On Automorphic Forms on GL 2 and Hecke Operators

Miyake, Toshitsune

doi:10.2307/1970741

Cited by 124 publications

(67 citation statements)

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“…Du fait que / est primitive, il existe une constante 'k ^ 0 telle que/(-l/Nz) = ^zf(z\ cf. [12] et [13]. Par transformation de Mellin, on en déduit…”

Section: N=0unclassified

Formes modulaires de poids $1$

Deligné¹,

Serre²

1974

Ann. Sci. École Norm. Sup.

361

240

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“…Du fait que / est primitive, il existe une constante 'k ^ 0 telle que/(-l/Nz) = ^zf(z\ cf. [12] et [13]. Par transformation de Mellin, on en déduit…”

Section: N=0unclassified

Formes modulaires de poids $1$

Deligné¹,

Serre²

1974

Ann. Sci. École Norm. Sup.

361

240

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“…Here we are only concerned with cusp forms of weight two for ro(n), where n is a nonzero ideal in the ring of integers R of an imaginary quadratic field K of class number one and discriminant D. The theory of these forms, Hecke operators, oldforms and newforms is entirely analogous to the classical 'Atkin-Lehner' theory over Q. For a complete discussion of the general case, see [16]; here we quote a summary of the facts we will need.…”

Section: Introductionmentioning

confidence: 99%

Periods of Cusp Forms and Elliptic Curves Over Imaginary Quadratic Fields

Cremona¹,

Whitley²

1994

Mathematics of Computation

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Abstract.In this paper we explore the arithmetic correspondence between, on the one hand, (isogeny classes of) elliptic curves E defined over an imaginary quadratic field K of class number one, and on the other hand, rational newforms F of weight two for the congruence subgroups r0(n), where n is an ideal in the ring of integers R of K . This continues work of the first author and forms part of the Ph.D. thesis of the second author. In each case we compute numerically the value of the L-series L(F, s) at s = 1 and compare with the value of L(E, 1 ) which is predicted by the Birch-Swinnerton-Dyer conjecture, finding agreement to several decimal places. In particular, we find that L(F, 1) = 0 whenever E(K) has a point of infinite order. Several examples are given in detail from the extensive tables computed by the authors.

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“…Proof. By virtue of results of Miyake [7] and Asai [1], W p gives the isomorphism between S k (Np, ψX) and S k (Np, ψχ …”

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confidence: 98%