A linear lower bound on the gonality of modular curves
نویسنده
چکیده
0.2. Remarks. The proof, which was included in the author’s thesis [א], follows closely a suggestion of N. Elkies. In the exposition here many details were added to the argument in [א]. We utilize the work [L-Y] of P. Li and S. T. Yau on conformal volumes, as well as the known bound on the leading nontrivial eigenvalue of the non-euclidean Laplacian λ1 ≥ 21 100 [L-R-S]. If Selberg’s eigenvalue conjecture is true, the constant 7/800 above may be replaced by 1/96. Since, by the Gauss Bonnet formula, the genus g(XΓ) is bounded by DΓ/12+1 (indeed the difference is o(DΓ)), we may rewrite the inequality above in the slightly weaker form
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