# SEMINAR:Divisibility by 2 on quartic models of elliptic curves...

**Speaker:** Tuğba Yesin, Sabancı University

**Title:** Divisibility by 2 on quartic models of elliptic curves and rational Diophantine *D*(q)-quintuples

**Date/Time:**23 November 2022 / 14:40-15:30

**Zoom Link:**https://sabanciuniv.zoom.us/j/99963877622?pwd=UklXR3FUMDlQcGI0bm1oL0pkamRxQT09

**Meeting ID**: ** **999 6387 7622

**Passcode**: algebra

**Abstract:**Let C be a smooth genus one curve described by a quartic polynomial equation over the rational field Q with P ∈ C(Q). In this talk, we will give an explicit criterion for the divisibility-by-2 of a rational point on the elliptic curve (C, P). This generalizes the classical criterion of the divisibility-by-2 on elliptic curves described by Weierstrass equations.

We also employ this criterion to investigate the question of extending a rational D(q)-quadruple to a quintuple. We give concrete examples to which we can give an affirmative answer. One of these results implies that although the rational D(16t + 9)-quadruple {t, 16t + 8, 225t + 14, 36t + 20} can not be extended to a polynomial D(16t + 9)-quintuple using a linear polynomial, there are infinitely many rational values of t for which the aforementioned rational D(16t+9)-quadruple can be extended to a rational D(16t+9)-quintuple. Moreover, these infinitely many values of t are parametrized by the rational points on a certain elliptic curve of positive Mordell-Weil rank.

stem Security Symposium in 2015 and USENIX Security in 2017.