Number theory/Advanced topics

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 $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ . 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

Elliptic curves

An elliptic curve is the set of points $(x,y)$ 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: $y^2=x^3+g_2x+g^3.$ Technically, one should consider all pairs $(x,y)$ 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 $x\to \infty$ , the derivative $\frac{dy}{dx}$ 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 $\mathcal{O}$ .

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 $P$ and $Q$ on the elliptic curve. The line through $P$ and $Q$ cuts the curve in a third point $R$ . (One needs to take some care when $P=Q$ or when this line is tangent to the curve, and hence cuts it in only two points.) We define $P+Q$ to be the reflection of $R$ in the $x$ -axis. It takes some effort showing that this defines a group, but it can be done. The point at infinity $\mathcal{O}$ is the identity for this group, and an inverse is obtained by reflecting a point in the $y$ -axis. We may thus summarize the group law by saying $P+Q+R=\mathcal{O}$ if and only if $P,Q$ and $R$ lie on a line.

Modular forms

Denote by $\mathcal{H}$ the upper half plane (those complex numbers with positive imaginary part). Then there are functions $G_4$ and $G_6$ defined by $G_k(z)=\sum_{(c,d)\in Z^2\backslash 0} (cz+d)^{-k}$ called Eisenstein series. If we set $g_2(z)=60G_4(z)$ and $g_3(z)=140G_6(z)$ , then there is a natural association with $z$ of the elliptic curve defined by $y^2=x^3+g_2(z)x+g_3(z)$ . Then every elliptic curve over the complex numbers is isomorphic to one given by some $z$ , and two such curves, associated to $z$ and $z'$ are isomorphic if and only if there is a relation $z'=\frac{az+b}{cz+d},\quad a,b,c,d\in\mathbb{Z}, ad-bc=1.$ This encourages us to define an action of the matrix group $SL_2(\mathbb{Z})$ on $\mathcal{H}$ by setting $\left(\begin{array}{cc}a&b\\c&d\end{array}\right)z=\frac{az+b}{cz+d}.$ A modular form $f$ is a function such that for all $z\in\mathcal{H}$ and all $\gamma\in SL_2(\mathbb{Z})$ we have $f(\gamma z)=(cz+d)^{-k}f(z)$ and such that $f$ is holomorphic on $\mathcal{H}$ and holomorphic at infinity. This last condition means that $f$ can be written as an expansion in the parameter $q=e^{2\pi i z}$ with no negative exponents: $f(z)=\sum_{n\ge 0}a_nq^n.$ As an example, the Eisenstein series $G_4$ and $G_6$ are modular forms of weight 4 and 6 respectively.