Bielliptic modular curves X₁(N)
Daeyeol Jeon, Chang Heon Kim (2004)
Acta Arithmetica
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Daeyeol Jeon, Chang Heon Kim (2004)
Acta Arithmetica
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Andreas Enge, Reinhard Schertz (2005)
Acta Arithmetica
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François Brunault (2008)
Acta Arithmetica
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Matija Kazalicki, Koji Tasaka (2014)
Acta Arithmetica
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Kaneko and Sakai (2013) recently observed that certain elliptic curves whose associated newforms (by the modularity theorem) are given by the eta-quotients can be characterized by a particular differential equation involving modular forms and Ramanujan-Serre differential operator. In this paper, we study certain properties of the modular parametrization associated to the elliptic curves over ℚ, and as a consequence we generalize and explain some of their findings. ...
Tsz Ho Chan, Igor E. Shparlinski (2010)
Acta Arithmetica
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Daeyeol Jeon, Chang Heon Kim (2007)
Acta Arithmetica
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D. Choi (2006)
Acta Arithmetica
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Masataka Chida (2005)
Acta Arithmetica
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Jing Yu (1980)
Mathematische Annalen
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Francesc Bars, Aristides Kontogeorgis, Xavier Xarles (2013)
Acta Arithmetica
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We determine all modular curves X(N) (with N ≥ 7) that are hyperelliptic or bielliptic. We also give a proof that the automorphism group of X(N) is PSL₂(ℤ/Nℤ), whence it coincides with the normalizer of Γ(N) in PSL₂(ℝ) modulo ±Γ(N).
Besser, Amnon (1997)
Documenta Mathematica
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W. Barth, J. Michel (1993)
Mathematische Annalen
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Heima Hayashi (2006)
Acta Arithmetica
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Arjune Budhram (2002)
Acta Arithmetica
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Hidegoro Nakano (1968)
Studia Mathematica
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Nobuhiko Ishida, Noburo Ishii (2002)
Acta Arithmetica
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Andrew Wiles (1980)
Inventiones mathematicae
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(2013)
Acta Arithmetica
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The classical modular equations involve bivariate polynomials that can be seen to be univariate in the modular invariant j with integer coefficients. Kiepert found modular equations relating some η-quotients and the Weber functions γ₂ and γ₃. In the present work, we extend this idea to double η-quotients and characterize all the parameters leading to this kind of equation. We give some properties of these equations, explain how to compute them and give numerical examples.