Abstract
We develop a strategy for bounding from above the height of rational points of modular curves with values in number fields, by functions which are polynomial in the curve’s level. Our main technical tools come from effective Arakelov descriptions of modular curves and jacobians. We then fulfill this program in the following particular case:
If is a not-too-small prime number, let be the classical modular curve of level over . Assume Brumer’s conjecture on the dimension of winding quotients of . We prove that there is a function (depending only on ) such that, for any quadratic number field , the -height of points in which are not lifts of elements of is less or equal to .
Citation
Pierre Parent. "Heights on squares of modular curves." Algebra Number Theory 12 (9) 2065 - 2122, 2018. https://doi.org/10.2140/ant.2018.12.2065
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