Distribution of Fourier coefficients of primitive forms

Transcription

1 Distribution of Fourier coefficients of primitive forms Jie WU Institut Élie Cartan Nancy CNRS et Nancy-Université, France Clermont-Ferrand, le 25 Juin 2008

2 2 Presented work [1] E. Kowalski, O. Robert & J. Wu, Small gaps in coefficients of L-functions and B-free numbers in short intervals, Revista Mat. Iberoamericana 23 (2007), No. 1, [2] Y.-K. & J. Wu, The number of Hecke eigenvalues of same signs, Preprint 2008.

3 3 1. Motivation Ramanujan s function τ(n) For =m z > 0, define (z) := e 2πiz Y (1 e 2πinz ) 24 =: X τ(n)e 2πinz. n=1 n=1 Ramanujan s conjecture τ(n) 6 d(n)n 11/2 (n > 1) where d(n) is the divisor function. This conjecture has been proved by Deligne (1974).

4 4 Lehmer s conjecture τ(n) 6= 0 (n > 1) This is open! Two partial results on Lehmer s conjecture Lehmer (1959) : τ(n) 6= 0 for n Serre (1981) : {n 6 x : τ(n) 6= 0} αx (x, α > 0). Questions (i) α = 1? (Nonvanishing of τ(n)) 1 X (ii) lim 1 = 1? (Signs of τ(n)) x x 2 n6x, τ(n) 0

5 5 2. Modular forms Notation k := integer > 2, N := squarefree integer, log r := the r-fold iterated logarithm, χ 0 := trivial Dirichlet character (mod N), χ := Dirichlet character (mod N) such that χ( 1) = ( 1) k. Cusp forms Denote by S k (N, χ) the set of all cusp forms of weight k and of level N with nebentypus χ.

6 6 Decomposition of S k (N, χ) S k (N, χ) equipped with the Petersson inner product h, i is a finite dimensional Hilbert space and we have S k (N, χ) = S k(n, [ χ) S ] k (N, χ), where S k [(N, χ) is the linear subspace of S k(n, χ) spanned by all forms of type f(dz), where d N and f S k (N 0, χ 0 ) with N 0 < N and dn 0 N. Here χ 0 (mod N 0 ) is the character which induces χ. S ] k (N, χ) is the linear subspace of S k (N, χ) orthogonal to S k [ (N, χ) with respect to h, i. Newforms Denote by Sk (N, χ) the set of all newforms in S] k (N, χ). It constitutes a base of S ] k (N, χ). We can prove (z) S 12 (1) := S 12 (1, χ 0 ).

7 7 Fourier development of f Sk (N, χ) at X f(z) = λ f (n)n (k 1)/2 e 2πinz (=m z > 0), n=1 where λ f (n) has the following properties: (i) λ f (1) = 1, (ii) T n f = λ f (n)n (k 1)/2 f for any n > 1, (iii) for all integers m > 1 and n > 1, λ f (m)λ f (n) = X d (m,n) where T n is the nth Hecke s operator. χ(d)λ f µ mn d 2 (iv) Further if f S k (N, χ 0), then λ f (n) R for n > 1.,

8 Deligne s inequality 8 Deligne (1974): If f S k (N, χ), then p, α f (p), β f (p) such that ( αf (p) = ±p 1/2, β f (p) = 0 if p N and α f (p) = 1, α f (p)β f (p) = χ(p) if p - N λ f (p ν ) = α f (p) ν + α f (p) ν 1 β f (p) + + β f (p) ν ( ν > 0). In particular More generally λ f (p ν ) 6 ν + 1 ( p and ν > 0). λ f (n) 6 d(n) (n > 1) where d(n) is divisor function.

