ON THE LIFTING OF HERMITIAN MODULAR FORMS TAMOTSU IEDA Notation Let be an imaginary quadratic field with discriminant D = D. We denote by O = O the ring of integers of. The non-trivial automorphism of is denoted by x x. The primitive Dirichlet character corresponding to /Q is denoted by χ = χ D. We denote by O = ( D) 1 O the inverse different ideal of /Q. The special unitary group G = SU(m, m) is an algebraic group defined over Q such that ( ) ( )} 0m 1 G(R) = g SL 2m (R ) g m 1 m 0 tḡ 0m 1 = m m 1 m 0 m for any Q-algebra R. We put Γ (m) = G(Q) GL 2m(O). The hermitian upper half space H m is defined by H m = Z M m (C) 1 2 1 (Z t Z) > 0}. Then G(R) acts on H m by ( ) A B g Z = (AZ + B)(CZ + D) 1, Z H m, g =. C D We put Λ m (O) = h = (h ij ) M m () h ii Z, h ij = h ji O, i j}, Λ m (O) + = h Λ m (O) h > 0}. We set e(t ) = exp(2π 1tr(T )) if T is a square matrix with entries in C. For each prime p, the unique additive character of Q p such that e p (x) = exp( 2π 1x) for x Z[p 1 ] is denoted e p. Note that e p is of order 0. We put e (x) = e(x ) p< e p(x p ) for an adele x = (x v ) v A. 1
2 TAMOTSU IEDA Let χ = v χ v be the Hecke character of A /Q determined by χ. Then χ v is the character corresponding to Q v ( D)/Q and given by ( ) D, t χ v (t) =. Let Q D be the set of all primes which divides D. For each prime q Q D, we put D q = q ordqd. We define a primitive Dirichlet character χ q by χ(n ) if (n, q) = 1 χ q (n) = 0 if q n, Q v where n is an integer such that n n mod D q, 1 mod D 1 q D Then we have χ = q D χ q. Note that ( ) χq ( 1)D q, n χ q (n) = = ( ) χq ( 1)D q, n Q q Q p p n for q n, n > 0. One should not confuse χ q with χ q. 1. Fourier coefficients of Eisenstein series on H m In this section, we consider Siegel series associated to non-degenerate hermitian matrices. Fix a prime p. Put ξ p = χ(p), i.e., 1 if D (Q p ) 2 ξ p = 1 if Q p ( D)/Q p is unramified quadratic extension 0 if Q p ( D)/Q p is ramified quadratic extension. For H Λ m (O), det H 0, we put γ(h) = ( D) [m/2] det(h) ζ p (H) = χ p (γ(h)) m 1. The Siegel series for H is defined by b p (H, s) = e p (tr(br))p ordp(ν(r))s, Re(s) 0. R Her m( p)/her m(o p) Here, Her m ( p ) (resp. Her m (O p )) is the additive group of all hermitian matrices with entries in p (resp. O p ). The ideal ν(r) Z p is defined
as follows: Choose a coprime pair C, D}, C, D M 2n (O p ) such that C t D = D t C, and D 1 C = R. Then ν(r) = det(d)o p Z p. We define a polynomial t p (/Q; X) Z[X] by t p (/Q; X) = [(m+1)/2] (1 p 2i X) (1 p 2i 1 ξ p X). [m/2] There exists a polynomial F p (H; X) Z[X] such that F p (H; p s ) = b p (H, s)t p (/Q; p s ) 1. This is proved in [9]. Moreover, F p (H; X) satisfies the following functional equation: F p (H; p 2m X 1 ) = ζ p (H)(p m X) ordpγ(h) F p (H; X). This functional equation is a consequence of [7], Proposition 3.1. We will discuss it in the next section. The functional equation implies that degf p (H; X) = ord p γ(h). In particular, if p γ(h), then F p (H; X) = 1. Put F p (H; X) = X ordpγ(h) F p (H; p m X 2 ). Then following lemma is a immediate consequence of the functional equation of F (H; X). 3 Lemma 1. We have F p (H; X 1 ) = F p (H; X), if m is odd. F p (H; ξ p X 1 ) = F p (H; X), if m is even and ξ p 0. Let k be a sufficiently large integer. Put n = [m/2]. The Eisenstein series E (m) (Z) of weight 2k + 2n on H m is defined by E (m) (Z) = det(cz + D) 2k 2n, C,D}/ where C, D}/ extends over coprime pairs C, D}, C, D M 2n (O) such that C t D = D t C modulo the action of GLm (O). We put m E (m) (Z) = A 1 m L(1 + i 2k 2n, χ i 1 )E (m) (Z). Here A m = 2 4n2 4n D 2n2 +n if m = 2n + 1, ( 1) n 2 4n2 +4n D 2n2 n if m = 2n.
