By Nikos Tzanakis
This ebook provides in a unified method the gorgeous and deep arithmetic, either theoretical and computational, on which the categorical resolution of an elliptic Diophantine equation is predicated. It collects various effects and strategies which are scattered in literature. a few effects are even hidden in the back of a couple of workouts in software program applications, like Magma. This e-book is acceptable 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
R//, as required. 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.
P r X D iD1 r X 2 i ni iD1 nT n D n2i D m2i mT QT Qm D mT m c1 max m2i , iD1 1ÄiÄr as claimed. 3. Let E : y 2 D x 3 C Ax C B with A, B 2 Q be an elliptic curve model. log 2 C h. 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 .
E. P/. p/ . 8). The proof is now complete. K/, where K is a number field, is defined as follows. x0 : x1 : p2PK and the infinite-prime factor of the K-height is defined by Y : xn / D max¹jxi jd º. x0 : x1 : : xn /. ˛x0 : ˛x1 : : ˛xn /, which shows that the K-height of a projective point is independent from the choice of its projective coordinates. x0 : x1 : same. 1. x0 : x1 : field K containing the coordinates x0 , x1 , : : : , xn . Proof. x0 : x1 : : xn /1=ŒK:Q . Actually, this is true for both the “finite-prime” and the “infinite-prime” factor separately.
Elliptic Diophantine Equations: A Concrete Approach Via the Elliptic Logarithm by Nikos Tzanakis