Algebra & Number Theory

Heights on squares of modular curves

Pierre Parent

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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 p is a not-too-small prime number, let X0(p) be the classical modular curve of level p over . Assume Brumer’s conjecture on the dimension of winding quotients of J0(p). We prove that there is a function b(p)=O(p5 logp) (depending only on p) such that, for any quadratic number field K, the j-height of points in X0(p)(K) which are not lifts of elements of X0+(p)() is less or equal to b(p).

Article information

Algebra Number Theory, Volume 12, Number 9 (2018), 2065-2122.

Received: 15 July 2017
Revised: 29 May 2018
Accepted: 15 July 2018
First available in Project Euclid: 5 January 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35]
Secondary: 14G05: Rational points 14G40: Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30]

modular curves Arakelov geometry


Parent, Pierre. Heights on squares of modular curves. Algebra Number Theory 12 (2018), no. 9, 2065--2122. doi:10.2140/ant.2018.12.2065.

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