Contents Acknowledgements iii Introduction Preliminaries on Maass forms for

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Nathan McDonald
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3 L σ = 1 L L Γ 0 (q) q

4 Γ 0 (q) GL(2) L

8 a n α S(α) = a n e(nα), e(θ) = e 2πiθ S(α) α R/Z α 1,..., α R δ R/Z α r α s δ r s R S(α r ) 2 r=1 ( N + 1 ) a n 2. δ

9 1 ψ(x; q, a) = n x n a q Λ(n), Λ(n) Λ(n) = { p n = p k, p k 1, 0. A B Q x1/2 ( x) B (a,q)=1 y x q Q ψ(y; q, a) y ϕ(q) x( x) A. L (a,q)=1 y x ψ(y; q, a) y ϕ(q) x1/2 2 x,

10 Q x 1/2 ( x) B Q = x ϑ ϵ ϑ > 1/2 ϵ > 0 n (p n+1 p n ) <, p n n ϑ = 1 n (p n+1 p n ) 12. ϑ = 1/2 q n (p n+1 p n ) < q Q χ q a n χ(n) 2 (N + Q 2 ) a n 2.

11 N 2 N 2 N χ(n) χ µ(n)χ(n) µ L Q M = q Q q D = d(q) d(q) q N(α, T ; χ) q Q s = σ + it L(s, χ) α σ 1, t < T τ(χ) a q q Q 1 ϕ(q) χ(a)e(a/q) χ q τ(χ) 2 N(α, T ; χ) DT (M 2 + MT ) 4(1 α) (3 2α) 10 (M + T ), Q 1 2 α 1 T 2

12 T c q Q χ q N(α, T ; χ) (Q 7 T 4 ) 1 α α c (Q + T ). N(α, T ; χ) s = σ + it L(s, χ) α σ 1, t < T c q T χ q N(α, T ; χ) T c(1 α). X < X 1 δ δ L L GL(1, A Q ) L GL(n)

13 L L L L GL(2) GL(2) GL(2) L T 2 T 0 = T 2/3 ϵ > 0 T +T0 T ( ) 1 4 ζ 2 + it dt T 0 T ϵ.

14 GL(2) L L q 1 {f j } j 1 Γ 0 (q) f j = ( 1 + t 4 j 2 )f j f j T n f j (z) = a j (n) yk itj (2π n y)e(nx) n 0 N j (α, T ) s = σ + it α σ 1, t T L(s, f j ) c ϵ > 0 α t j T a j (1) 2 πt j N j (α, T ) T c(1 α)+ϵ. q t2 L 2 (Γ 0 (q)\h) q N(α, T )

15 T c(1 α) c T 2(1 α)/α ( T ) C T T L GL(n) M(Q, α) Q Q p ( Q) α ( p) ϵ > 0 M(Q, α) Q 8 α +ϵ. GL(3) GL(3) SL(3, Z) N 3 2 +ϵ N 1+ϵ GL(2)

16 Γ 0 (q) SL(2, Z)

17 Γ 0 (q) GL(2) GL(2) h {x + iy x R, y > 0} SL(2, R) h ( ) a b z = az + b c d cz + d. h = h Q

18 SL(2, Z) SL(2, R) h ( ) a b = c d { a c c 0, c = 0, ( ) a b m c d n = { cm + dn = 0, am+bn cm+dn. q Γ 0 (q) {( ) a b Γ 0 (q) = c d } SL(2, Z) c 0 q. Γ 0 (q) Q Γ 0 (q) SL(2, Z) f : h C a Q f a σ a SL(2, R) σ a = a f(σ a (x + iy)) = O(y N ) N y k 0 k Γ 0 (q) f : h C ( ) a b γ = Γ c d 0 (q) f(γz) = (cz + d) k f(z) z h f Γ 0 (q)

19 f g k Γ 0 (q) f g f, g k = Γ 0 (q)\h f(z)g(z)y k dxdy y 2, Γ 0 (q) h k f, f < L 2 (Γ 0 (q)\h, k) n f Γ 0 (q) T n T n f(z) = 1 n ad=n (a,n)=1 d ( ) az + b f. d b=1 T 1 f(z) = f ( ) 1. Nz

20 T m T n = d (m,n) (d,n)=1 T mn d 2, T n (n, N) = 1 T n L 2 (Γ 0 (q)\h, k) T n (n, N) = 1 T n ( ) = y 2 2 x y 2 SL(2, R) Γ 0 (q) Γ 0 (q) f : h C f 0 Γ 0 (q) f = ( t2 )f 1 4 +t2 t

