Abstract
We explicitly write down the Eisenstein elements inside the space of modular symbols for Eisenstein series with integer coefficients for the congruence subgroups \(\Gamma _0(N)\) with N odd square-free. We also compute the winding elements explicitly for these congruence subgroups. Our results are explicit versions of the Manin-Drinfeld Theorem [Thm. 6]. These results are the generalization of the paper [1] results to odd square-free level.
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References
Debargha Banerjee and Srilakshmi Krishnamoorthy. The eisenstein elements of modular symbols for level product of two distinct odd primes. Pacific Journal of Mathematics, 281(2):257–285, 2016.
B. Mazur. Modular curves and the Eisenstein ideal. Inst. Hautes Études Sci. Publ. Math., 47:33–186 (1978), 1977.
Loïc Merel. L’accouplement de Weil entre le sous-groupe de Shimura et le sous-groupe cuspidal de \(J_0(p)\). J. Reine Angew. Math., 477:71–115, 1996.
Loïc Merel. Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math., 124(1-3):437–449, 1996.
Frank Calegari and Matthew Emerton. On the ramification of Hecke algebras at Eisenstein primes. Invent. Math., 160(1):97–144, 2005.
Loïc Merel. Intersections sur des courbes modulaires. Manuscripta Math., 80(3):283–289, 1993.
Amod Agashe. The Birch and Swinnerton-Dyer Conjecture for modular abelian varities of analytic rank zero. PhD thesis, University of California at Berkeley, 2000.
Debargha Banerjee. A note on the eisenstein elements of prime square level. Proceedings of the American Mathematical Society., 12:3675–3686, 2014.
Ju. I. Manin. Parabolic points and zeta functions of modular curves. Izv. Akad. Nauk SSSR Ser. Mat., 36:19–66, 1972.
V. G. Drinfeld. Two theorems on modular curves. Funkcional. Anal. i Priložen., 7(2):83–84, 1973.
L. Merel. Homologie des courbes modulaires affines et paramétrisations modulaires. In Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993), Ser. Number Theory, I, pages 110–130. Int. Press, Cambridge, MA, 1995.
B. Mazur. On the arithmetic of special values of \(L\) functions. Invent. Math., 55(3):207–240, 1979.
Fred Diamond and Jerry Shurman. A first course in modular forms, volume 228 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2005.
Rick Miranda. Algebraic curves and Riemann surfaces, volume 5 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1995.
Glenn Stevens. Arithmetic on modular curves, volume 20 of Progress in Mathematics. Birkhäuser Boston, Inc., Boston, MA, 1982.
Glenn Stevens. The cuspidal group and special values of \(L\)-functions. Trans. Amer. Math. Soc., 291(2):519–550, 1985.
Acknowledgements
The author would like to thank Loïc Merel, Debargha Banerjee, Narasimha Kumar and Joseph Oesterlé for their helpful discussion, suggestions and comments. Part of this work was done at IMJ-PRG, France. The author would like to thank the mathematics department for the hospitality. The author would like to thank IISER, TVM for providing the excellent working conditions.
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Communicated by C. S. Rajan.
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Krishnamoorthy, S. The Eisenstein and winding elements of modular symbols for odd square-free level. Indian J Pure Appl Math 54, 713–724 (2023). https://doi.org/10.1007/s13226-022-00289-8
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DOI: https://doi.org/10.1007/s13226-022-00289-8