On Siegel modular forms. II: On Siegel modular forms. I
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| Název: | On Siegel modular forms. II: On Siegel modular forms. I |
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| Autoři: | Runge, Bernhard |
| Zdroj: | Nagoya Math. J. 138 (1995), 179-197 |
| Informace o vydavateli: | De Gruyter, Berlin, 1995. |
| Rok vydání: | 1995 |
| Témata: | ring of Siegel modular forms of small genus, dimension formulae, injectivity of double covering, 11F46, Complex multiplication and moduli of abelian varieties, partial Fourier transformation, decomposition of Bruhat type, Siegel modular groups, Siegel and Hilbert-Siegel modular and automorphic forms, Poincaré series, invariant theory, Siegel modular forms, Cohen-Macaulay, Modular and automorphic functions, moduli space of abelian varieties, dimension formulas, theta constants, theta squares, Theta series, Weil representation, theta correspondences, generators, Gorenstein, Hilbert-Samuel and Hilbert-Kunz functions, Actions of groups on commutative rings |
| Popis: | The author studies the ring of Siegel modular forms of small genus \(A(\Gamma_ g)\). Some of the results contained in the paper are classical, but they are obtained in a simpler way. The method used in this paper is similar to Igusa's going down process [\textit{cf. J. Igusa}, Am. J. Math. 86, 392-412 (1964; Zbl 0133.33301)]. The main difference between the two processes is the starting point. Whenever Igusa begins from the ring of classical Thetanullwerte \(\mathbb{C}[\vartheta_ m(\tau)]\), Runge uses the ring \(\mathbb{C}\bigl[\vartheta {\alpha \brack 0}(2\tau)\bigr]\). Then acting with finite groups, he obtains Poincaré series and generators of the ring of modular forms relative to \(\Gamma_ g\) and some related subgroups for \(g=1\) and 2. In a forthcoming paper the author will get these results in the genus 3 case, this will be possible since, as he remarks, in this case the ring \(\mathbb{C}\bigl[ \vartheta {\alpha \brack 0}(2\tau)\bigr]\) is integrally closed. As a consequence of this fact, the author proves that \(A(\Gamma_ 3)\) is Cohen-Macaulay. Finally, let \({\mathcal A}_{2,4}\) be the moduli space of abelian varieties with level 2 theta structure, the reviewer proves in [Moduli space of ppav with level 2 (theta) structure, Am. J. Math. (to appear)]\ that the map defined from the \(\vartheta {\alpha \brack 0}(2\tau)\) is injective; using this result the author proves the injectivity of a double covering (\(g\) even) of this moduli space. |
| Druh dokumentu: | Article Other literature type |
| Popis souboru: | application/xml; application/pdf |
| DOI: | 10.1515/crll.1993.436.57 |
| DOI: | 10.1017/s0027763000005237 |
| Přístupová URL adresa: | http://projecteuclid.org/euclid.nmj/1118775400 |
| Přístupové číslo: | edsair.dedup.wf.002..23da06e62b4397ea60b8083cef21c012 |
| Databáze: | OpenAIRE |
Nájsť tento článok vo Web of Science | Abstrakt: | The author studies the ring of Siegel modular forms of small genus \(A(\Gamma_ g)\). Some of the results contained in the paper are classical, but they are obtained in a simpler way. The method used in this paper is similar to Igusa's going down process [\textit{cf. J. Igusa}, Am. J. Math. 86, 392-412 (1964; Zbl 0133.33301)]. The main difference between the two processes is the starting point. Whenever Igusa begins from the ring of classical Thetanullwerte \(\mathbb{C}[\vartheta_ m(\tau)]\), Runge uses the ring \(\mathbb{C}\bigl[\vartheta {\alpha \brack 0}(2\tau)\bigr]\). Then acting with finite groups, he obtains Poincaré series and generators of the ring of modular forms relative to \(\Gamma_ g\) and some related subgroups for \(g=1\) and 2. In a forthcoming paper the author will get these results in the genus 3 case, this will be possible since, as he remarks, in this case the ring \(\mathbb{C}\bigl[ \vartheta {\alpha \brack 0}(2\tau)\bigr]\) is integrally closed. As a consequence of this fact, the author proves that \(A(\Gamma_ 3)\) is Cohen-Macaulay. Finally, let \({\mathcal A}_{2,4}\) be the moduli space of abelian varieties with level 2 theta structure, the reviewer proves in [Moduli space of ppav with level 2 (theta) structure, Am. J. Math. (to appear)]\ that the map defined from the \(\vartheta {\alpha \brack 0}(2\tau)\) is injective; using this result the author proves the injectivity of a double covering (\(g\) even) of this moduli space. |
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| DOI: | 10.1515/crll.1993.436.57 |