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Eisenstein Series of Small Weight VECTOR-VALUED EISENSTEIN SERIES OF SMALL WEIGHT BRANDON WILLIAMS Abstract. We study the (mock) Eisenstein series of weight k ≤ 2 for the Weil representation on an even lattice. We describe the transformation laws in general and give examples where the coefficients contain interesting arithmetic information. Key words and phrases: Modular forms; Mock modular forms; Eisenstein series; Weil representation 1. Introduction In [4], Bruinier and Kuss give an expression for the Fourier coefficients of the Eisenstein series Ek of weight k ≥ 5=2 for the Weil representation attached to a discriminant form. These coefficients involve special values of L-functions and zero counts of polynomials modulo prime powers, and they also make sense for k 2 f1; 3=2; 2g and sometimes k = 1=2: Unfortunately, the q-series obtained in this way often fail to be modular forms. In particular, in weight k = 3=2 and k = 2, the Eisenstein series may be a mock modular form that requires a real-analytic correction in order to transform as a modular form. Many examples of this phenomenon of the Eisenstein series are well-known (although perhaps less familiar in a vector-valued setting). We will list a few examples of this: Example 1. The Eisenstein series of weight 2 for a unimodular lattice Λ is the quasimodular form 1 X n 2 3 4 E2(τ) = 1 − 24 σ1(n)q = 1 − 24q − 72q − 96q − 168q − ::: n=1 P where σ1(n) = djn d, which transforms under the modular group by aτ + b 6 E = (cτ + d)2E (τ) + c(cτ + d): 2 cτ + d 2 πi 2 Example 2. The Eisenstein series of weight 3=2 for the quadratic form q2(x) = x is essentially Zagier's mock Eisenstein series: 2 3 3=4 7=4 11=4 E3=2(τ) = 1 − 6q − 12q − 16q − ::: e0 + − 4q − 12q − 12q − ::: e1=2; n=4 in which the coefficient of q en=2 is −12 times the Hurwitz class number H(n): It transforms under the modular group by r Z i1 aτ + b 3=2 ∗ a b h 3 i −3=2 i E3=2 = (cτ + d) ρ E3=2(τ) − (τ + t) #(t) dt ; cτ + d c d π 2 d=c where # is the theta series X n2=4 #(τ) = q en=2: n2Z 2 Example 3. In the Eisenstein series of weight 3=2 for the quadratic form q3(x) = −12x ; the components of e1=12, e5=12; e7=12 and e11=12 are 23=24 47=24 71=24 95=24 119=24 143=24 − 3q − 5q − 7q − 8q − 10q − 10q − ::: eγ ; γ 2 f1=12; 5=12; 7=12; 11=12g: We verified by computer that the coefficient of qn−1=24 above is (−1) times the degree of the n-th partition class polynomial considered by Bruinier and Ono [5] for 1 ≤ n ≤ 750, which is not surprising in view of 2010 Mathematics Subject Classification. 11F27, 11F30, 11F37. 1 example 2 since this degree also counts equivalence classes of certain binary quadratic forms. This Eisenstein series is not a modular form. 2 2 2 Example 4. The Eisenstein series of weight 3=2 for the quadratic form q4(x; y; z) = x + y − z is a mock modular form that is related to the functions considered by Bringmann and Lovejoy [1] in their work on overpartitions. More specifically, the component of e(0;0;0) in E3=2 is 1 X 1 − 2q − 4q2 − 8q3 − 10q4 − ::: = 1 − jα(n)jqn; n=1 where α(n) is the difference between the number of even-rank and odd-rank overpartitions of n. Similarly, the M2-rank differences considered in [1] appear to occur in the Eisenstein series of weight 3=2 for the 2 2 2 quadratic form q5(x; y; z) = 2x + 2y − z , whose e(0;0;0)-component is 1 − 2q − 4q2 − 2q4 − 8q5 − 8q6 − 8q7 − ::: Example 5. Unlike the previous examples, the Eisenstein series of weight 3=2 for the quadratic form 2 2 2 q6(x; y; z) = −x − y − z is a true modular form; in fact, it is the theta series for the cubic lattice and the Fourier coefficients of its e(0;0;0)-component count the representations of integers as sums of three squares. The results of this note imply that we should expect different behavior in example 4 and example 5 because there are many more isotropic vectors for q4 and q5 modulo large powers of 2 than there are for q6. Among negative-definite lattices of small dimension there are lots of examples where the Eisenstein series