Now work to define the terms in the Taniyama-Shimura conjecture is complete, part 4 presents the famous Taniyama-Shimura conjecture which led to a proof of Fermat’s Last Theorem.

We’ve done a lot of work so far just to try to define the terms in the Taniyama-Shimura conjecture, but today we should finally make it. Our last piece of information is to write down what the L-function of a modular form is. Since I don’t want to build a whole bunch of theory needed to define the special class of modular forms we’ll be considering, I’ll just say that we actually need to restrict our definition of “modular form” to “normalized cuspidal Hecke eigenform”. I’ll point out exactly why we need this, but it doesn’t change anything in the conjecture except that every elliptic curve actually corresponds to an even nicer type of modular form.

Let $latex {f\in S_k(\Gamma_0(N))}&fg=000000$ be a weight $latex {k}&fg=000000$ cusp form with $latex {q}&fg=000000$-expansion $latex {\displaystyle f=\sum_{n=1}^\infty a_n q^n}&fg=000000$. Since this is an analytic function on the disk, we have the tools and theorems…

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