Vector valued Siegel's modular forms of degree two and the associated Andrianov L-functions

Arakawa, Tsuneo

doi:10.1007/bf01166080

Cited by 47 publications

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“…For these facts on the Siegel operator we refer to Arakawa's paper [6]. The Siegel operator is multiplicative:…”

Section: Conjecture 242 Let (L M) Be Regular Then the Endoscopic mentioning

confidence: 99%

“…In principle our database allows for the calculation of the traces of the Hecke operators T (p) with p ≤ 37 on the spaces S j,k for all values j, k. In the cases at hand these numbers tend to be 'smooth', i.e., they are highly composite numbers as we illustrate with the two 1-dimensional spaces S j,k for (j, k) = (8,8) and (12,6) (where the trace equals the eigenvalue of T (p)).…”

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confidence: 99%

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Siegel Modular Forms and Their Applications

Geer

2008

The 1-2-3 of Modular Forms

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“…For these facts on the Siegel operator we refer to Arakawa's paper [6]. The Siegel operator is multiplicative:…”

Section: Conjecture 242 Let (L M) Be Regular Then the Endoscopic mentioning

confidence: 99%

mentioning

confidence: 99%

Siegel Modular Forms and Their Applications

Geer

2008

The 1-2-3 of Modular Forms

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“…Moreover maps in the free resolutions can be given explicitly. In other words, M 0 Sym(10) (Γ (2) ) is generated by 13 modular forms of determinant weights 6,8,10,10,12,12,14,14,14,16,16,18,20 and they satisfy two fundamental relations. M 1 Sym(10) (Γ (2) ) is generated by 13 modular forms of determinant weights 9,11,13,15,15,15,17,17,17,19,19,21, 23 and they satisfy two fundamental relations.…”

Section: Introductionmentioning

confidence: 99%

“…where M k (Γ (2) ) = M k,0 (Γ (2) ). For j ∈ Z ≥0 and = 0, 1, we define the graded A ev module of vector valued Siegel modular forms as…”

Section: Introductionmentioning

confidence: 99%

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Structure theorems for vector valued Siegel modular forms of degree 2 and weight detk ⊗Sym(10)

Takemori

2016

Int. J. Math.

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Mellin Transforms of Vector Valued Theta Series Attached to Quaternion Algebras

Böcherer

Schulze-Pillot

1994

Mathematische Nachrichten

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IntroductionWe continue in this article our investigation of a lifting of automorphic forms that has been first studied by H. YOSHIDA IY1, Y2]. This lifting associates to a pair of automorphic forms cpl , cpz on the adelization D i of a definite quaternion algebra D over Q that are right invariant under a prescribed compact subgroup of D i a Siegel modular form of degree 2. The lifted form F = Y ( 2 ) ( y , , cp, ) is a linear combination of the Siegel theta series of the ideals in D with respect to maximal orders (or more general Eichler orders of square free level N), equipped with the reduced norm of the algebra as quadratic form. By EICHLEK'S work on the so called basis problem this lifting can also be viewed as associating a Siegel modular form of degree 2 to a pair of elliptic modular forms (newforms) f , , ,fz of Haupttypus for the group T,(N). Using Shimura's lifting one can moreover transfer the pair ( f l , f,) to a pair of modular forms of half integral weight g,, g, with respect t o T0(4N), and if one of these pairs consists of Hecke eigenforms all of them do and have the same eigenvalues. In the special case that f, and f z are of weight 2 (equivalently: vl and cpz are right invariant under DG = ( D 0 IR)") we showed in [B-S 31 that the Mellin transform (or Dirichlet series of Koecher and MaaB) of Y(2)(cpl, q z ) is essentially proportional to the Rankin convolution of the Mellin transforms of g,, g,. Using a result of ARAKAWA on the poles and residues of this Dirichlet series we could (generalizing a result of GROSS [Gr]) describe the linear relations between the theta series in certain genera of ternary quadratic forms and give a necessary and sufficient criterion for a cusp form g of weight 3/2 to be a linear combination of these theta series.The purpose of the present article is to generalize these results to arbitrary weights of ,fl, f z (or D2-representation types of cpl, qZ). By EICHLER'S results one sees that this leads one to work with Siegel theta series with spherical harmonics which, by work of KASHIWARA/~ERGNE [K-V], are vector valued unless one of , f l , j , has weight 2 (this latter case was the situation in which YOSHIDA had defined his lifting). In joint work with M. FURUSAWA we will show in a separate article (using our relation for the Mellin transform mentioned above) that Y (21(ql, cp, ) can be further lifted to an automorphic form on the metaplectic cover CST,(A) of GSp2(A) whose Whittaker coefficients are related to the products of the Fourier coefficients of g,, g,. This will confirm a conjecture of BUMP,

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Vector valued Siegel's modular forms of degree two and the associated Andrianov L-functions

Cited by 47 publications

References 5 publications

Siegel Modular Forms and Their Applications

Siegel Modular Forms and Their Applications

Structure theorems for vector valued Siegel modular forms of degree 2 and weight detk ⊗Sym(10)

Mellin Transforms of Vector Valued Theta Series Attached to Quaternion Algebras

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