A. Atkin and J. Lehner. Hecke operators on \Gamma 0 (m). Math. Ann. , (185):134--160, 1970.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....d une courbe elliptique quelquonque E=Q est le produit des signes locaux w p , o u p est un diviseur premier du conducteur et de w1 = Gamma1. Ces signes d ependent du type de r eduction. Si la courbe E a une r eduction multiplicative en p, alors w p = Gamma i Gammac 6 p j (voir [1]) o u c 6 est le covariant habituel de la courbe. En particulier, si E 1 admet une r eduction lisse en p = 2, on a w 2 = 1, et si E 1 admet une r eduction multiplicative en p = 2, on a u 0 = 2 et c 6 = A 2 B 2 ) A 2 Gamma 2B 2 ) 2A 2 Gamma B 2 ) 2 j B 2 (mod 2) et donc w 2 = Gamma Gamma ....

A.O.L. Atkin and J. Lehner, Hecke operators on \Gamma 0 (m), Math. Ann. 185 (1970), 134--160.

....and e Theta therefore not only preserve Laplace eigenvalues, but also Hecke eigenvalues. 7. Newforms. In this section, we will investigate to which extent the theta lifts of waveforms in L 2 0 (XO ) are so called newforms or not. First we recall the oldform newform formalism as developed in [1] for holomorphic modular forms. However, the concepts and principal results carry over to Maa# cusp forms, see for example [9, #8.5] Let a; m 2 N, m d, be such that amjd, and take some h 2 Cm . The inclusion Gamma 0 (d) ae Gamma 0 (m) implies that Cm ae C d so that h 2 C d , but also h (a) ....

....j ( k ) t k (m) Theta j ( k ) j = 1; 2 : Therefore, Theta 1 ( k ) and Theta 2 ( k ) have the same eigenvalues for all m with (m; d(O 2 ) 1. By the non holomorphic analogue of [1, Th.5] this implies that Theta 1 ( k ) and Theta 2 ( k ) are in the same class in the sense of [1]. Since the rst Fourier coeOEcients c 1 = 4 k (z 0 ) 6= 0 agree, this implies that Theta 1 ( k ) Theta 2 ( k ) The result then follows by the linearity of Theta 1 and Theta 2 . In the case of Proposition 7.2, we now denote the theta lift of the intersection L 2 0 (XO1 ) L 2 0 (XO2 ) ....

A. O. L. Atkin and J. Lehner, Hecke operators on \Gamma 0 (m), Math. Ann., 185 (1970), pp. 134160.

....aspects of the geometry of X 0 (N ) The complex analytic (Riemann surface) theory gives links to the theory of modular forms, while the fact that X 0 (N ) is an algebraic curve is implicit in our quest for good projective models. 2. 2 Involutions We introduce the Atkin Lehner involutions (see [1]) In this section we will write these as matrices in SL 2 (R) though in the applications we usually write them as elements of GL 2 (Z) and therefore the normalisation is implicit. For each prime ljN , let ff be such that l ff kN , and choose a; b; c; d 2 Zsuch that l ff ad Gamma (N=l ....

....many different choices of n which would give the same Wn , for example if N = 2 3 3 2 5 then W 6 = W 12 = W 18 = W 24 = W 36 = W 72 . For theoretical purposes one may assume that (n; N=n) 1 without any loss of generality. The crucial property of the matrices Wn is the following. Lemma 1 ([1] Lemma 8) For any two choices Wn and W 0 n we have Wn Gamma 0 (N )W 0 n = Gamma 0 (N ) 2.3) Proof. That Wn Gamma 0 (N )W 0 n Gamma 0 (N ) is a simple calculation. The equality follows from the fact that, for any Wn , the matrix W Gamma1 n is also of the same form. Thus, for all ....

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A. O. L. Atkin, J. Lehner, Hecke Operators on \Gamma 0 (N ), Math. Ann., 185, (1970) p. 134\Gamma160

....Siegel theta functions as kernels. As a consequence, theta lifts preserve eigenvalues of the hyperbolic Laplacian. In [2] we also showed that in the case of so called Eichler orders the eigenvalues of Hecke operators also remain unchanged. Furthermore in the language of the Atkin Lehner formalism [1], theta lifts of a Hecke basis for L 2 (O 1 nH) were found to be newforms of a level dividing d. Counting theta lifts and newforms suggested that in the case of maximal orders a Hecke basis for L 2 (O 1 nH) is lifted to a Hecke basis for the newspace of level d. In this paper, we continue ....

