Explicit bounds on the coefficients of modular polynomials for the elliptic $j$-invariant

Florian Breuer; Fabien Pazuki

Explicit bounds on the coefficients of modular polynomials for the elliptic $j$-invariant

Number Theory 2023-10-06 v2

Authors: Florian Breuer , Fabien Pazuki

Abstract

We obtain an explicit upper bound on the size of the coefficients of the elliptic modular polynomials $\Phi_N$ for any $N\geq1$ . These polynomials vanish at pairs of $j$ -invariants of elliptic curves linked by cyclic isogenies of degree $N$ . The main term in the bound is asymptotically optimal as $N$ tends to infinity.

Keywords

modular polynomials elliptic j-invariant explicit bounds cyclic isogenies coefficients

Cite

@article{arxiv.2211.06019,
  title  = {Explicit bounds on the coefficients of modular polynomials for the elliptic $j$-invariant},
  author = {Florian Breuer and Fabien Pazuki},
  journal= {arXiv preprint arXiv:2211.06019},
  year   = {2023}
}

Comments

9 pages, to appear in Proc. Amer. Math. Soc

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