21 f Γ 0 (q)\h f = λf λ 1/4 λ t { ( ) } 1 b Γ = ± b Z Γ (q) a σ a SL(2, R) σ a = a σa 1 Γ a σ a = Γ Γ a a Γ 0 (q) σ a g a Γ a Γ 0 (q) ( ) 1 1 σa 1 g a σ a = f Γ (q) f(σ a (z + 1)) = f ( ( ) ) 1 1 σ a z = f(g 0 1 a σ a z) = f(σ a z), z f(σ a z) x f(σ a z) = n Z c an (y)e(nx), e(θ) = e 2πiθ Γ 0 (q) f Γ 0 (q) f, f < c a0 (y) a f = ( t2 )f

22 c an (y) n 0 y 2 c an(y) = ( 4π 2 n 2 y 2 1 t 2) c an (y). c an (y) = a(n) yk it (2π n y), K K s (y) = e y(t+t 1 )/2 t s dt t, f f(z) = n 0 a f (n) yk it (2π n y)e(nx). a f (n) L f SL(2, Z) z f(nz) Γ 0 (q) L

23 f f(z) = g(dz) g Γ 0 (M) M N d N M T n n (n, N) = 1 T n L L f(z) = n Z a f (n) yk it (2π n y)e(nx) λ f (n) = a f(n) a f (1). λ f (1) = 1 λ f (n) T n f L f R(s) > 1 L(s, f) = λ f (n)n s. n=1 T m T n = T mn d 2 d (m,n) (d,n)=1

24 λ f (mn) = λ f (m)λ f (n) (m, n) = 1 L L(s, f) = p q p q ( 1 λf (p)p s) 1 ( 1 α1,f (p)p s) 1 ( 1 α2,f (p)p s) 1, ( 1 λf (p)p s) 1 ( 1 λf (p)p s + p 2s) 1 = p q p q α i,f (p) α 1,f (p) + α 2,f (p) = λ f (p) α 1,f (p)α 2,f (p) = 1 α i,f (p) λ f (n) α i,f (p) = 1 α i,f (p) p 7/64. L L ( ) s + ϵ + it Λ(s, f) = q s/2 π s Γ Γ 2 ( s + ϵ + it 2 ) L(s, f),

25 1 4 + t2 f ϵ = { 0 T 1 f = f, 1 T 1 f = f. Λ(s, f) = ( 1) ϵ Λ(1 s, f). L(σ + it, f) 0 < σ < 1 σ = 1/2 {u j } j 1 Γ 0 (q) {u j } u j T n (n, N) = 1 {u j } u j T n n λ j u j = λ j u j λ 1 λ 2 u 0 u 0

26 L 2 (Γ 0 (q)\h) a Γ 0 (q) R(s) > 1 z h E a (z, s) = (σ a γz) s, γ Γ a\γ 0 (q) Γ a a σ a SL(2, R) σ a = a σa 1 Γ a σ a = Γ E a (z, s) b E α (σ b z, s) = δ ab y s + π 1/2 Γ(s 1) 2 φ ab0 (s)y 1 s + 2y1/2 π s Γ(s) Γ(s) n 0 n s 1 2 φabn (s)k s 1 (2π n y)e(nx), 2 φ abn (s) = c>0 c 2s d e ( ) nd, c d ( c) ( ) a b c d σa 1 Γ 0 (q)σ b. E a (z, s) s Ea(z, s) a s 1 s

27 E a (z, ir) u 0 L 2 (Γ 0 (q)\h) f L 2 (Γ 0 (q)\h) {u j } f(z) = j 0 f, u j u j (z) + 1 4π a f, E a (, 12 + ir) E a (z, ir)dr, a Γ 0 (q) f, g L 2 (Γ 0 (q)\h) {u j } f, g = j 0 f, u j g, u j + 1 4π a f, E a (, 12 + ir) g, E a (, 12 + ir) dr.