A. O. L. Atkin and J. Lehner, Hecke operators on \Gamma 0 (m), Math. Ann., 185 (1970), pp. 134160.

.... if C is odd the shadow of the lattice is obtained by lifting the shadow of the code) Since all lattices arising in this way share the common sublattice ( p 2A 2 ) n , they are rationally equivalent to (C (3) N , where C (3) Z Phi p 3Zarises from the code C with generator matrix [1]. Thus these lattices all satisfy the hypothesis of Theorem 2. In particular, the hexacode (with n = 6, d = 4) 15, p. 82] gives rise to the Coxeter Todd lattice K 12 . There are two related additive self dual codes, the shorter (n = 5; d = 3) and odd (n = 6; d = 3) hexacodes [9] 20] 42] The ....

.... b) cmz d) det Pi(d) Gamma1=2 c;d ( i p cmz d j dim Theta Pi(d) mz) m Gamma(dim ) 4 c;d ( i p cmz d j dim Theta p m Pi(m) z) c;d ( q m 1=2 cz m Gamma1=2 d dim Theta (z) The matrix Wm in (14) is called an Atkin Lehner involution [1] of order m. The next result combines known properties of these involutions with a slight generalization of a result of Nebe [33] on modularities. We omit the proof. Theorem 6. If Wm 1 and Wm 2 are Atkin Lehner involutions then Wm 1 Wm 2 is an Atkin Lehner involution of level m 1 m 2 =gcd(m 1 ; m ....

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A. O. L. Atkin and J. Lehner, Hecke operators on \Gamma 0 (m), Math. Ann. 185 (1970), 134-- 160.

....of forms with rational Fourier coefficients. We are interested here in rational newforms f : that is, forms f which have rational Fourier coefficients a(n; f ) are simultaneous eigenforms for all the Hecke operators, and which are also newforms in the sense of Atkin and Lehner (see [1]) We briefly recall the definition. For each proper divisor M of N and each g 2 S 2 (M ) the forms g(Dz) for divisors D of N=M are in S 2 (N ) The subspace S old 2 (N) of S 2 (N) spanned by all such forms is called the space of oldforms. There is also an inner product on S 2 (N ) called the ....

....N , and has L series L(E f ; s) P a(n; f)n Gammas where f = P a(n; f) exp(2inz) See [20] 19] 3] 5.2. Fourier coefficients and L series. The Fourier coefficients a(n; f) of a newform f(z) P a(n; f) exp(2inz) are obtained from the Hecke eigenvalues of f as follows (see [1]) Firstly, for a newform f we always have a(1; f) 6= 0, and we normalise so that a(1; f) 1. Then: If p is a prime not dividing N , and f jT p = a p f , then a(p; f) a p . By standard estimates, the integer a p satisfies ja p j 2 p p. If q is a prime dividing N , and f jW q = q f with ....

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A. O. L. Atkin and J. Lehner, Hecke operators on \Gamma 0 (m), Math. Ann. 185 (1970), 134--160.

....and (mn; q) 1 we can write (2.6) as (2.9) X f2F f f (m) f (n) ffi(m; n) p mn X cj0(modq) c Gamma2 S(m; n; c)J(2 p mn=c) where (2.10) f = 4) 1 Gammak Gamma(k Gamma 1)ja f (1)j 2 ; 2. 11) J(x) 4i k x Gamma1 J k Gamma1 (2x) According to the Atkin Lehner theory [AL] the sum (2.9) can be arranged into a sum over all primitive forms of level dividing q , but, of course, with slightly different coefficients. Precisely, a primitive form f appears with coefficient (2.12) f = 12 (k Gamma 1)q X ( q) 1 f ( 2 ) Gamma1 Gamma1 AE (kq) ....

A. Atkin and J.Lehner, Hecke operators on \Gamma 0 (m), Math. Ann. 185 (1970), 134--160.

....to the Petersson inner product. Hence S 2 (N) decomposes into a direct sum of simultaneous eigenspaces and all eigenvalues are real. For elements in S new 2 (N) we get more: Let E be a simultaneous eigenspace of S new 2 (N) under the Hecke algebra. Then dim(E) 1 (Multiplicity one theorem) [AL70]. Theorem 2.12 Let R be an commutative ring. The pairing TN Theta S 2 (N) R) Gamma R; T n ; f) 7 a 1 (T n (f) first Fourier coeff. of T n (f) has no left and right radical. Since we know that S 2 (N) C ) S 2 (N) Z) Omega C we get Corollary 2.13 Hom( Omega 1 (X 0 (N) C ) is a ....

....operator S 2 (N) Gamma S 2 (N ) f(z) 7 f(W r ) z) 8) is an involution on the space of cuspforms and Y pjN;p prime W r = 0 1 GammaN 0 ) WN is the Fricke involution. 2. One can define the Hecke and Atkin Lehner operators for cuspforms with nebentype and higher weights (see [AL70]) 3. Without using the modular interpretation, but the theory of parabolic cohomology Shimura [Shi71] proved the existence of a basis of modular (cusp) forms with rational Fourier coefficients. For S 2 (N) this follows from the existence of a non degenerate pairing H 1 (X 0 (N) Z) Theta S 2 ....

