By Nikos Tzanakis
This ebook offers in a unified approach the attractive and deep arithmetic, either theoretical and computational, on which the categorical resolution of an elliptic Diophantine equation is predicated. It collects a number of effects and strategies which are scattered in literature. a few effects are even hidden at the back of a couple of workouts in software program programs, like Magma. This e-book is appropriate for college kids in arithmetic, in addition to expert mathematicians.
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Extra info for Elliptic Diophantine Equations: A Concrete Approach Via the Elliptic Logarithm
X/. 2 =2/. mod ƒ/. 2 D i s, s 2 RC and n 2 Z. t / 1 The left-hand side is positive, therefore, n 0. We will show thatR n D 0. t/ fined on the interval . 1, e3 /. 0, . 12 C n/s/. 2 =2. 2 =2/ D e3 , a contradiction. 2 D 2i e3 1 Case > 0 Z e3 dt dt p p Di . 19) Let < 0. Now we cut the complex plane along the half-line on the real axis from e1 to C1 and along the segment joining e2 with e3 D e2 . x/ > 0 for x < e1 , the real function . 1, e1 3 x 7! x/ Œ0, C1/ is well defined and is extended analytically on C .
12]. Remark. ƒ/. Actually, sometimes we will use this more precise notation if we want to emphasise the role of ƒ; otherwise, for the sake of simplicity in our notation, we will omit the indication of ƒ. /3 , which is an odd function. 2 / e3 D }. 6) e1 D }. /, e2 D }. 10]. g2 , g3 / satisfying the condition g23 27g32 ¤ 0. The converse is also true. 9]. 2. 3). 4) is satisfied. ƒ/ D a3 . 1) will be called Weierstrass } function with parameters g2 D a2 and g3 D a3 . 1 ƒ . The simple equations below relate the values of g2 , g3 and that correspond to the lattices ƒ and ƒ .
X, y/ we do not know a priori how to choose " 2 ¹ 1, 1º. 5. 2, we intend to study Diophantine equations E : y 2 D x 3 C Ax C B, A, B 2 Q, 4A3 C 27B 2 ¤ 0. z// is an integral (or rational) point, where } is the Weierstrass function with parameters g2 D 4A, g3 D 4B. z// is a sought for point on E. ƒ/ are real numbers, forgetting for the moment that, in the context of our Diophantine study, these are actually rational numbers. 11) when A, B 2 R. z// for a unique z belonging to a fundamental parallelogram.
Categories: Algebraic Geometry