A height inequality for rational points on elliptic curves implied by the abc-conjecture

Abstract

Abstract. In this short note we show that the uniform abc-conjecture puts strong restrictions on the coordinates of rational points on elliptic curves. For the proof we use a variant of Vojta's height inequality formulated by Mochizuki. As an application, we generalize a result of Silverman on elliptic non-Wieferich primes.

Distribution of Non-Wieferich primes in certain algebraic groups under ABC

“…The same arguments could not be applied to other elliptic curves due to the unavailability of an inequality involving the height of points. This case was later settled by Kühn and Müller [7]. We shall discuss their result and prove the following Theorem 1.0.4.…”

Divisibility sequences of polynomials and heights estimates

“…Silverman in [27] established that for elliptic divisibility sequences over Q the number of non-primitive divisors is finite. This result was investigated further by several authors [4,5,10,12,15,29]. In another direction Streng [31] generalized the primitive divisor theorems for curves with complex multiplication.…”

Yang–Mills theory and the ABC conjecture

“…n =2, 4,5,8,11,13,16,17,19,22,23,25,29,31,32,34,37,38,41,43,44,46,47,53,58,59, 61, 62, 64, 65, 67 . .…”

“…Of the some 200 highest quality triples, we find 47 corresponding to Hall triples on E D . For reference, we tabulate 10 with the highest quality in 3 4 · 23 3 · 109 2 · 3 · 11 · 23 2 · 109 · 292561 3 · 23 2 · 109 1.62991 {1.9, 2.6} 2 9 · 3 2 · 5 2 · 7 · 23 2 2 · 3 2 · 19 · 23 · 227 · 22367 2 2 · 3 2 · 11 2 · 23 1.62599 {8.0, 22.8} 2 4 · 3 8 · 5 2 · 7 · 29 2 · 31 4 2 · 3 · 5 · 7 · 29 2 · 31 2 · 41579743 · 241510369 2 · 3 · 5 · 7 · 19 · 29 2 · 31 2 · 1307 1.62349 {0.0, 0.0} 2 4 · 3 4 · 5 5 · 13 2 · 17 2 · 3 2 · 5 2 · 13 2 · 2953 · 5588861 2 · 3 2 · 5 2 · 13 2 · 283 1.58076 {0.96, 0.9} 2 · 3 3 · 5 2 · 7 3 · 5 · 7 · 13 · 673 3 · 5 · 7 1.56789 {0.9, 0.8} 2 4 · 5 4 · 17 · 37 2 3 2 · 5 2 · 7 · 37 · 60925103 5 2 · 37 · 239 1.50284 {0.1, 0.04} 2 8 · 3 3 · 7 5 · 13 2 2 2 · 3 · 7 · 11 · 13 2 · 30949 · 569201 2 2 · 3 · 5 2 · 7 · 13 2 · 7937 1.49762 {0.0, 0.0} 3 2 · 5 3 · 13 4 · 17 · 139 2 · 151 · 4423 3 2 · 13 · 17 · 47 · 73 · 151 · 1399 · 4423 · 6691 · 438620957 2 · 3 2 · 11 · 13 · 17 · 151 · 4423 1.49243 {0.1, 0.0} 2 5 · 3 6 · 7 3 · 103 · 127 · 941 2 3 · 5 · 7 · 31 · 127 · 673 · 941 2 · 327823 · 349381 3 · 7 · 73 · 127 · 941 2 1.49159 {2.8, 4.6} 2 5 · 3 3 · 5 · 13 2 · 13 · 19 · 29 · 929 2 · 11 2 · 13 1.48887 {0.7, 0.6} Another statement, which follows from the uniform ABC conjecture for number fields, is an interesting bound on the actual points on an elliptic curve [37]. Adhereing to our notation in Appendix B, let P = (s/d 2 , t/d 3 ) be a point on an elliptic curve E in Weierstraß form, then there is the conjecture that for all > 0, there exists a constant K such that max 1 2 log |s| , log |d| ≤ (1 + ) log Rad(d) + K , (4.21)…”