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Rational points on elliptic curves by John Tate, Joseph H. Silverman

Rational points on elliptic curves John Tate, Joseph H. Silverman ebook
Page: 296
Format: djvu
Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
ISBN: 3540978259, 9783540978251

The first proposition is that an elliptic curve $y^2 = x^3 + A x + B$, with $A,B \in Z$, $A \geq 0$, cannot contain a rational torsion point of order 5 or 7. Moreover, it is a unirational variety: it admits a dominant rational map from a projective space. These finite étale coverings admit various symmetry properties arising from the additive and multiplicative structures on the ring Fl = Z/lZ acting on the l-torsion points of the elliptic curve. Then there is a constant B(d) depending only on d such that, if E/K is an elliptic curve with a K -rational torsion point of order N , then N < B(d) . Heavily on the fact that E has a rational point of finite rank. Theorem 5 (on page vi) of Diem's thesis states that the discrete logarithm problem in the group of rational points of an elliptic curves E( F_{p^n} ) can be solved in an expected time of \tilde{O}( q^{2 – 2/n} ) bit operations. Theorem (Uniform Boundedness Theorem).Let K be a number field of degree d . Henri Poincaré studied them in the early years of the 20th century. Since it is a degree two cover, it is necessarily Galois, and {C} has a hyperelliptic involution {\iota: C \rightarrow C} over {\mathbb{ P}^1} with those is an elliptic curve (once one chooses an origin on {C} ), and the hyperelliptic . Rational points on elliptic curves. Elliptic curves have been a focus of intense scrutiny for decades. Rational.points.on.elliptic.curves.pdf. In other words, it is a two-sheeted cover of {\mathbb{P}^1} , and the sheets come together at {2g + 2} points.