28 S(m, n; c) = ( ) md + nd e, c d c (c, d) = 1 d dd 1 c

29 (c 1, c 2 ) = 1 c 1 c 2 c 1 c 1 1 c 2 c 2 c 2 1 c 1 S(m, n; c 1 c 2 ) = S(c 2 m, c 2 n; c 1 )S(c 1 m, c 1 n; c 2 ). ( c2 md 1 + c 2 nd 1 e c 1 d 1 c 1 e d 1 c 1 d 2 c 2 = ) e d 2 c 2 ( ) c1 md 2 + c 1 nd 2 c 2 ( c2 c 2 md 1 + c 2 c 2 nd 1 + c 1 c 1 md 2 + c 1 c 1 nd 2 c 1 c 2 ). md 1 +nd 1 c 1 md 2 +nd 2 c 2 d c 1 c 2 (d, c 1 c 2 ) = 1 d 1 d 2 d c 1 c 2 md + nd c 1 c 2 d c 1 c 2 (d, c 1 c 2 ) = 1 (d 1, d 2 ) S(m, n; c 1 c 2 ) c

30 S(m, n; c) τ(c)(m, n, c) 1/2 c 1/2, τ(c) c (m, n, c) m n c T m T n = d (m,n) T mn d 2 SL(2, Z)\h S(m, n; c) = d (m,n,c) d S ( 1, mn d, c ). 2 d Z m,n (s) = S(m, n; c)c 2s. c=1 σ > 1/2 Z m,n (s) R(s) > σ

31 σ(1 σ) R(s) > 3/4 3/16 α 1,..., α R α r α s δ r s S(α) = a n e(nα). R r=1 S(α r ) 2 (N + 1 δ ) a n 2. α r r/c r c δ = 1/c r c ( a n e n r ) 2 c (N + c) a n 2.

32 B(θ, c, M, N) = m=m+1 ( ) 2 mn b m b n S(m, n, c)e θ, c N M θ > 0 c > M 1 ϵ B(θ, c, M, N) M ϵ (c + N) b n 2. α = N M 1 1 mn M η( mn ) M η(x) = 1 + α m, n { 1 1 x 1 + α, 0 x 1 α 2 x 1 + 2α. η(x)e(2θc 1 Mx) = 1 2πi 1+i 1 i R(s)x s ds, R(s) = 0 η(x)e(2θc 1 Mx)x s 1 dx. s = 1 + it t t 2 t > 16πθc 1 M

33 1+i 1 i ( ) ( ) mn 2 mn B(θ, c, M, N) = b m b n S(m, n, c)η e M c m=m+1 = 1 1+i ( ) s mn b m b n S(m, n, c)r(s) ds 2πi 1 i M m=m+1 ( ) md b m m s/2 ( ) nd e b n n s/2 e c c M s R(s)ds. d c m=m+1 d c m=m+1 ( ) md b m m s/2 e c b n n s/2 e ( ) nd c M 1 (N + c) b n 2. B(θ, c, M, N) 1+i 1 i R(s)ds (N + c) b n 2, R c > M 1 ϵ c B(θ, c, M, N) N M c M 1 ϵ 0 < θ < 2 B(θ, c, M, N) θ c 2 N 4 M 4 +ϵ b n 2.

34 B(θ, c, M, N) = d c d X ( θ, c d, M d, N ), d x n = b dn, q = c/d, K = M/d, L = N/d X (θ, q, K, L) = K<m,n K+L ( ) 2 mn x m x n S(1, mn; q)e. q X(θ, q, K, L) 2 = ( ( K<n K+L K<n K+L ) ( n ) x n 2 η K n Z x n 2 ) K<m 1,m 2 K+L K<m K+L x m1 x m2 ( ) 2 2 mn x m S(1, mn; q)e q ( ) h1 h 2 e f(n), q h 1,h 2 q n Z η (h 1 h 2, q) = 1 h i ( q) h i h i 1 q ( n ) ( h1 m 1 h 2 m 2 f(n) = η e n + 2( m 1 ) m 2 )θ n. K q q A B f(n) = η ( n K ) e (An + B n) n Z f(n) = u Z f(u),

35 f(u) = ( ) t ( η e (A u)t + B ) t dt. K u A A u 1 q t η ( t K ) 0 t > K/2 B 2 < θ K+L K/2 K t q 1 q p ( ) p (( 1 ( ) ) ) t 1 1 f(u) = η 2πi K A u + B 2 A u + B t 2 t ( e (A u)t + B ) t dt, f(u) K( A u K) p. u A f(u) K ( ) p K = M q d ( ) p M M c d M ϵp K 1 p u = A A h 1 m 1 h 2 m 2 q h 1,h 2 q ( ) h1 h 2 e (m 1 m 2, q), q h 1 m 1 h 2 m 2 q m 1 = m 2 f(a) L f(a) = ( ) t ( η e B ) t dt K L B q KL θ m 1 m 2.

