Trace formulas. Modular tools

732 Emmanuel Royer, Jie Wu by the expansion 67 Ls, Sym 2 f Ls, Sym m f z =: + ∞ X n=1 λ 1,z Sym 2 f,Sym m f nn −s . The preceding results imply the Proposition 9. Let N be a squarefree integer, f ∈ H ∗ 2 N ; let ν ≥ 0 and m 0 be integers and z be a complex number. Then λ 1,z Sym 2 f,Sym m f p ν =                    ν X ν ′ =0 τ z p ν ′ λ f p mν ′ p ν ′ −ν if p | N max2,mν X m ′ =0 µ 1,z,ν Sym 2 ,Sym m ,Sym m′ λ f p m ′ if p ∤ N . Moreover, λ 1,z Sym 2 f,Sym m f p ν ≤ τ m+1 |z|+3 p ν µ 1,z,0 Sym 2 ,Sym m ,Sym m′ = δm ′ , 0 µ 1,z,1 Sym 2 ,Sym m ,Sym m′ = zδm ′ , m + δm ′ , 2 µ 1,z,ν Sym 2 ,Sym 2m ,Sym 2m′+1 = 0, and max2,mν X m ′ =0 µ 1,z,ν Sym 2 ,Sym m ,Sym m′ ≤ m + 1 |z| + 2 + ν ν .

2.4. Trace formulas.

In this section, we establish a few mean value results for Dirichlet coefficients of the different L–functions we shall en- counter. Let f ∈ H ∗ 2 N . Denote by ε f N := εSym 1 f the sign of the functional equation satisfied by Ls, f . We have 68 ε f N = −µN √ N λ f N ∈ {−1, 1}. The following trace formula is due to Iwaniec, Luo Sarnak [ILS00, Corollary 2.10]. Lemma 10. Let N ≥ 1 be a squarefree integer and m ≥ 1, n ≥ 1 two integers satisfying m, N = 1 and n, N 2 | N. Then X f ∈H ∗ 2 N ω ∗ f λ f mλ f n = δm, n + OErr Special values of symmetric power L-functions and Hecke eigenvalues 733 with Err := τ N 2 log 2 3N N mn 14 τ 3 [m, n] p n, N log2mnN . We shall need a slightly different version of this trace formula we actually only remove the condition n, N = 1 from [ILS00, Proposition 2.9]. Lemma 11. Let N ≥ 1 be a squarefree integer and m ≥ 1, n ≥ 1 two integers satisfying m, N = 1 and n, N 2 | N. Then X f ∈H ∗ 2 N ω ∗ f [1 + ε f N ] λ f mλ f n = δm, n + OErr with Err := δn, mN √ N + τ N 2 log 2 3N N 34 mn 14 p n, N log2mnN τ 3 [m, n] N 14 + τ [m, n] p n, N . Proof. By Lemma 10, it suffices to prove that X f ∈H ∗ 2 N ω ∗ f ε f N λ f mλ f n ≪ δn, mN √ N + τ N 2 log 2 3N N 34 mn 14 n, N τ [m, n] log2mnN . Since ε f N = −µN √ N λ f N , we shall estimate R := √ N X f ∈H ∗ 2 N ω ∗ f λ f mλ f nλ f N . The multiplicativity relation 30 and equation 31 give R = √ N X f ∈H ∗ 2 N ω ∗ f λ f mλ f n N λ f n N 2 λ f N n N = √ N n N X d |m,n N X f ∈H ∗ 2 N ω ∗ f λ f mn N d 2 λ f N n N . Then, Lemma 10 leads to the result since M N N d 2 = Nn N implies N = n N , m = n N and d = m. We also prove a trace formula implying the Dirichlet coefficients of the symmetric power L-functions. 734 Emmanuel Royer, Jie Wu Lemma 12. Let N be a squarefree integer, m, n, q be nonnegative integers and z be a complex number. Then X f ∈H ∗ 2 N ω ∗ f [1 + ε f N ] λ z Sym m f nλ f q = w z m n, q + OErr with 69 w z m n, q := τ z n N n m N q N p n m N q N Y 1 ≤j≤r X ≤ν ′ j ≤mν j p ν′1 1 ···p ν′r r =q N µ z,ν j Sym m ,Sym ν′ j where n N = r Y j=1 p ν j j , p 1 · · · p j and Err := τ N 2 log 2 3N N 34 n m4 τ m+1 |z| nτ qq 14 log2N nq. The implicit constant is absolute. Proof. Let S := X f ∈H ∗ 2 N ω ∗ f [1 + ε f N ] λ z Sym m f nλ f q. Writing n Q N M N = g 2 h with h squarefree, equation 31 and Proposition 2 give S = τ z n N g X ν ′ i 1≤i≤r ∈X r i=1 [0,mν i ]   r Y j=1 µ z,ν j Sym m ,Sym ν′ j   × X d | „ q N , Q r j=1 p ν′ j j « X f ∈H ∗ 2 N ω ∗ f [1 + ε f N ]λ f hλ f   q N d 2 r Y j=1 p ν ′ j j   . Then, since h | N, Lemma 11 gives S = P + E with P = τ z n N g r Y j=1 mν j X ν ′ j =0 µ z,ν j Sym m ,Sym ν′ j X d | „ q N , Q r j=1 p ν′j j « q N p ν′1 1 ···p ν′r r d 2 =h 1 Special values of symmetric power L-functions and Hecke eigenvalues 735 and E ≪ τ N 2 log 2 3N N 34 n m4 τ |z| n N n m2 N q 14 τ q q 12 N log2N nq g 12 r Y j=1 mν j X ν ′ j =0 µ z,ν j Sym m ,Sym ν′ j . Using 51, we obtain E ≪ τ N 2 log 2 3N N 34 n m4 q 14 τ q log2N nqτ m+1 |z| n. We transform P as the announced principal term since q N p ν ′ 1 1 · · · p ν ′ r r d 2 = h implies p ν ′ 1 1 · · · p ν ′ r r = q N = d and h = 1. Similarly to Lemma 12, we prove the Lemma 13. Let k, N , m, n be positive integers, k even, N squarefree. Let z ∈ C. Then X f ∈H ∗ 2 N ω ∗ f λ 1,z Sym 2 f,Sym m f n = w 1,z 2,m n + O k,m Err with Err := τ N 2 log 2 3N N n max2,mν4 r 1,z 2,m n log2nN where w 1,z 2,m and r 1,z 2,m are the multiplicative functions defined by w 1,z 2,m p ν :=                ν X ν ′ =0 τ z p ν ′ p mν ′ p ν −ν ′ +mν ′ 2 if p | N µ 1,z,ν Sym 2 ,Sym m ,Sym if p ∤ N and r 1,z 2,m p ν :=                ν X ν ′ =0 τ |z| p ν ′ p ν −ν ′ +mν ′ 2 if p | N m+1 |z|+ν+2 ν if p ∤ N . 736 Emmanuel Royer, Jie Wu

2.5. Mean value formula for the central value of Ls, f . Using the