ACTA ARITHMETICA LXIII.3 (1993) The Eichler Commutation Relation for theta series with spherical harmonics by Lynne H. Walling (Boulder, Colo.) It is well known that classical theta series which are attached to positive definite rational quadratic forms yield elliptic modular forms, and linear combinations of theta series attached to lattices in a fixed genus can yield both cusp forms and Eisenstein series whose weight is one-half the rank of the quadratic form. In contrast, generalized theta series—those augmented with a spherical harmonic polynomial—will always yield cusp forms whose weight is increased by the degree of the spherical harmonic. A recent demonstration of the far-reaching importance of generalized theta series is Hijikata, Pizer and Shemanske’s solution to Eichler’s Basis Problem [4] (cf. [2]) in which character twists of such theta series are used to provide a basis for the space of newforms. In this paper we consider theta series with spherical harmonics over a totally real number field. We show that such theta series are Hilbert modular cusp forms whose weight is integral or half-integral, depending on the rank of the associated lattice. We explicitly describe the action of the Hecke operators on these theta series in terms of other theta series, yielding a generalization of the well-known Eichler Commutation Relation. Finally, we use these theta series to construct Hilbert modular forms which are invariant under a subalgebra of the Hecke algebra. We are able to show that if the quadratic form has rank m and the spherical harmonic has degree l, then the theta series attached to the genus of a lattice is identically zero whenever l is small relative to m; in particular, the associated collection of theta series are linearly dependent. 1. The transformation formula. Here we derive the transformation formula for theta series with spherical harmonics. Our approach is similar to that of Eichler in [3], but unlike the derivation in [3], we first derive an Partially supported by NSF DMS 9103303. 234 L. H. Walling inversion formula by identifying our theta series with a Siegel modular form. Let K be a totally real number field of degree n over Q, and let O and ∂ denote its ring of integers and its different (respectively). Let V be an m- dimensional vector space over K with B : V × V → K a symmetric bilinear form such that Q(x) def= B(x, x) is totally positive for all nonzero x ∈ V . Take L to be a lattice on V (so L is a rank m O-module contained in V ). We let L# denote the dual of L (i.e. L# = {x ∈ V : B(x, L) ⊆ O }) and 1 nL the norm of L (i.e. nL is the fractional O-ideal generated by 2 Q(L)). Then as shown in [3] (cf. [7]), (nL)−1(nL#)−1 is an integral ideal. For later convenience, we take IL to be the smallest fractional ideal of O such that 2 2 IL ⊇ nL, and we put N(L) = nL in case m is even, and N(L) = IL in case m is odd. We also put N equal to the stufe of L as defined in [7]. Thus if m is even, N is the product of (nL)−1(nL#)−1 and perhaps some dyadic # −1 −2 primes; if m is odd, N = (nL ) IL . Viewing L as a Z-module, we take {λ1, . , λmn} to be a Z-basis for L. Having fixed a K-basis {v1, . , vm} for V , we associate vectors of V with m × 1 column vectors whose entries lie in K; hence for T = (λ1 . λmn) ∈ mn Mmn,m(K), L = T Z . We remark here that the matrix T (1) . . T (n) is invertible (where the superscript of (j) denotes the image under the jth embedding of K/Q into R). To see this, notice that hv, wi = Tr(tv · w) is a symmetric bilinear form on the rational vector space V (here we still identify v and w with m × 1 column vectors). Let {κ1, . , κmn} be a dual basis for {λ1, . , λmn} with respect to h∗, ∗i (see [5]); thus n 1 if i = j , hλ , κ i = i j 0 otherwise and so t (1) t (n) (1) (1) κ1 ... κ1 λ1 . λmn . . .. .. = (hκi, λji) = Imn . t (1) t (n) (n) (n) κmn ... κmn λ1 . λmn Set F = (B(vi, vj)); then since Q is a totally positive quadratic form, . (1) n .. T def X t (j) t (1) t (n) (j) . Z = ( TFT ) zj = ( T ... T ) F zj . j=1 . (n) .. T n is in Siegel upper half-space whenever z = (z1, . , zn) ∈ H (where H denotes the complex upper half-plane). Given w = (w1, . , wn) with wj ∈ Eichler Commutation Relation 235 m mn t (1) (n) t C , we can find some u ∈ C such that (T ...T )u = (w1 . wn). Let Zhhxii denote txZx; then for any x ∈ Qmn with v = T x ∈ V , n n X t (j) X (j) (j) Zhhx + uii = ( TFT ) hhx + uiizj = F hhv + wjiizj; j=1 j=1 (j) note that if w0 ∈ V and wj = w0 then Zhhx + uii = Tr(Q(v + w0)z) . P Set ϑ(Z; u) = mn exp{πiZhhx + uii}; by Lemma 1.3.1 of [1] we have x∈Z X (1) (det(−iZ))1/2ϑ(Z; u) = exp{πi(−Z−1hhxii + 2 txu)} mn x∈Z where the square-root function is defined as in the above cited lemma. Given our choice of Z, we find that det(−iZ) = (−i)mn/2zm/2pΦ(L) where n n m/2 Y m/2 X t (j) z = zj and Φ(L) = det ( TFT ) . j=1 j=1 Also, n X 1 Z−1 = (tSF −1S)(j) z j=1 j . .. S(1) t (1) t (n) (F (j))−1 1 . = ( S ... S ) zj . . (n) .. S where S = (κ1 . κmn); thus X exp{πi(−Z−1hhxii + 2 txu)} mn x∈Z w1 X −1 t t (1) t (n) . = exp πi −Z hhxii + 2 x( S ... S ) . x∈ mn Z wn n X X 1 −1 (j) (j) t (j) = exp πi − (F ) hhx ii + 2 x wj mn zj x∈SZ j=1 n X X 1 = exp πi − F (j)hhx(j)ii + 2 tx(j)F (j)w . z j −1 mn j x∈F SZ j=1 236 L. H. Walling Notice that F −1SZmn is Le, the complement of L, which is the lattice dual to L with respect to the bilinear form TrK/Q ◦ B. Then with a slight abuse of notation, formula (1) gives us the “inversion formula”: imn/2 X 1 (2) θ(L, w; z) = e − Q(y) + 2B(y, w) zm/2pΦ(L) z y∈Le P where θ(L, w; z) = x∈L e{Q(x + w)z} and e{αz} means exp{πiTr(αz)}. (When w 6∈ V we identify x and w with column vectors and interpret Q(x + w) as F hhx + wii.) From this inversion formula we derive an inversion formula for inhomo- geneous theta series with spherical harmonics. Now, a degree l (l ∈ Z+) spherical harmonic P relative to the lattice L is a linear combination of Qn (j) (j) l m functions on L of the form Pζ,l(x) = j=1(x F ζj) where ζj ∈ C such t (j) that ζjF ζj = 0 (here x is again identified with an m × 1 column vector using the basis {v1, . , vm} for V ). Thus it suffices to derive an inversion (j) formula with P = Pζ,l. To derive this formula, we take wj = v + σjζj m t (j) where v ∈ V , σj is a scalar, ζj ∈ C and ζjF ζj = 0; we then differentiate equation (2) l times with respect to each σj. Setting the σj = 0 then gives us l X z Pw,l(x + v)e{Q(x + v)z} x∈L imn/2 X 1 = Pw,l(y)e − Q(y) + 2B(y, v) . zm/2pΦ(L) z y∈Le Thus for any degree l spherical harmonic P relative to L and the inhomo- geneous theta series def X θ(L, P, v; z) = P (x + v)e{Q(x + v)z} x∈L we have imn/2 X 1 (3) θ(L, P, v; z) = P (y)e − Q(y) + 2B(y, v) . zm/2+lpΦ(L) z y∈Le (j) For wj = w0 with w0 ∈ Le this is the inversion formula from [3]. Note, however, that i has an exponent of mn/2 and not −mn/2; J. Stopple informs us that in carefully following Eichler’s proof he found a minor oversight which accounts for this difference in sign. Without loss of generality, we will assume herein that l is even. For, if l is odd then θ(L, P ; z) def= θ(L, P, 0; z) = 0 since the coefficient of e{2ζz} is P x∈L,Q(x)=2ζ P (x) and P (x) + P (−x) = 0 when l is odd. Eichler Commutation Relation 237 The arguments of [3] or of [7] can now be used to derive the transfor- mation formula; for completeness, we will sketch here a simplified version a b of this proof. Take such that a, d ∈ O, d 6= 0, b ∈ (N(L))−1∂−1, c d c ∈ N N(L)∂ and ad − bc = 1. Then as in [7] we have az + b θ L, P ; cz + d X az + b = P (x)e Q(x) cz + d x∈L X b X Q(x + x0) = e Q(x ) P (x + x )e d 0 0 d(d/z + c) x0∈L/dL x∈dL and by (3), imn/2 = (d(d/z + c))m/2+l pΦ(dL) X b X × e Q(x ) P (y)e{−Q(y)d(d/z + c) + 2B(y, x )} d 0 0 x ∈L/dL 0 y∈d−1Le imn/2 = (d/z + c)m/2+lN(d)l−m/2 pΦ(L) X b 2 c 1 × e Q(x )+ B(y, x )− Q(y) N(d)−lP (y)e −Q(y) d 0 d 0 d z x0∈L/dL y∈Le b 2 c c −1 and since d Q(x0) + d B(y, x0) − d Q(y) ≡ − d Q(bx0 + y) (mod 2∂ ) imn/2 = (d/z + c)m/2+lN(d)−m/2 pΦ(L) X c 1 × e − Q(bx + y) P (y)e − Q(y) .