Arxiv:2104.10817V1 [Math.NT] 22 Apr 2021 the Odco of Conductor Eo.Mroe,I the If Moreover, Below
LOWER BOUNDS FOR THE MODIFIED SZPIRO RATIO ALEXANDER J. BARRIOS Abstract. Let E/Q be an elliptic curve. The modified Szpiro ratio of E is the quan- 3 2 tity σm(E) = log max c4 ,c6 / log NE where c4 and c6 are the invariants associated to a global minimal model of E, and NE denotes the conductor of E. In this article, we show that for each of the fifteen torsion subgroups T allowed by Mazur’s Torsion Theorem, there is a rational number lT such that if T ֒→ E(Q)tors, then σm(E) > lT . We also show that this bound is sharp if the ABC Conjecture holds. 1. Introduction By an ABC triple, we mean a triple (a,b,c) where a,b,c are relatively prime positive integers with a + b = c. The quality of an ABC triple (a,b,c) is the quantity log c q(a,b,c)= . log(rad(abc)) In this notation, Masser and Oesterl´e’s ABC Conjecture [16] states that for each ǫ> 0, there are finitely many ABC triples with q(a,b,c) > 1+ ǫ. In [6, Theorem 4], Browkin, Filaseta, Greaves, and Schinzel showed that if the ABC Conjecture holds, then the set of limit points of q(a,b,c) (a,b,c) is an ABC triple { | } 1 1 15 is 3 , 1 . In fact, they showed unconditionally that the set of limit points contains 3 , 16 . In this article, we conjecture an analogous statement in the context of the modified Szpiro conjecture and establish a lower bound on the modified Szpiro ratio.
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