36 K<m 1,m 2 K+L x m1 x m2 q(q + L) h 1,h 2 q θ 1 qk ϵ KL K<m K+L ( ) h1 h 2 e f(n) q K<m K+L n Z x m 2 + θ 1 q KL x m 2. K<m 1,m 2 K+L m 1 m 2 (m 1 m 2, q) x m1 x m2 m 1 m 2 ( ) 2 X(θ, q, K, L) 2 θ 1 qk ϵ KL x n 2. K<m K+L d c Γ 0 (q) m 0 P m (z, s) = (γz) s e(mγz). γ Γ \Γ 0 (q) m = 0 R(s) > 1 P m (γz, s) = P m (z, s) P m (, s) L 2 (Γ 0 (q)\h)

37 P m (, s 1 ), P n (, s 2 ) P m (z, s) P m (z, s) R(s) > 1 m 0 P m (z, s) = n Z B n (m, y, s)e(nx), B n (m, y, s) = δ mn y s e 2πny + y s c>0 c 0 q c 2s S(m, n; c) ( (x 2 + y 2 ) s e nx ) m dx. c 2 (x + iy) Z m,n (s) = c=1 S(m, n; c)c 2s R(s) > 3/4 P m (z, s) R(s) > 3/4 m, n 1 P m (, s 1 ), P n (, s 2 ) = 0 B n (m, y, s 1 )y s 2 2 e 2πny dy.

38 D Γ 0 (q) h P m (, s 1 ), P n (, s 2 = = D P m (z, s 1 ) γ Γ \Γ 0 (q) γd = E γ Γ \Γ 0 (q) (γz) s 2 e(nz) dxdy y 2 P m (γ 1 z, s 1 )y s 2 e(nz) dxdy y 2 P m (z, s 1 )e( nx)y s 2 2 e 2πny dxdy, E Γ 0 x < 1, y > 0 P m (z, s 1 ) P m (, s 1 ), P n (, s 2 ) = k Z B k (m, y, s 1 )e(kx)e( nx)y s2 2 e 2πmy dxdy. x k = n B k (m, y, s 1 ) P m (, s 1 ), P n (, s 2 ) R(s 1 ) > 3/4 R(s 2 ) > 3/4 s 1 = 1+it s 2 = 1 it m, n 1 P m (, 1 + it), P n (, 1 + it) = δ ( mn n ) it 4πn 2i S(m, n; c) i m c 2 c>0 i c 0 q ( ) 4π mn dv K 2it v c v. P m (, s 1 ), P n (, s 2 )

39 P m (, s) L 2 (Γ 0 (q)\h) P m m f(z) = n 0 a f (n) yk it (2π n y)e(nx) Γ 0 (q) t2 m 1 R(s) > 1 P m (, s), f = (4πm) 1 2 s π 1 2 af (m) Γ(s 1 + it)γ(s 1 it) 2 2. Γ(s) P m (, s), f = = D γ Γ \Γ 0 (q) γ Γ \Γ 0 (q) γd = = E 1 0 = a f (m) (γz) s e(mγz)f(z) dxdy y 2 (z) s e(mz)f(γ 1 z) dxdy y 2 f(z)e(mx)y s 2 e 2πmy dxdy a f (n)k it (2π n y)e( nx)e(mx)y s 3 2 e 2πmy dxdy 0 n 0 0 K it (2πmy)y s 3 2 e 2πmy dy. 0 K it (y)y s 3 2 e y dy = π s Γ(s it)γ(s 1 2 it) Γ(s)

40 m 1 R(s) > 1 a Γ 0 (q) P m (, s), E a (, 12 + ir ) = 2 2 2s m 1 2 s ir π 3 2 s ir φ a m ( ir ) Γ(s ir)γ(s 1 2 ir) Γ(s)Γ( 1 2 ir). u j (z) = n 0 a j(n) yk it (2π n y)e(nx) u j t2 j m, n 1 ( πm 1 2 s 1 n 1 2 s 2 P m (, s 1 ), P n (, s 2 ) = (4π) s 1+s 2 a 1 j (m)a j (n)λ(s 1, s 2 ; t j ) Γ(s 1 )Γ(s 2 ) j 1 + ( n ) ( ) ( ) ) ir 1 1 φa m a m 2 + ir φ a n 2 + ir Λ(s1, s 2 ; r) Γ( 1 dr, + ir) 2 2 Λ(s 1, s 2 ; r) = Γ (s 1 12 ) + ir Γ (s 1 12 ) ir Γ (s 2 12 ) + ir Γ (s 2 12 ) ir. s 1 = 1 + it s 2 = 1 it

