BSD system:sage

Nonsingular Plane Curves

$$F(X,Y) = 0$$

A rational point is $(x,y) \in \mathbf{Q}\times\mathbf{Q}$ such that $F(x,y) = 0$.

Ancient Theorem: A curve of degree $\leq 2$ has no rational points ($x^2 + y^2 = -1$) or infinitely many rational points ($x^2 + y^2 = 1$), and there is a way to decide which and enumerate all solutions.

Faltings Theorem (1985): A curve of degree $\geq 4$ has finitely rational points.

Birch and Swinnerton-Dyer Conjecture (1960s): A curve of degree 3 has either finitely rational points ($x^3 + y^3 = 1$) or infinitely many rational points $y^2 + y = x^3 - x$. he BSD Conjecture provides a way to decide which and enumerate all solutions.

{{{id=1| /// }}}

The Congruent Number Problem

Open Problem: Give an algorithm to decide whether or not an integer $n$ is the area of a right triangle with rational side lengths.

This is a 1000-year old open problem, perhaps the oldest open problem in mathematics.

{{{id=3| T = line([(0,0), (3,0), (3,4), (0,0)],rgbcolor='black',thickness=2) lbl = text("3",(1.5,-.5),fontsize=28) + text("4",(3.2,1.5),fontsize=28) lbl += text("5",(1.5,2.5),fontsize=28) lbl += text("Area $n = 6$", (2.1,1.2), fontsize=28, rgbcolor='red') show(T+lbl, axes=False) /// }}}

Congruent Numbers and the BSD Conjecture

Theorem: The Birch and Swinnerton-Dyer Conjecture provides a solution to the congruent number problem.

Proof: Suppose $n$ is a positive integer. Consider the cubic curve $y^2 = x^3 - n^2 x$. Using algebra (see next slide), one sees that this cubic curve has infinitely many rational points if and only if there are rationals $a,b,c$ such that $n=ab/2$ and $a^2 + b^2 = c^2$. The Birch and Swinnerton-Dyer conjecture gives an algorithm to decide whether or not any cubic curve has infinitely many solutions.

{{{id=13| /// }}} {{{id=12| /// }}} {{{id=5| n = 5 var('x,y') C = EllipticCurve(y^2 == x^3 - n^2 * x) show(C) ///

y^2 = x^3 - 25x

}}} {{{id=8| P = C.gens()[0]; P [n*P for n in [1..5]] # infinitely many solutions... /// [(-4 : 6 : 1), (1681/144 : -62279/1728 : 1), (-2439844/5094049 : 39601568754/11497268593 : 1), (11183412793921/2234116132416 : 1791076534232245919/3339324446657665536 : 1), (-50674456250230065124/79467131846613549025 : -2805376007832772561194783839874/708404494466557080860839692625 : 1)] }}} {{{id=9| (-62279/1728)^2 == (1681/144)^3 - 25*(1681/144) /// True }}} {{{id=10| /// }}}