A note on a question of J. Nekovár and the Birch and Swinnerton-Dyer Conjecture
نویسندگان
چکیده
منابع مشابه
The Birch-swinnerton-dyer Conjecture
We give a brief description of the Birch-Swinnerton-Dyer conjecture which is one of the seven Clay problems.
متن کاملThe conjecture of Birch and Swinnerton-Dyer
This essay starts by first explaining, for elliptic curves defined over Q, the statement of the conjecture of Birch and Swinnerton-Dyer. Alongside, it contains a discussion of some results that have been proved in the direction of the conjecture, such as the theorem of Kolyvagin-Gross-Zagier and the weak parity theorem of Tim and Vladimir Dokchitser. The second, third and fourth part of the ess...
متن کاملThe Birch and Swinnerton-Dyer Conjecture
A polynomial relation f(x, y) = 0 in two variables defines a curve C. If the coefficients of the polynomial are rational numbers then one can ask for solutions of the equation f(x, y) = 0 with x, y ∈ Q, in other words for rational points on the curve. If we consider a non-singular projective model C of the curve then over C it is classified by its genus. Mordell conjectured, and in 1983 Falting...
متن کاملThe Birch and Swinnerton-Dyer Conjecture
In this talk I shall attempt to introduce some of the main features of the Birch and Swinnerton-Dyer conjecture, (BSD). The congruent number problem, deciding whether an integer D is the area of a right angle triangle with rational sides, is not easy. It turns out that the problem is equivalent to finding out if a certain elliptic curve has an infinite number of rational points. In 1983 Tunnell...
متن کاملOn a Conjecture of Birch and Swinnerton-Dyer
Let E/Q be an elliptic curve defined by the equation y 2 = x 3 + ax + b. For a prime p, p ∤ ∆ = −16(4a 3 + 27b 2) = 0, define N p = p + 1 − a p = |E(F p)|. As a precursor to their celebrated conjecture, Birch and Swinnerton-Dyer originally conjectured that for some constant c, p≤x,p∤∆ N p p ∼ c(log x) r , x → ∞. Let α p and β p be the eigenvalues of the Frobenius at p. Define ˜ c n = α k ...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 1998
ISSN: 0002-9939,1088-6826
DOI: 10.1090/s0002-9939-98-04465-7