with k = l + 1 (see [IK, Chapter 3] or [Iw, Chapter 12]). Moreover, f is a Hecke newform with Hecke eigenvalue

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1 L 4 -NORMS OF THE HOLOMORPHIC DIHEDRAL FORMS OF LARGE LEVEL SHENG-CHI LIU Abstract. Let f be an L -normalized holomorphic dihedral form of prime level q and fixed weight. We show that, for any ε > 0, for some δ > 0. f 4 4 ε q 3 δ+ε,. Introduction Let f be an L -normalized holomorphic Hecke newform or Maass-Hecke eigenform for the congruence subgroup Γ 0 (q). It is of great interest to understand the mass distribution of f as the Laplacian eigenvalue, weight or level of f goes to infinity. One way to examine this is bounding the L 4 -norm of f. This direction has been investigated in many works (see [Bl], [BKY], [BuK], [Lu], [Sa], for example). The L 4 -norm problem has a special feature that connects to triple product L-functions via Watson s formula. One may predict the optimal bounds of the L 4 -norms by assuming the Lindelöf hypothesis for the related L-functions. However, the behavior of the L-functions are quite different in terms of the spectral, weight and level. In the spectral aspect, Sarnak and Watson ([Sa]) have announced a proof of the optimal bound and Luo [Lu] proved it for the dihedral Maass forms by establishing the Lindelöf hypothesis on average for a family of triple product L-functions. In the weight or level aspects, it is very difficult to prove the optimal bound due to large conductors of the L-functions comparing with the size of families under averages. We refer to the nice discussion in Blomer s paper [Bl]. In this work, we will prove a bound for L 4 -norms in the level aspect for holomorphic dihedral forms. Let q be a prime such that q is a fundamental discriminant. Let K = Q( q) and let O denote its ring of integers. Let l be a fixed positive integer. Let ψ be a ( ) l α Hecke grossencharacter of K such that ψ((α)) = for α O. Then α f(z) := a O ψ(a)(na) l/ e(zna) S k (Γ 0 (q), χ q ), with k = l + (see [IK, Chapter 3] or [Iw, Chapter ]). Moreover, f is a Hecke newform with Hecke eigenvalue λ f (n) = n k ψ(a). Na=n 00 Mathematics Subject Classification. F, M99. Key words and phrases. dihedral form; grossencharacter; L 4 -norm. The author was supported by AMS-Simons Travel Grant.

2 SHENG-CHI LIU Let f be the L -normalized newform corresponding to f; that is, f(z) y k dxdy =. y The main result in this paper is established in the following theorem. Theorem.. Let δ > 0. Suppose that the subconvexity bound L(, g) ε q 4 δ+ε holds, for all Hecke newforms g S k (Γ 0 (q)), and any ε > 0. Then we have f 4 4 := f(z) 4 k dxdy y y ε q 3 δ+ε, for any ε > 0. Our result and method also hold for dihedral Hecke-Maass forms of prime level q. The Lindelöf hypothesis asserts that δ = /4, which gives the best possible bound f 4 4 q +ε. The best-known subconvexity bound is δ = /9, which is due to Duke, Friedlander, and Iwaniec [DFI]. For a L -normalized Hecke-Maass f of prime level q with bounded Laplace eigenvalue, Blomer [Bl] showed the optimal bound on average for such forms, and also noted that f 4 4 q /3+ε by interpolating the non-trivial sup-norm bound of Harcos and Templier [HT]. It was proved f 4 4 q /+ε by the author, Masri and Young in [LMY]. For a holomorphic Hecke newform of prime level q, Buttcane and Khane [BuK] showed f 4 4 q 3/4 δ+ε with a restriction on the weight of f.. Preliminaries For h, h S k (Γ 0 (q)), the Petersson inner product is defined by k dxdy (.) h, h := h (z)h (z)y. y An orthonormal Hecke basis of S k (Γ 0 (q)) is given by (see [ILS, Proposition.6] and [Lu, pp ]) B := B q B (q) B where B q is an orthonormal Hecke basis for the newforms Sk New (Γ 0 (q)), B (q) is a Hecke basis of S k (SL (Z)), which is orthonormal with respect to the inner product (.), and where (.) B = {g q : g B (q) } ( ) / ( ) g q (z) = qλ g(q) q k g(qz) q/ λ g (q) (q + ) q + g(z), and λ g (n) is the nth Hecke eigenvalue of g.

