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Quadratic modular symbols. (English) Zbl 1329.11065

Summary: We study a special kind of homology cycles of the modular curve \(X_0(N)\). For a newform of weight 2 for \(\Gamma_0(N)\), we construct a \(p\)-adic \(L\)-function by using these cycles. If the newform is defined over \(\mathbb{Q}\), this \(p\)-adic \(L\)-function gives rise to algebraic points of the attached elliptic curve.

MSC:

11G18 Arithmetic aspects of modular and Shimura varieties
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
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References:

[1] Arenas, A., Lario, J.-C.: Sistema minimal de generadors de {\(\Gamma\)}0(N). In: Bayer, P., Travesa, A. (eds.) Corbes modulars: taules. Notes del Seminari de Teoria de Nombres (UB-UAB-UPC), vol. 1, pp. 165–168, Barcelona (1992)
[2] Bertolini M., Darmon H.: Heegner points on Mumford–Tate curves. Invent. Math. 126, 413–456 (1996) · Zbl 0882.11034
[3] Birch, B.: Heegner points: the beginnings. In: Darmon, H., Zhang, S.W. (eds.) Heegner points and Rankin L-series. Mathematical Sciences Research Institute Publications, vol. 49, pp. 1–10. Cambridge University Press, Cambridge (2004) · Zbl 1073.11001
[4] Chuman Y.: Generators and relations of {\(\Gamma\)}0(N). J. Math. Kyoto Univ. 13, 381–390 (1973) · Zbl 0269.20028
[5] Mazur B., Tate J., Teitelbaum J.: On p-adic analogues of the conjecture of Birch and Swinnerton-Dyer. Invent. Math. 84, 1–48 (1986) · Zbl 0699.14028
[6] Manin, J.: Parabolic points and zeta functions of modular curves. Izv. Akad. Nauk SSSR Ser. Mat. 36, 19–66 (1972) · Zbl 0243.14008
[7] Rademacher H.: Über die Erzeugende von Kongruenzuntergruppen der Modulgruppe. Abh. Math. Seminar Hamburg 7, 134–138 (1929) · JFM 55.0083.02
[8] Robert, A.M.: A course in p-adic analysis. In: Graduate Texts in Mathematics, vol. 198. Springer, New York (2000) · Zbl 0947.11035
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