A. Atkin and J. Lehner. Hecke operators on \Gamma 0 (m). Math. Ann. , (185):134--160, 1970.

....and e Theta therefore not only preserve Laplace eigenvalues, but also Hecke eigenvalues. 7. Newforms. In this section, we will investigate to which extent the theta lifts of waveforms in L 2 0 (XO ) are so called newforms or not. First we recall the oldform newform formalism as developed in [1] for holomorphic modular forms. However, the concepts and principal results carry over to Maa cusp forms, see for example [9, x8.5] 30 JENS BOLTE AND STEFAN JOHANSSON Let a; m 2 N, m d, be such that amjd, and take some h 2 Cm . The inclusion Gamma 0 (d) ae Gamma 0 (m) implies that Cm ae C ....

....j ( k ) t k (m) Theta j ( k ) j = 1; 2 : Therefore, Theta 1 ( k ) and Theta 2 ( k ) have the same eigenvalues for all m with (m; d(O 2 ) 1. By the non holomorphic analogue of [1, Th.5] this implies that Theta 1 ( k ) and Theta 2 ( k ) are in the same class in the sense of [1]. Since the first Fourier coefficients c 1 = 4 k (z 0 ) 6= 0 agree, this implies that 34 JENS BOLTE AND STEFAN JOHANSSON Theta 1 ( k ) Theta 2 ( k ) The result then follows by the linearity of Theta 1 and Theta 2 . In the case of Proposition 7.2, we now denote the theta lift of the ....

A. O. L. Atkin and J. Lehner, Hecke operators on \Gamma 0 (m), Math. Ann., 185 (1970), pp. 134--160.

....(mn; q) 1 we can write (2.6) as (2.9) X f2F f f (m) f (n) ffi(m; n) p mn X cj0(mod q) c Gamma2 S(m; n; c)J(2 p mn=c) where (2.10) f = 4 ) 1 Gammak Gamma(k Gamma 1)ja f (1)j 2 ; 2. 11) J(x) 4 i k x Gamma1 J k Gamma1 (2 x) According to the Atkin Lehner theory [AL] the sum (2.9) runs over all primitive forms of level dividing q, because f = 0 if f is not such a form. If f is primitive then (2.11) f AE (kq) Gamma1 Gamma for any 0, the implied constant depending only on . Remarks. The lower bound (2.11) was established in [I4] Actually if f is ....

A. Atkin and J.Lehner, Hecke operators on \Gamma 0 (m), Math. Ann. 185 (1970), 134-160.

....the same is true for f 0 . But f 0 is by assumption an eigenvector of T r with eigenvalue r . Thus r = a(r) and in particular, a(r) 6= Sigma1. This is a contradiction, because Gammaa(r) is the eigenvalue of an involution on S 2 (N 0 r) namely the Atkin Lehner involution at the prime r ([1], p. 147, Thm. 3(iii) AN ELEMENTARY CASE OF SERRE S CONJECTURE 303 Put = oe Gamma1 l , where oe l 2 Gal(Q =Q) is some fixed Frobenius element at l. Case 1. 0. Let h = g 2 . Then h 2 S 2 (Nr) and h j P b(n) 2 q 2n (mod l) Consequently h j P b(n)q 2n (mod l) But h ....

A.O.L. Atkin and J. Lehner, Hecke operators on \Gamma 0(m), Math. Ann., 185 (1970), 134-160.

....Siegel theta functions as kernels. As a consequence, theta lifts preserve eigenvalues of the hyperbolic Laplacian. In [2] we also showed that in the case of so called Eichler orders the eigenvalues of Hecke operators also remain unchanged. Furthermore in the language of the Atkin Lehner formalism [1], theta lifts of a Hecke basis for L 2 (O 1 nH) were found to be newforms of a level dividing d. Counting theta lifts and newforms suggested that in the case of maximal orders a Hecke basis for L 2 (O 1 nH) is lifted to a Hecke basis for the newspace of level d. In this paper, we continue ....

A. O. L. Atkin and J. Lehner, Hecke operators on \Gamma 0 (m), Math. Ann., 185 (1970), pp. 134--160.

No context found.

A. Atkin and J. Lehner. Hecke operators on \Gamma 0 (m). Math. Ann. , (185):134--160, 1970.

No context found.

Atkin, A., Lehner, J.: Hecke Operators on \Gamma 0 (m). Math. Ann. 185 (1970), 134-160

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