9 9 3. Nonvanishing of λ f (n) Notation f S k (N, χ) : P f := {p : λ f (p) = 0}. Forms of CM type Ribet (1977): If f Sk cm(n, χ) (the subspace of S k(n, χ) spanned by all CM forms), then Ø ØP f [1, x] Ø = x µ x 2 log x + O (log x) 2. By using Landau s method, we can prove α > 0 such that for x, X 1 αx X, 1 x. log x n6x, λ f (n)6=0 n6x, λ f (n)=0

10 Lang-Trotter s conjecture If f S k (N, χ)rscm k 10 (N, χ), then Ø ØP f [1, x] Ø Ø ø f Forms of non-cm type x/ log x if k = 2, log 2 x if k = 3, 1 if k > 4. Serre (1981): If f Sk (N, χ)rscm k (N, χ), then δ < 1 2, Ø (1) ØP f [1, x] Ø x ø f,δ (log x) 1+δ. From this, we deduce that there are constants C > c > 0 such that X (2) cx Cx for x > x 0 (f). n6x, λ f (n)6=0

11 11 Serre s function i f (n) For f S k (N, χ)rsk cm (N, χ), we define i f (n) := max{i : λ f (n + j) = 0 (0 < j 6 i)}. The inequalities (2) imply that (3) i f (n) ø f n (n > 1), since 0 = X n<m6n+i f (n) λ f (m)6=0 1 > c n + i f (n) Cn = ci f (n) (C c)n. Serre s question (1981) Find constant θ < 1 such that (4) i f (n) ø f,θ n θ ( n > 1).

12 12 Remark 1. Balog & Ono (2001) remarked that Serre s question has been resolved before proposing it! There is two methods! Rankin-Selberg convolution (The first method) Rankin (1939) and Selberg (1940): For f Sk (N, χ), we have X λ f (n) 2 = A f x + O f (x 3/5 ) (A f > 0). n6x This implies trivially (4) with θ = 3 5 X, X since 0 = λ f (m) 2 = λ f (m) 2 X n<m6n+i f (n) m6n+i f (n) m6n = A f n + if (n) A f n + O (n + i f (n)) 3/5 + n 3/5 = A f i f (n) + O(n 3/5 ) (via (3)). λ f (m) 2

13 13 Erdős B-free numbers (The second method) Let B = {b k N : 1 < b 1 < < b k < } such that (5) X i=1 1 b i < and (b i, b j ) = 1 (i 6= j). An integer n > 1 is called B-free if b - n for any b B. The set of all B-free numbers is denoted by A = A(B). Taking B = {p 2 : p prime} =: P 2, then A(P 2 ) is the set of all squarefree integers.

14 14 (6) Erdős (1966): θ < 1 such that for x > x 0 (B, θ), Ø x < n 6 x + x θ : n is B free Ø Ø B,θ x θ. The records on θ: θ = 1 2 θ = 9 20 θ = 5 12 θ = θ = θ = θ = ε = ε (Szemerédi, 1973), = ε (Bantle & Grupp, 1986), = ε (Wu, 1990), = ε (Wu, 1993), = ε (Wu, 1994) and (Zhai, 2000), = ε (Sargos & Wu, 2000), (Conjecture), where ε is an arbitrarily small positive number.

15 15 Application of B-free numbers Take P f := {p : λ f (p) = 0}, B f := P f {p 2 : p / P f }. The multiplicativity of λ f (m) implies m is B f free λ f (m) 6= 0 In fact, if m is B f -free, we have (i) if p m, then λ f (p) 6= 0, (ii) m = p 1 p j with λ f (p i ) 6= 0 for 1 6 i 6 j and p 1 < < p j, (iii) λ f (m) = λ f (p 1 ) λ f (p j ) 6= 0.

16 16 In view of Serre s (1), B f satisfies (5). Thus Erdős (6) implies : θ < 1 such that for x > x 0 (f, θ) X X 1 > 1 f,θ x θ, x<m6x+x θ λ f (m)6=0 from which we deduce (taking x = n) for n > x 0 (f, θ). Thus x<m6x+x θ m is B f free i f (n) < n θ i f (n) ø f,θ n θ (n > 1). In particular, the value θ = ε is admissible.