41 ( P m (, 1 + it), P n (, 1 + it) = 1 ( n ) it 4 πt π mn m t j 1 + a ( n m a j (m)a j (n) π(t j t) π(t j + t) ) ir φ a m ( ir) φ a n ( ir) π(r t) π(r + t) πrdr ). ( 1 4 n mn m ) it πt t {u j } Γ 0 (q) 1 4 +t2 j u j (z) = n 0 a j(n) yk it (2π n y)e(nx) m, n 1 t R 2it πt c>0 c 0 q 4π mn i S(m, n; c) c 2 = π a j (m)a j (n) π(t j 1 j t) π(t j + t) + a i ( ) 4π mn dv K 2it v c ( n m v + δ t mn π πt ) ir φ a m ( ir) φ a n ( ir) π(r t) π(r + t) πrdr. a Γ 0 (q) {u j }

42 {b n } M N 1 πt j t j K b n a j (n) 2 (K 2 + M ϵ N) b n 2. t πte (t/k)2 b m b n t j 1 t j + 1 (tj/k)2 e πt j K 3 + a b n c c>0 c 0 q b n a j (n) 2 ( r + 1)e (r/k)2 m=m+1 b n n ir φ a n ( ir ) 2 dr 4π mn b m b n S(m, n; c)φ c ( 4π mn c ), Φ (x) = t 2 e (t/k)2 i i K 2it (xv) dv v. Φ Φ (x) = πik 3 e (ξk)2 ξ ξ (x ξ)dξ 1 0 xφ (x) = ( πik 3 e (ξk)2 1 ξ ξ 2ξ 2 K 2) dξ (x ξ) ξ xk 2. 0 c > K 2 M c K 2 M ϕ(c, K, M, N) = m=m+1 4π mn b m b n S(m, n; c)φ c ( 4π mn c ).

43 c > M 2 Φ(c, K, M, N) c 1 2 +ϵ M 2 b n 2. M 1 ϵ < c M 2 Φ(c, K, M, N) M ϵ N b n 2. K 2 M < c M 1 ϵ ξ 1 ξ 1 Φ(c, K, M, N) M ϵ (Ke K2 c 1 MN + K 2 c M 4 N 4 ) b n 2. c K 2 M Φ(c, K, M, N) M ϵ (Ke K2 c 1 MN + c M 4 N 4 ) b n 2. c>0 c 0 q 1 c ϕ(c, K, M, N) KN(M ϵ + MKe K2 ) b n 2,

44 t j + 1 πt j t j K b n a j (n) 2 K(K 2 + M ϵ N + MNKe K2 ) b n 2. K K K + M ϵ t j + 1 πt j t j K b n a j (n) 2 K(K 2 + M ϵ N) b n 2, 1 πt j t j K b n a j (n) 2 (K 2 + M ϵ N) b n 2.

45 L L L L L L GL(2) L

46 L(s, f) L Γ 0 (q) w = 1 + iv L L (s, f) = ρ 1 + O( q( v + 2)), s ρ ρ L(s, f) ρ w 1 L L(s, f) Q(f, v, r) L(s, f) s (1 + iv) r Q(f, v, r) r q( v + 2) + 1. m 0 {b j } z 1 z 2 z n. ν m + 1 ν m + n ( ) n b 1 z1 ν + b 2 z2 ν + + b n zn ν z 1 ν n 24e 2 (m + 2n) b b j. 1 j n

47 L(s, f) w = 1 + iv v T r 0, A, B, C L(s, f) s w r 1 T r r 0 x T A x B x x<p y a f (p) p a f (1)p w dy y x Cr 2 x, p s w 1/2 L (s, f) = L ρ w O( T ). s ρ s w 1/4 d k 1 L (s, f) = k! ds k ( 1)k L ρ w 1 1 (s ρ) k+1 + O(4k T ). s = w+r r < 1/4 λ r λ 1/4 ρ 2 j λ < ρ w 2 j+1 λ 2 j λ T ρ 1 (s ρ) k+1 (2 j λ) (k+1) ρ w > λ (2 j λ) k T λ k T. j=0 λ 1/4 ρ w > λ d k 1 L (s, f) = k! ds k ( 1)k L ρ w λ 1 (s ρ) k+1 + O(λ k T ).

