Algebraic Number Theory
Algebraic number theory studies number theory from the perspective of abstract algebra. In particular, heavy use is made of ring theory and Galois theory. Algebraic methods are particularly well-suited to studying properties of individual prime numbers. From an algebraic perspective, number theory can perhaps best be described as the study of . Famous problems in algebraic number theory include the Birch and Swinnerton-Dyer Conjecture and Fermat's Last Theorem.
Analytic Number Theory
Analytic number theory studies number theory from the perspective of calculus, and in particular real analysis and complex analysis. The techniques of analysis and calculus are particularly well-suited to studying large-scale properties of prime numbers. The most famous problem in analytic number theory is the Riemann Hypothesis.
Elliptic Curves and Modular Forms
An elliptic curve is the set of points satisfying some two variable third degree equation. Using certain affine transformations it can be shown that it is sufficient to consider those equations which are in Weierstrass form: Technically, one should consider all pairs of complex numbers satisfying such an equation, but often one can study the set of points where both coordinates lie in some subfield (like the reals or the rationals). One also needs to add a limit point, called the point at infinity. As , the derivative tends to infinity as well, and this should serve as some motivation to consider the point infinitely far vertically upward as the point at infinity. We denote it by .
The most important aspect of studying elliptic curves is the fact that there is a natural abelian group structure on its points. This means that given 2 points on the curve, they can be added in a way that satisfies the normal laws of addition, like associativity, commutativity and the existence of an identity and inverses.
The addition can be described as follows. Take two points and on the elliptic curve. The line through and cuts the curve in a third point . (One needs to take some care when or when this line is tangent to the curve, and hence cuts it in only two points.) We define to be the reflection of in the -axis. It takes some effort showing that this defines a group, but it can be done. The point at infinity is the identity for this group, and an inverse is obtained by reflecting a point in the -axis. We may thus summarize the group law by saying if and only if and lie on a line.
Denote by the upper half plane (those complex numbers with positive imaginary part). Then there are functions and defined by called Eisenstein series. If we set and , then there is a natural association with of the elliptic curve defined by . Then every elliptic curve over the complex numbers is isomorphic to one given by some , and two such curves, associated to and are isomorphic if and only if there is a relation This encourages us to define an action of the matrix group on by setting A modular form is a function such that for all and all we have and such that is holomorphic on and holomorphic at infinity. This last condition means that can be written as an expansion in the parameter with no negative exponents: As an example, the Eisenstein series and are modular forms of weight 4 and 6 respectively.
The connection between elliptic curves and modular forms
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