In this paper we compute the gonality over Q of the modular curve X1(N) for all N ≤ 40 and give upper bounds for each N ≤ 250. This allows us to determine all N for which X1(N) has infinitely points of degree ≤8. We conjecture that the modular units of Q(X1(N)) are freely generated by f2, . . . , f⌊N/2⌋ +1 where f k is obtained from the equation for X1(k).
Gonality of the modular curve X1(N) / M. Derickx, M. van Hoeij. - In: JOURNAL OF ALGEBRA. - ISSN 0021-8693. - 417(2014 Nov 30), pp. 52-71. [10.1016/j.jalgebra.2014.06.026]
Gonality of the modular curve X1(N)
M. DerickxPrimo
;
2014
Abstract
In this paper we compute the gonality over Q of the modular curve X1(N) for all N ≤ 40 and give upper bounds for each N ≤ 250. This allows us to determine all N for which X1(N) has infinitely points of degree ≤8. We conjecture that the modular units of Q(X1(N)) are freely generated by f2, . . . , f⌊N/2⌋ +1 where f k is obtained from the equation for X1(k).
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