n, χ : n 7 8; Hn r = + χn σ 1 n, χ References Na=n b a b a [1] Kathrin Bringmann and Ben Kane. Sums of class numbers and mixed mock modular forms. Preprint, 013. URL arxiv: [] Kathrin Bringmann and Jeremy Lovejoy. Overpartitions and class numbers of binary quadratic forms. Proc. Natl. Acad. Sci. USA, 10614: , 009. ISSN doi: /pnas URL [3] Brittany Brown, Neil Calkin, Timothy Flowers, Kevin James, Ethan Smith, and Amy Stout. Elliptic curves, modular forms, and sums of Hurwitz class numbers. J. Number Theory, 186: , 008. ISSN X. doi: /j.jnt URL [4] Jan Bruinier and Michael Kuss. Eisenstein series attached to lattices and modular forms on orthogonal groups. Manuscripta Math., 1064: , 001. ISSN doi: /s URL [5] Raemeon Cowan, Daniel Katz, and Lauren White. A new generating function for calculating the Igusa local zeta function. Adv. Math., 304:355 40, 017. ISSN doi: /j.aim URL 10

11 [6] Masanobu Kaneko and Don Zagier. A generalized Jacobi theta function and quasimodular forms. In The moduli space of curves Texel Island, 1994, volume 19 of Progr. Math., pages Birkhäuser Boston, Boston, MA, doi: / URL _6. [7] Jeremy Lovejoy. Rank and conjugation for a second Frobenius representation of an overpartition. Ann. Comb., 11: , 008. ISSN doi: /s URL http: //dx.doi.org/ /s [8] Yves Martin. Multiplicative η-quotients. Trans. Amer. Math. Soc., 3481: , ISSN doi: /S URL S [9] Michael Mertens. Mock modular forms and class number relations. Res. Math. Sci., 1:Art. 6, 16, 014. ISSN doi: / URL [10] Nils-Peter Skoruppa. Über den Zusammenhang zwischen Jacobiformen und Modulformen halbganzen Gewichts, volume 159 of Bonner Mathematische Schriften [Bonn Mathematical Publications]. Universität Bonn, Mathematisches Institut, Bonn, Dissertation, Rheinische Friedrich-Wilhelms- Universität, Bonn, [11] Brandon Williams. Vector-valued Eisenstein series of small weight. Preprint, 017. URL arxiv: [1] Brandon Williams. Poincaré square series of small weight. Preprint, 017. URL arxiv: Department of Mathematics, University of California, Berkeley, CA address: btw@math.berkeley.edu 11