equals the theta series. (Note that we find theta series for negative-definite lattices instead of positive-definite because we consider the dual Weil representation ρ∗.) When the lattice is even-dimensional this immediately leads to formulas for representation numbers in terms of twisted divisor sums. These formulas are well-known from the theory of scalar modular forms of higher level but the vector-valued derivations of these formulas seem more natural. We give several examples of this throughout the note. Acknowledgments: I thank Kathrin Bringmann for discussing the examples involving overpartition rank differences with me. Contents 1. Introduction 1 2. Notation 2 3. Background 3 4. The real-analytic Eisenstein series 5 5. Weight 1=2 7 6. Weight 1 9 7. Weight 3=2 11 8. Weight 2 13 References 15 2. Notation Λ denotes an even lattice with nondegenerate quadratic form q. The signature of Λ is (b+; b−) and its 0 0 0 dimension is e. The dual lattice is Λ : The natural basis of the group ring C[Λ =Λ] is denoted eγ , γ 2 Λ =Λ: 0 ∗ We denote by ρ the Weil representation of Mp2(Z) on the group ring C[Λ =Λ] and ρ its (unitary) dual. 0 Elements of Λ =Λ are usually denoted γ or β and eγ ; eβ denotes the corresponding element in the natural 0 0 basis of C[Λ =Λ]: We denote the denominator of γ 2 Λ =Λ by dγ ; i.e. dγ 2 N is minimal such that dγ γ 2 Λ: Ek(τ) denotes the weight-k (mock) Eisenstein series, which is the result of Bruinier and Kuss's formula [4] for Ek;0 naively evaluated at k ≤ 2: (However, we divide their formula by 2 such that the result has 2 ∗ constant term e0.) Ek (τ; s) is the weight-k real-analytic Eisenstein series described in section 4. For a discriminant D (an integer congruent to 0 or 1 mod 4), we denote by χD the Dirichlet character mod jDj given by the Kronecker symbol, D χ (n) = : D n The associated L-function is 1 X −s L(s; χD) = χD(n)n ; n=1 converging when Re[s] > 1 and extended to all s 2 C by analytic continuation. We denote by σk(n; χD) the twisted divisor sum X k σk(n; χD) = χD(n=d)d : djn We use the abbreviation e(x) = e2πix and the notation e(τ) = q; where τ = x + iy is a complex variable confined to the upper half-plane H: 3. Background In this section we review some facts about the metaplectic group and vector-valued modular forms, as well as Dirichlet L-functions, which will be useful later. Recall that the metaplectic group Mp2(Z) is the double cover of SL2(Z) consisting of pairs (M; φ); a b p where M = 2 SL ( ), and φ is a branch of cτ + d on the upper half-plane c d 2 Z H = fτ = x + iy 2 C : y > 0g: We will usually omit φ. Mp2(Z) is generated by the elements 1 1 0 −1 p T = ; 1 ;S = ; τ ; 0 1 1 0 with defining relations S8 = I and S2 = (ST )3: Let Λ be a lattice (which we can always take as Λ = Ze for some e 2 N) with an even quadratic form q :Λ ! Z, and let 0 n Λ = fv 2 Q : hv; wi 2 Z for all w 2 Λg 0 0 be the dual lattice. We denote by eγ , γ 2 Λ =Λ the natural basis of the group algebra C[Λ =Λ]. The Weil representation of Mp2(Z) attached to Λ is the map 0 ρ : Mp2(Z) −! Aut C[Λ =Λ] defined by p b−−b+ i X ρ(T )eγ = e q(γ) eγ ; ρ(S)eγ = e − hγ; βi eβ: pjΛ0=Λj β2Λ0=Λ In particular, b−−b+ 2 3 ρ(Z)eγ = i e−γ ; where Z = (−I; i) = S = (ST ) : Here we use e(x) to denote e2πix; and (b+; b−) is the signature of Λ: We will usually consider the dual representation ρ∗ of ρ (which also occurs as the Weil representation itself, for the lattice Λ and quadratic form −q). A modular form of weight k for ρ∗ is a holomorphic function f : H ! C[Λ0=Λ] with the properties: (i) f transforms under the action of Mp2(Z) by a b f(M · τ) = (cτ + d)kρ∗(M)f(τ);M = 2 Mp ( ); c d 2 Z 3 (ii) f is holomorphic in 1. This means that in the Fourier expansion X X n f(τ) = c(n; γ)q eγ ; 0 γ2Λ =Λ n2Z−q(γ) all coefficients c(n; γ) are zero for n < 0: If N is the smallest natural number such that Nhγ; βi and Nq(γ) 2 Z for all β; γ 2 Λ0=Λ; then ρ∗ factors through SL2(Z=NZ) if e = dim Λ is even, and through a double cover of SL2(Z=NZ) if e is odd.
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