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Lower bounds for the modified Szpiro ratio

Volume 208 / 2023

Alexander J. Barrios Acta Arithmetica 208 (2023), 51-68 MSC: Primary 11G05; Secondary 11D75, 11J25. DOI: 10.4064/aa220417-31-3 Published online: 12 June 2023


Let $E/\mathbb Q$ be an elliptic curve. The modified Szpiro ratio of $E$ is the quantity $\sigma_{m}(E) =\log\max\{\vert c_{4}^{3}\vert ,c_{6}^{2}\}/\log N_{E}$, where $c_{4}$ and $c_{6}$ are the invariants associated to a global minimal model of $E$, and $N_{E}$ denotes the conductor of $E$. We show that for each of the fifteen torsion subgroups $T$ allowed by Mazur’s Torsion Theorem, there is a rational number $l_{T}$ such that if $T\hookrightarrow E(\mathbb Q) _{\text{tors}}$, then $\sigma_{m}(E) \gt l_{T}$. We also show that this bound is sharp.


  • Alexander J. BarriosDepartment of Mathematics
    University of St. Thomas
    St. Paul, MN 55105, USA

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