17 17 Theorem 1 (Kowalski, Robert & Wu, 2007). Suppose that Ø (7) ØP f [1, x] Ø ø f x ρ /(log x) Θ ρ (x > 2) for any f Sk (N, χ)rscm k (N, χ), where ρ [0, 1] and Θ ρ R such that Θ 1 > 1. Then for any f S k (N, χ)rsk cm (N, χ) we have i f (n) ø f,ε n θ(ρ)+ε, where θ(ρ) = 1/4 if 0 6 ρ 6 1/3, 10ρ/(19ρ + 7) if 1/3 < ρ 6 9/17, 3ρ/(4ρ + 3) if 9/17 < ρ 6 15/28, 5/16 if 15/28 < ρ 6 5/8, (22ρ/(24ρ + 29) if 5/8 < ρ 6 9/10, 7ρ/(9ρ + 8) if 9/10 < ρ 6 1.

18 18 Corollary 2 (KRW, 2007). For any form f S k (N, χ)rsk cm (N, χ), we have i f (n) ø f,ε n 7/17+ε for all n > 1. [(1) (7) with ρ = 1 and Θ 1 = 1 + δ] θ(1) = improves very slightly Balog & Ono s Corollary 3 (KRW, 2007). For any f S k (N, χ)rsk cm (N, χ), Lang- Trotter s conjecture implies i f (n) ø f,ε n 10/33+ε for all n > 1. [Lang-Trotter s conjecture implies (7) with ρ = 1 2 and Θ 1/2 = 1] θ( 1 2 ) = improves considerably Alkan s Corollary 4 (KRW, 2007). Let E/Q be an elliptic curve without CM and f the associated newform. Then i f (n) ø E,ε n 33/94+ε for all n > 1. [For elliptic curve, Elkies proved (7) with ρ = 3 4 and Θ 3/4 = 0] θ( 3 4 ) = improves considerably Alkan s

19 19 4. Further improvements Expected result θ(0) = 0 in Theorem 1 Defect of the constraint squarefree θ(0) = 1 4 6= 0 in Theorem 1 Squarefree integers in short intervals Filaseta & Trifonov (1992) : c > 0 such that Ø x < n 6 x + cx 1/5 log x : n is squarefree Ø Ø x 1/5 log x. Remark 2. With the multiplicative constraint squarefree, we have few chance for obtaining θ(0) = 0.

20 20 Treatment without the constraint squarefree Lemma 1 (KRW, 2007). Let f S k (N, χ). Then ν f such that for any p - N: either λ f (p ν ) 6= 0 (ν > 0) or ν 6 ν f such that λ f (p ν ) = 0. Lemma 2 (KRW, 2007). For f S k (N, χ)rscm k (N, χ), we define P f,ν := p : λ f (p ν ) = 0 and λ f (p j ) 6= 0 (0 6 j < ν), B f := P f,1 P f,2 P f,νf. Then for any δ < 1 2, we have Ø ØB f [1, x] Ø Ø ø f,δ x (log x) 1+δ. Remark 3. This contains Serre s (1) and leads us to propose a generalized Lang-Trotter s conjecture.

21 21 Generalized Lang-Trotter s conjecture Conjecture 1. If f Sk (N, χ)rscm k (N, χ), then Ø ØB f [1, x] Ø x/ log x if k = 2, ø f log 2 x if k = 3, 1 if k > 4. Lemma 3 (KRW, 2007). If f S k (N) such that λ f (n) Z for any n > 1 (for example for elliptic curves), then Conjecture 1 is equivalent to Lang-Trotter s conjecture. Relation between B f -free and λ f (n) 6= 0 n is B f -free λ f (n) 6= 0

22 22 Theorem 5 (KRW, 2007). Suppose that Ø ØB f [1, x] Ø x ρ ø f (log x) Θ ρ (x > 2) for any f Sk (N, χ)rscm k (N, χ), where ρ [0, 1] and Θ ρ R such that Θ 1 > 1. Then for any f S k (N, χ)rsk cm (N, χ) and ε > 0, we have i f (n) ø f,ε n θ(ρ)+ε (n > 1), where θ(ρ) = ρ/(1 + ρ). In particular θ(0) = 0. Corollary 6 (KRW, 2007). Let k > 3. Suppose that Conjecture 1 holds for all f Sk (N, χ)rscm k (N, χ). Then for any ε > 0 and all f S k (N, χ)rsk cm (N, χ), we have i f (n) ø f,ε n ε (n > 1).

