By Nikos Tzanakis
This e-book provides in a unified approach the attractive and deep arithmetic, either theoretical and computational, on which the specific answer of an elliptic Diophantine equation relies. It collects quite a few effects and strategies which are scattered in literature. a few effects are even hidden in the back of a few workouts in software program programs, like Magma. This e-book is appropriate for college students in arithmetic, in addition to specialist mathematicians.
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Additional resources for Elliptic Diophantine Equations: A Concrete Approach Via the Elliptic Logarithm
41) where and j are, respectively, the discriminant and j -invariant of the model E, b2 D a12 C 4a2 , 2 D 1 or 2 according to whether b2 vanishes or not, respectively, and logC is defined for any real ˛ > 0 by logC ˛ D log max¹1, ˛º. P // 2 Ä . 42) Proof. Note that the two models D and E have equal j -invariants, while their corresponding discriminants 1 and are related by 1 D Ä 12 . 40). That theorem requires that the ai ’s be algebraic integers. In our proposition, these coefficients are 3 One can easily see that there always exists such a model D.
X0 : x1 : : xn /ŒL:K . 3 Heights: Absolute and logarithmic First we note that every embedding : K ,! C has exactly ŒL : K distinct extensions L ,! C. If is an extension of we write j . xi /jº o 2EK : xn /ŒL:K . x0 : as required. K/, where K is a number field. 1. x0 : x1 : : xn /. 1, this definition is independent from K. x0 : x1 : : xn /. 1 : ˛/. 2. ˛/. 17) 16 Chapter 2 Heights Proof. Let K be any number field containing ˛ and ˇ. Denote by P the set of prime ideals of ZK and by EK the set of all embeddings K ,!
Z// and this is either z D z1 or z D z2 . 1. Let A, B 2 C such that 4A3 C 27B 2 ¤ 0 and consider the elliptic curve model E : y 2 D x 3 CAxCB. z1 / D 0, a contradiction. C/ 7 ! C=ƒ 33 period parallelogram) and let ! be the only pole of } in F. z// if z ¤ ! C/ F3z7 ! O if z D ! C/ 7 ! 1. 1, E is just one out of many models of an elliptic curve E, therefore, any point on E, being a representative of a certain abstract point P (say) of E, should be denoted by P E . However, since in the present chapter we will not make use of any other model of E except for E, and for the sake of simplifying the notation, we will omit the superscript E on P , simply writing P instead of P E .
Elliptic Diophantine Equations: A Concrete Approach Via the Elliptic Logarithm by Nikos Tzanakis