Bayer, Pilar; Blanco-Chacón, Iván Quadratic modular symbols. (English) Zbl 1329.11065 Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 106, No. 1, 429-441 (2012). 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. Cited in 1 ReviewCited in 1 Document MSC: 11G18 Arithmetic aspects of modular and Shimura varieties 11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols Keywords:\(p\)-adic \(L\)-functions; modular symbols; Diophantine geometry PDFBibTeX XMLCite \textit{P. Bayer} and \textit{I. Blanco-Chacón}, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 106, No. 1, 429--441 (2012; Zbl 1329.11065) Full Text: DOI 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 · doi:10.1007/s002220050105 [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 · doi:10.1007/BF01388731 [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 · doi:10.1007/BF02941169 [8] Robert, A.M.: A course in p-adic analysis. In: Graduate Texts in Mathematics, vol. 198. Springer, New York (2000) · Zbl 0947.11035 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.