23 23 5. Katz conjecture and nonvanishing problem Question Is there a cusp form f such that λ f (n) 6= 0 for any n > 1? Katz conjecture (1972) Let S(1, 1; p) be the sum of Kloosterman. Then L(s, Kl) := Y 1 S(1, 1; p)p s + p 1 2s 1 X =: λ Kl (n)n s p is L-function of a non-holomorphic cusp form of weight 2 over SL 2 (Z). n>1 Theorem 7 (KRW, 2007). We have λ Kl (n) 6= 0 for any n > 1. Remark 4. This gives an affirmative answer conditionally. But very probably Katz conjecture should not hold.

24 24 6. Sign changes of Hecke s eigenvalues Sato-Tate s conjecture For any 2 6 α 6 β 6 2 and any f Sk (N) := S k (N, χ 0), we have {p 6 x : α 6 λ f (p) 6 β} x log x Z β α 4 t 2 2π dt (x ). Here 4 t 2 /(2π) dt is called Sate-Tate s measure. Hecke s eigenvalues of same signs For f Sk (N), define N ± f (x) := X 1. n6x, (n,n)=1 λ f (n) 0

25 It is natural to conjecture lim x 25 N ± f (x) Kohnen, Lau & Shparlinski (2007): N ± f (x) x f (log x) 17 x = 1 2. (x > x 0 (f)). Wu (2008): The exponent 17 can be reduced to 1 1/ and 2 16/(3π) if we assume Sato-Tate s conjecture. Theorem 8 (Lau & Wu, 2008). For any f Sk (N), we have for all x > x 0 (f). N ± f (x) f x

26 26 Sign changes of Hecke s eigenvalues Kohnen, Lau & Shparlinski (2007): there are absolute constants η < 1 and A > 0 such that for any f Sk (N) we have N ± f (x + xη ) N ± f (x) > 0 (x > (kn)a ). Theorem 9 (Lau & Wu, 2008). Let f Sk (N). There is an absolute constant c > 0 such that for any ε > 0 and x > cn 1+ε x 0 (k), we have N ± f (x + cn 1/2+ε x 1/2 ) N ± f (x) ε x 1/4 ε, where x 0 (k) is a suitably large constant depending on k and the implied constant in ε depends only on ε. Remark 5. λ f (n) has a sign-change in interval x, x+cn 1/2+ε x 1/2 for all sufficiently large x.

27 27 7. Ideas of the proof of Theorem 1 Recall B f := P f {p 2 : p / P f }. The existence of B f -free numbers in short intervals is a problem of sieve. (a) System of weights for detecting B f -free numbers; (b) Exponential sum for controlling the error terms in the sieve. Alkan s condition on θ δ 1 (θ) + θ + θ/ρ > 1, where δ 1 (θ) is increasing and given by exponential sums of type II. Our condition on θ δ 2 (θ) + 2θ/ρ > 1. where δ 2 (θ) is increasing and given by exponential sums of type I.

28 28 Two improvements (i) By exploiting {p 2 : p / P f }, we improve θ + θ/ρ into 2θ/ρ. (ii) Our system of weights allows us to bring back to estimate bilinear forms of type I : X X ψ n r mn (x, x θ ), M6m<2M N6n<2N instead of type II (as in the work of Alkan) X X φ m ψ n r mn (x, x θ ), M6m<2M N6n<2N where φ m 6 1, ψ n 6 1 and r d (x, y) := {x < n 6 x + y : d n} y/d. Thus we have δ 2 (θ) > δ 1 (θ).

29 29 Multiple exponential sums : By the Fourier analyse, the estimate of bilinear forms can be transformed into estimate of multiple exponential sums X H6h<2H X M6m<2M X N6n<2N φ h ψ n e µx hα m β n γ H α M β N γ where e(t) := e 2πit, φ h 6 1, ψ n 6 1, α, β, γ R. According the size of ρ, we use the methods of Fouvry Iwaniec (with Robert Sargos refinement) and of Heath-Brown to estimate this sum.,