3 L 4 -NORMS OF THE HOLOMORPHIC DIHEDRAL FORMS OF LARGE LEVEL 3 By Watson s formula [Wa], for g B q, we have f, g = f(z) k dxdy g(z)y (.3) y q L(, g)l(, sym f g) L(, sym f) L(, sym g), = 4q Λ(, f f g) Λ(, sym f) Λ(, sym g) where Λ(, f f g) is the completed triple product L-function (see [Wa, Chapter 4]). For a dihedral form f corresponding to the Hecke grossencharacter ψ, the L-function L(, sym f g) has a further factorization: (.4) L(, sym f g) = L(, F g)l(, g χ q) where F is the dihedral form corresponding to the Hecke grossencharacter ψ. We will make use of the following well-known large sieve inequality (see [DFI]). Lemma.. Let N. Then for any complex numbers {a n } N n=, we have a L(, sym n λ g (n) k(q + N) a n. g) n N n N g B 3. Proof of Theorem. Since f S k (Γ 0 (q)), we have f (z) = g B f, g g(z). Hence f 4 4 = f, f = g B Theorem. follows from Lemmas 3. and 3. below. f, g. Lemma 3.. We have g B (q) B f, g q. Proof. For g B (q), we have sup y k g(z) q /, which comes from the normalization with z H respect to the inner product (.). In view of (.), the same bound holds for g B. Thus for g B (q) B, we have f, g q / f y k dxdy y = q /. Using the fact B (q) B, we complete the proof. Lemma 3.. Under the same assumption as in Theorem., we have f, g ε q 3 δ+ε, for any ε > 0.

4 4 SHENG-CHI LIU Proof. By (.3) and (.4), we have f, g L(, g)l(, F g)l(, g χ q). q L(, sym f) L(, sym g) Applying Hölder s inequality with exponents 6,, and 3, respectively, we obtain f, g q L(, sym f) M /6 M / M /3 3, where M = L(, g) 6 L(, sym g), M = L(, F g) L(, sym g), M 3 = L(, g χ q) 3. L(, sym g) By our assumption, we have M q δ+ε L(, g) 4 L(, sym g). Using the approximate functional equation (see [IK, Theorem 5.3 and Proposition 5.4]), we have, for any ε > 0, L(, g) = a(n)λ g (n) + O(q 00 ), n / n q +ε L(, F g) = b(n)λ g (n) + O(q 00 ) n / n q +ε for certain a(n), b(n) q ε. Then the large sieve inequality [Lemma.] gives us M q 3/ δ+ε and M q +ε. For M 3, Conrey and Iwaniec [CI] showed M 3 ε q +ε. The Lemma follows from the estimates of M, M and M 3. Note that a different Hölder s inequality was used in [Lu], which produces the optimal bound for L 4 -norms in the spectral aspect. The optimal bound for L 4 -norms in the level aspect will give a strong subconvexity bound for L(, f f g), which seems out of reach with present technique. References [Bl] V. Blomer, On the 4-norm of an automorphic form, J. Eur. Math. Soc. 5 (03), [BKY] V. Blomer, R. Khan and M. Young, Distribution of mass of holomorphic cusp forms, Duke Math. J. 6 (03), [BuK] J. Buttcane and R. Khan, L 4 -norms of Hecke newforms of large level, Math. Ann., to appear. [CI] J.B. Conrey, and H. Iwaniec, The cubic moment of central values of automorphic L-functions, Ann. of Math. () 5 (000), no. 3, [DFI] W. Duke, J.B. Friedlander, and H. Iwaniec, Bounds for automorphic L-functions. II, Invent. Math. 5 (994), no., [HT] G. Harcos and N. Templier, On the sup-norm of Maass cusp forms of large level. III, Math. Ann. 356 (03), [Iw] H. Iwaniec, Topics in classical automorphic forms, Grad. Stud. Math., vol 7, Amer. Math. Soc., 997. [IK] H. Iwaniec and E. Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, 53. American Mathematical Society, Providence, RI, 004.

5 L 4 -NORMS OF THE HOLOMORPHIC DIHEDRAL FORMS OF LARGE LEVEL 5 [ILS] H. Iwaniec, W. Luo and P. Sarnak, Low lying zeros of families of L-functions. Publ. Math. Inst. Hautes Études Sci. (000) no. 9, [Lu] W. Luo, L 4 -norm of the dihedral Maass forms, Int. Math. Res. Notices (04) 04 (8): [Lu] W. Luo, Special L-values of Rankin-Selberg convolutions, Math. Ann. 34 (999), [LMY] S.-C. Liu, R. Masri and M. Young, Subconvexity and equidistribution of Heegner points in the level aspect, Compositio Mathematica 49 (03), No. 7, [Sa] P. Sarnak, Spectra of hyperbolic surfaces, Bull. Amer. Math. Soc. (N.S.) 40 (003), no. 4, [Wa] T. Watson, Rankin triple products and quantum chaos, preprint, available at arxiv: Department of Mathematics, Washington State University, Pullman, WA , U.S.A. address: scliu@math.wsu.edu