On Modular Forms of Half Integral Weight

Shimura, Goro

doi:10.2307/1970831

Cited by 844 publications

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“…Both results follow from Theorem 3 on the nonvanishing of the Fourier coefficients in the case of half-integralweight cusp forms. The works of Shimura and Waldspurger [Sh2,W] show that the coefficients of a half-integral-weight cusp form g(z), which is an eigenform but not a theta function, interpolate many central critical values of the quadratic twists of the modular L-function associated to the Shimura correspondent of g(z). Although the Shimura correspondence is not surjective, it is shown in [OSk,Sect.…”

Section: Proofs Of Corollariesmentioning

confidence: 99%

The Chebotarev Density Theorem in Short Intervals and Some Questions of Serre

Balog

Ono

2001

Journal of Number Theory

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Section: Proofs Of Corollariesmentioning

confidence: 99%

The Chebotarev Density Theorem in Short Intervals and Some Questions of Serre

Balog

Ono

2001

Journal of Number Theory

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“…Around ten years ago, Shimura [12] discovered a relationship between modular forms of arbitrary even weight 2k and modular forms of halfintegral weight k + 1/2. This was studied subsequently by many other authors.…”

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confidence: 99%

Modular points, modular curves, modular surfaces and modular forms

Zagier

1985

Lecture Notes in Mathematics

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“…, and U l V p θ 3 are modular forms of weight 3/2 on Γ 0 (4lp) with character 4lp · (see [10]). To prove Theorem 1.1, we need the following lemmas.…”

Section: Proof Of Resultsmentioning

confidence: 99%

A note on basic Iwasawa λ-invariants of imaginary quadratic fields and congruence of modular forms

Byeon

1999

Acta Arith.

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A note on basic Iwasawa λ-invariants of imaginary quadratic fields and congruence of modular forms by Dongho Byeon (Seoul)1. Introduction and statement of results. For a number field k and a prime number l, we denote by h(k) the class number of k and by λ l (k) the Iwasawa λ-invariant of the cyclotomic Z l -extension of k, where Z l is the ring of l-adic integers.Let l be an odd prime number. Using the Kronecker class number relation for quadratic forms, Hartung [3] proved that there exist infinitely many imaginary quadratic fields k whose class numbers are not divisible by l. For the case l = 2, this is an immediate consequence of Gauss' genus theory. On the other hand, using the idea of Hartung and Eichler's trace formula combined with the l-adic Galois representation attached to the Jacobian variety J = J 0 (l) of the modular curve X = X 0 (l), Horie [4] proved that there exist infinitely many imaginary quadratic fields k such that l does not split in k and l does not divide h(k). Later Horie and Onishi [5] obtained more refined results. By a theorem of Iwasawa [6], these results imply that there exist infinitely many imaginary quadratic fields k with λ l (k) = 0. For the case l = 2, this is also an immediate consequence of Gauss' genus theory. For the case l = 3, by refining Davenport and Heilbronn's result [2], Nakagawa and Horie [8] gave a positive lower bound on the density of imaginary quadratic fields k and real quadratic fields k with λ l (k) = 0.

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On Modular Forms of Half Integral Weight

Cited by 844 publications

References 11 publications

The Chebotarev Density Theorem in Short Intervals and Some Questions of Serre

The Chebotarev Density Theorem in Short Intervals and Some Questions of Serre

Modular points, modular curves, modular surfaces and modular forms

A note on basic Iwasawa λ-invariants of imaginary quadratic fields and congruence of modular forms

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