Geodesic
Duality in Siegel's theorem on representation by a genus of quadratic forms, and the averaging operator
Sbornik. Mathematics, Tome 50 (1985) no. 1, pp. 1-10 
A. N. Andrianov. Duality in Siegel's theorem on representation by a genus of quadratic forms, and the averaging operator. Sbornik. Mathematics, Tome 50 (1985) no. 1, pp. 1-10. http://geodesic.mathdoc.fr/item/SM_1985_50_1_a0/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $S$ and $T$ be two integral positive definite quadratic forms in the same number of variables, and let $S_1,\dots,S_H$ and $T_1,\dots,T_h$ be complete systems of representatives of the different classes in the genus of the form $S$ and $~T$, respectively. The author proves, in particular, that $$ \bigg(\sum_{i=1}^He(S_i)^{-1}\bigg)^{-1}\sum_{i=1}^He(S_i)^{-1}r(S_i,T)=\bigg(\sum_{j=1}^he(T_j)^{-1}\bigg)^{-1}\sum_{j=1}^he(T_j)^{-1}r(S,T_j), $$ where $r(S',T')$ denotes the number of integral representations of the form $T'$ by the form $S'$, and $e(S') = r(S',S')$. Bibliography: 6 titles.

[1] Siegel C. L., “Über die analytische Theorie der quadratischen Formen”, Ann. Math., 36 (1935), 527–606 | DOI | MR | Zbl

[2] Andrianov A. N., Maloletkin G. N., “Povedenie teta-ryadov roda $n$ pri modulyarnykh podstanovkakh”, Izv. AN SSSR. Ser. matem., 39:2 (1975), 243–258 | MR | Zbl

[3] Freitag E., “Holomorphie Differentialformen zu Kongruenzgruppen der Siegelschen Modulgruppe”, Inventiones Math., 30 (1975), 181–196 | DOI | MR | Zbl

[4] Andrianov A. N., “Multiplikativnaya arifmetika zigelevykh modulyarnykh form”, UMN, 34 (1979), 67–135 | MR | Zbl

[5] Andrianov A. N., “Modular Descent and the Saito–Kurokawa Conjecture”, Inventiones Math., 53 (1979), 267–280 | DOI | MR | Zbl

[6] Kurokawa N., “Examples of Eigenvalues of Hecke Operators on Siegel Cusp Forms of Degree Two”, Inventiones Math., 49 (1978), 149–165 | DOI | MR | Zbl