48 A 1 λ T A 1 s w r ρ s ρ 2r z j 1 s ρ b j = 1 z 1 1 2r m A 1λ T k m k 2m 1 (s ρ) k+1 (Dr) (k+1) ρ w λ D λ = A 2 Dr r 0 λ 1/4 A 2 O(λ k T ) r 1 T 1 k! d k L ds k L (s, f) (Dr) (k+1), m k 2m m A 1 λ T = Er T E = A 1 A 2 D L L (s, f) = Λ f (n)n s, n=1 Λ f (p j ) = { λ f (p) j p p q, (α f,1 (p) j + α f,2 (p) j ) p p q Λ f (n) = 0 n Λ f (p) = λ f (p) p Λ(f)(p j ) p 7 64 j+ϵ n=1 Λ f (n) n p k(r n) D k w r,

49 p k (u) = e u u k. k! B 1 B 2 p k p k (u) (2D) k u B 1 k p k (u) (2D) k e u/2 u B 1 k A = B 1 E x T A m = B 1 1 r x m Er T k m k 2m B = 2B 2 /B 1 Λ f (n) n p k(r n) (2D) k Λ f (n) w n n x n x Λ f (n) n p k(r n) (2D) k Λ f (n) w n 1+ r n>x B n>x B 2 (2D) k k r, (2D) k. r L(s, f) n x Λ f(n) x n x n > x B Λ f (n) n p k(r n) D k w r. x<n x B Λ(f)(p j ) p 7 64 j+ϵ p j j 2 S(y) = x<p y Λ f (p) p w = x<p y λ f (p) p p w = x<p y a f (p) p a f (1)p w. x B x p k (r y)ds(y) = p k (r x B )S(x B ) x B x S(y)p k(r y)r dy y.

50 (2D) k k p r k 1 x B x S(y) dy y D k r 2 x Cr 2 x. S(t) = n=1 a n n it T T S(t) 2 dt T 2 0 y<n ye 1/T a n 2 dy y. {u j } Γ 0 (q) u j t2 j u j (z) = n 0 a j(n) yk it (2π n y)e(nx) {b n } b n a j (n) n=1 1 πt j t j K T T b n a j (n)n 2dt it n=1 (K 2 T + n 1+ϵ ) b n 2. n=1

51 T 2 1 πt j 0 t j K ye 1/T b n a j (n) y 2 dy y. 1 πt j t j K ye 1/T 2 b n a j (n) ( K 2 + y ϵ (e 1/T 1) ) ye1/t b n 2. y y b n 2 T 2 K 2 n ne 1/T dy y + T ( 2 e 1/T 1 ) n ne 1/T y ϵ dy K 2 T + T 2 ( e 1/T 1 ) ( 1 e 1/T ) 1+ϵ n 1+ϵ K 2 T + n 1+ϵ T 1 2(1 α) r T 0 1 α 1 T α r = 2(1 α) x = T (A,3) L(s, f j )

52 s w r x B x Cr 2 x x x p y a j (p) p a j (1)p w dy y 1, p c = 4C (A, 3) T c(1 α) 3 T x B x x p y a j (p) p a j (1)p w 2 dy y 1. s w 2(1 α) ((1 α) T ) 1 α w 1 α w = 1 + iv N j (α, T ) T c(1 α) 2 T x B T x T x p y a j (p) p a j (1)p 1+iv 2 dv dy y. {f j } {u j } a j (1) 2 N j (α, T ) 1 x B T T c(1 α) 2 T πt j πt j x T t j T t j T x p y a j (p) p p 1+iv 2 dv dy y, {f j } {u j } b p = p p b n = 0 t j T a j (1) 2 x B N j (α, T ) T c(1 α) 2 T πt j x x p y ( ) 2 p (T 3 + p 1+ϵ dy ) p y.

53 x T 3 T x p x B p ϵ 2 p T ϵ,

54 Γ 0 (m) GL(3) σ = 1 L GL(n, R) L

55 GL(3) L L L GL 4 GL 2 ax 2 + by 2 + cz 2 + dt 2 L

56 L L L L GL 3

An application of the projections of C automorphic forms

ACTA ARITHMETICA LXXII.3 (1995) An application of the projections of C automorphic forms by Takumi Noda (Tokyo) 1. Introduction. Let k be a positive even integer and S k be the space of cusp forms of weight