Smirnov, A. L. Eisenstein’s program and modular forms. (English. Russian original) Zbl 1451.11032 J. Math. Sci., New York 249, No. 1, 104-111 (2020); translation from Zap. Nauchn. Semin. POMI 479, 160-170 (2019). Summary: The paper gives an identity for a sum of theta-series related to an imaginary quadratic field. This sum is expressed in terms of a certain Eisenstein series. The identity obtained is used in a new proof of the formula for the number of integral points in a system of ellipses. Such formulas are of interest because of their relations to the arithmetic Riemann-Roch theorem. MSC: 11F27 Theta series; Weil representation; theta correspondences 11F11 Holomorphic modular forms of integral weight 11P21 Lattice points in specified regions 11E41 Class numbers of quadratic and Hermitian forms Keywords:arithmetic Riemann-Roch theorem; theta-series PDFBibTeX XMLCite \textit{A. L. Smirnov}, J. Math. Sci., New York 249, No. 1, 104--111 (2020; Zbl 1451.11032); translation from Zap. Nauchn. Semin. POMI 479, 160--170 (2019) Full Text: DOI References: [1] Eisenstein, G., Geometrischer Beweis des Fundamentaltheorems für die quadratischen Reste, J. reine angew. Math., 28, 246-248 (1844) · ERAM 028.0828cj [2] Smirnov, AL, On exact formulas for the number of integral points, Zap. Nauchn. Semin. POMI, 413, 173-182 (2013) [3] Artushin, DA; Smirnov, AL, Eisenstein formula and Dirichlet correspondence, Zap. Nauchn. Semin. POMI, 469, 7-31 (2018) [4] H. L. S. Orde, “On Dirichlet’s class number formula,” J. London Math. Soc. (2), 18, No. 3, 409-420 (1978). · Zbl 0399.10023 [5] B. Schoeneberg, Elliptic Modular Functions (Grundl. Math. Wiss., 203), Springer-Verlag (1974). [6] E. T. Hecke, Vorlesungen über die Theorie der algebraischen Zahlen [Russian translation], Moscow-Leningrad (1940). · JFM 49.0106.10 [7] F. Diamond and J. Shurman, A First Course in Modular Forms (Grad. Texts Math., 228), Springer-Verlag (2005). · Zbl 1062.11022 [8] J. Milnor and D. Husemoller, Symmetric Bilinear Forms [Russian translation], Nauka, Moscow (1986). · Zbl 0292.10016 [9] T. M. Apostol, Introduction to Analytic Number Theory (Undergrad. Texts Math.), Springer, New York (1976). · Zbl 0335.10001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.