4 TAMOTSU IEDA Then the H-th Fourier coefficient of E (2n+1) (Z) is equal to γ(h) 2k 1 F p (H; p 2k 2n ) = γ(h) k (1/2) F p (H; p k+(1/2) ) = γ(h) k (1/2) F p (H; p k (1/2) ) for any H Λ 2n+1 (O) + and any sufficiently large integer k. The H-th Fourier coefficient of E (2n) (Z) is equal to γ(h) 2k F p (H; p 2k 2n ) = γ(h) k F p (H; p k ) for any H Λ 2n (O) + and any sufficiently large integer k. 2. Main theorems We first consider the case when m = 2n is even. Let f(τ) = N=1 a(n)qn S 2k+1 (Γ 0 (D), χ) be a primitive form, whose L-function is given by L(f, s) = a(n)n s N=1 = (1 a(p)p s + χ(p)p 2k 2s ) 1 a(q)q p D q D(1 s ) 1. For each prime p D, we define the Satake parameter α p, β p } = α p, χ(p)αp 1} by (1 a(p)x + χ(p)p 2k X 2 ) = (1 p k α p X)(1 p k β p X). For q D, we put α q = q k a(q). Put A(H) = γ(h) k F p (H, α p ), H Λ 2n (O) + F (Z) = A(H)e(HZ), Z H 2n. H Λ 2n (O) + Then our first main theorem is as follows: Theorem 1. Assume that m = 2n is even. Let f(τ), A(H) and F (Z) be as above. Then we have F S (Γ (2n) ). Moreover, F is a Hecke eigenform. F = 0 if and only if f(τ) comes from a Hecke character of and n is odd.
Now we consider the case when m = 2n + 1 is odd. Let f(τ) = N=1 a(n)qn S 2k (SL 2 (Z)) be a normalized Hecke eigenform, whose L-function is given by L(f, s) = a(n)n s N=1 = (1 a(p)p s + p 2k 1 2s ) 1 p 5 For each prime p, we define the Satake parameter α p, αp 1} by (1 a(p)x + p 2k 1 X 2 ) = (1 p k (1/2) α p X)(1 p k (1/2) αp 1 X). Put A(H) = γ(h) k (1/2) F p (H, α p ), H Λ 2n+1 (O) + F (Z) = H Λ 2n+1 (O) + A(H)e(HZ), Z H 2n+1. Theorem 2. Assume that m = 2n + 1 is odd. Let f(τ), A(H) and F (Z) be as above. Then we have F S (Γ (2n+1) ). Moreover, F is a non-zero Hecke eigenform. References [1] J. Arthur, Unipotent automorphic representations: conjectures, Astérisque 171-172 (1989), 13 71. [2] S. Böcherer, Über die Fourier-Jacobi-Entwicklung Siegelscher Eisensteinreihen. I, Math. Z. 183 (1983), 21 46. [3] S. Breulmann and M. uss, On a conjecture of Duke-Imamoglu, Proc. Amer. Math. Soc 107 (2000), [4] M. Eichler and D. Zagier, The theory of Jacobi forms, Progress in Mathematics 55 Birkhäuser Boston, Inc., Boston, Mass. 1985. [5] T. Ikeda, On the theory of Jacobi forms and the Fourier-Jacobi coefficients of Eisenstein series, J. Math. yoto Univ. 34 (1994), 615 636. [6], On the lifting of elliptic cusp forms to Siegel cusp forms of degree 2n, Ann. of Math. 154 (2001), 641 681. [7] S. udla and W. J. Sweet Jr. Degenerate principal series representations for U(n, n). Israel J. Math. bf 98 (1997), 253 306. [8] G. Shimura Introduction to the arithmetic theory of automorphic functions, Publ. Math. Soc. Japan 11 Iwanami Shoten and Princeton University Press, 1971. [9], Euler products and Eisenstein series, CBMS Regional Conference Series in Mathematics 93 the American Mathematical Society, Providence, RI, 1997.