Every elliptic curve over Q is modular

Number theory 1 Minute

You need to know: Basic complex analysis, infinite series, irreducible polynomial, and either “rational map” or function field.

Background: Let {\mathbb H} = \{z \in {\mathbb C}, \text{Im}(z) > 0\}, j:{\mathbb H}\to{\mathbb C} be given by j(z) = 1728\frac{20 G_4(z)^3}{20 G_4(z)^3-49G_6(z)^2}, where G_k(z)=\sum\limits_{(m,n)\neq (0,0)}(m+nz)^{-k}. For every positive integer n, there exists a non-zero irreducible polynomial P_n(x,y) with integer coefficients such that P_n(j(nz),j(z))=0, \, z\in {\mathbb H}. The set X_0(n) of pairs of complex numbers (x,y) such that P_n(x,y)=0 is called the classical modular curve.

Elliptic curve E over {\mathbb Q} is the set of solutions to the equation y^2=x^3+ax+b, where a,b \in {\mathbb Q} are such that 4a^3+27b^2 \neq 0. It is called modular if it can be obtained via a rational map with integer coefficients from X_0(n) for some positive integer n. Equivalently, E is modular if the field of functions on E (given by Q(x)[y]/(y^2-(x^3+ax+b))) is contained in the field of functions on X_0(n) for some n (given by given by Q(x)[y]/P_n(x,y)).

The Theorem: On 28th February 2000 Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor submitted to the Journal of the AMS a paper in which they proved that every Elliptic curve over {\mathbb Q} is modular.

Short context: The Theorem confirms conjecture of Taniyama and Shimura from 1961. In 1995, Wiles proved a special case of this conjecture and deduced Fermat’s Last Theorem, which was probably the most famous open problem in the whole mathematics for over 300 years. The Theorem confirms Taniyama-Shimura conjecture in full, has the name “the modularity theorem”, and is considered by many as one of the greatest achievements in the modern mathematics.

Links: The original paper is available here.

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Published by Bogdan Grechuk

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