Geodesic
Euler expansions of theta-transforms of Siegel modular forms of degree $n$
Sbornik. Mathematics, Tome 34 (1978) no. 3, pp. 259-300 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $F(Z)$ be a Siegel modular form of degree $n$, weight $k$ and character $\chi$ for the congruence subgroup $\Gamma_0^n(q)$ of the Siegel modular group $\Gamma^n$. Suppose that $F$ is an eigenfunction for all Hecke operators with index relatively prime to $q$. It is proven that for each fixed, symmetric, semi-integral, positive definite matrix $N$ of order $n$ and for each Dirichlet character $\psi$, equal to zero on all prime divisors of $q\operatorname{det}2N$, the Dirichlet series $$ \sum_{M\in\operatorname{SL}_n(\mathbf Z)\setminus M_n^+(\mathbf Z)}\frac{\psi(\operatorname{det}M)f(MN^tM)}{(\operatorname{det}M)^s}, $$ where $f(N')$ are the Fourier coefficients of $F$ and $M_n^+(\mathbf Z)$ is the set of integral matrices of order $n$ with positive determinant, has an expansion as an Euler product which can be explicitly calculated. Bibliography: 13 titles. 
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     title = {Euler expansions of theta-transforms of {Siegel} modular forms of degree~$n$},
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A. N. Andrianov. Euler expansions of theta-transforms of Siegel modular forms of degree $n$. Sbornik. Mathematics, Tome 34 (1978) no. 3, pp. 259-300. http://geodesic.mathdoc.fr/item/SM_1978_34_3_a0/

[1] G. Shimura, “On the holomorphy of certain Dirichlet series”, Proc. London Math. Soc., 31 (1975), 79–98 | DOI | MR | Zbl

[2] A. H. Andrianov, “Simmetricheskie kvadraty dzeta-funktsii zigelevykh modulyarnykh form roda 2”, Trudy Matem. in-ta im. V. A. Steklova, CXLII, 1976, 22–45 | MR

[3] I. Satake, “Theory of spherical functions on reductive algebraic group over $p$-adic fields”, Publ. Math. IHES, 1963, no. 18, 1–69 | MR

[4] G. Shimura, “Arithmetic of alternating forms and quaternion hermitian forms”, J. Math. Soc. Japan, 1963, no. 15, 33–65 | MR | Zbl

[5] A. H. Andrianov, “O razlozhenii mnogochlenov Gekke dlya simplekticheskoi gruppy roda $n$”, Matem. sb., 104 (146), 390–427 | MR | Zbl

[6] A. N. Andrianov, “Sfericheskie funktsii dlya $GL_n$ nad lokalnymi polyami i summirovanie ryadov Gekke”, Matem. sb., 83(125):3(11) (1970), 429–451 | MR | Zbl

[7] Seminaire H. Cartan, 1957/58, Fonctions automorphes, Secrétariat mathématique, Paris, 1958

[8] H. Maass, “Siegel's Modular Forms and Dirichlet Series”, Lecture notes in Math., 216, Springer-Verlag Lecture notes in Math., Berlin, 1971 | MR | Zbl

[9] T. Tamagawa, “On the $\zeta$-functions of a division algebra”, Ann. Math., 77 (1963), 387–405 | DOI | MR | Zbl

[10] E. Artin, Geometricheskaya algebra, izd-vo “Nauka”, Moskva, 1969 | MR

[11] L. E. Dickson, Linear Groups with an Exposition of the Galois Fields Theory, New York, 1958 | MR

[12] L. Carlitz, “Gauss sums over finite fields of order $2^n$”, Acta Arithm., 15 (1969), 247–265 | MR | Zbl

[13] S. A. Evdokimov, “Eilerovy proizvedeniya dlya kongruents-podgrupp zigelevoi gruppy roda $2$”, Matem. sb., 99(141) (1976), 483–513 | MR | Zbl