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Taniyama-Shimura 3: L-Series where it will be crucial in the definition of modularity. A Mind for Madness

For today, we assume our $latex {d}&fg=000000$-dimensional variety $latex {X/\mathbb{Q}}&fg=000000$ has the property that its middle etale cohomology is 2-dimensional. It won’t hurt if you want to just think that $latex {X}&fg=000000$ is an elliptic curve. We will first define the L-series via the Galois representation that we constructed last time. Fix $latex {p}&fg=000000$ a prime not equal to $latex {\ell}&fg=000000$ and of good reduction for $latex {X}&fg=000000$. Let $latex {M=\overline{\mathbb{Q}}^{\ker \rho_X}}&fg=000000$. By definition the representation factors through $latex {{Gal} (M/\mathbb{Q})}&fg=000000$. For $latex {\frak{p}}&fg=000000$ a prime lying over $latex {p}&fg=000000$ the decomposition group $latex {D_{\frak{p}}}&fg=000000$ surjects onto $latex {{Gal} (\overline{\mathbf{F}}_p/\mathbf{F}_p)}&fg=000000$ with kernel $latex {I_{\frak{p}}}&fg=000000$. One of the subtleties we’ll jump over to save time is that $latex {\rho_X}&fg=000000$ acts trivially on $latex {I_{\frak{p}}}&fg=000000$ (it follows from the good reduction assumption), so we can lift the generator of $latex {{Gal} (\overline{\mathbf{F}}_p/\mathbf{F}_p)}&fg=000000$ to get a conjugacy class $latex {{Frob}_p}&fg=000000$ whose image under…

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Taniyama-Shimura 2: Galois Representations where the standard modern approach to defining modularity for other types of varieties. A Mind for Madness

Fix some proper variety $latex {X/\mathbb{Q}}&fg=000000$. Our goal today will seem very strange, but it is to explain how to get a continuous representation of the absolute Galois group of $latex {\mathbb{Q}}&fg=000000$ from this data. I’m going to assume familiarity with etale cohomology, since describing Taniyama-Shimura is already going to take a bit of work. To avoid excessive notation, all cohomology in this post (including the higher direct image functors) are done on the etale site.

For those that are intimately familiar with etale cohomology, we’ll do the quick way first. I’ll describe a more hands on approach afterwards. Let $latex {\pi: X\rightarrow \mathrm{Spec} \mathbb{Q}}&fg=000000$ be the structure morphism. Fix an algebraic closure $latex {v: \mathrm{Spec} \overline{\mathbb{Q}}\rightarrow \mathrm{Spec}\mathbb{Q}}&fg=000000$ (i.e. a geometric point of the base). We’ll denote the base change of $latex {X}&fg=000000$ with respect to this morphism $latex {\overline{X}}&fg=000000$. Suppose the dimension of $latex {X}&fg=000000$ is $latex {n}&fg=000000$.

Let…

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Great post on understanding the statement of the famous Taniyama-Shimura conjecture that led to the proof of Fermat’s Last Theorem. A Mind for Madness

It’s time to return to plan A. I started this year by saying I’d post on some fundamental ideas in arithmetic geometry. The local system thing is hard to get motivated about, since the way I was going to use it in my research seems irrelevant at the moment. My other option was to blog some stuff about class field theory, since there is a reading group on the topic that I belong to this quarter.

The first goal of this new series is to understand the statement of the famous Taniyama-Shimura conjecture that led to the proof of Fermat’s Last Theorem. A lot of people can probably mumble something about the conjecture if they have any experience in algebraic/arithmetic geoemtry or any of the number theory type fields, but most people probably can’t say anything precise about what the conjecture says (I’ll continue to call it a “conjecture” even…

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Elliptic curves are especially important in number theory, and constitute a major area of current research; for example, they were used in the proof, by Andrew Wiles, of Fermat’s Last Theorem. More concretely, an elliptic curve is the set of zeros of a cubic polynomial in two variables. Where $ax^{3}+bx^{2}y+cxy^{2}+dy^{3}+ex^{2}+fxy+gy^{2}+hx+iy+j=0$ is the equation of a general cubic polynomial. A famous example being $\displaystyle x^{3}+y^{3}=1$

or in homogeneous form, $\displaystyle X^{3}+Y^{3}=Z^{3}$.

This is the first non-trivial case of Fermat’s Last Theorem.

A modular elliptic curve is an elliptic curve $E$ that admits a parametrization $X_{0}(N) \rightarrow E$ by a modular curve. This is not the same as a modular curve that happens to be an elliptic curve, and which could be called an elliptic modular curve. The modularity theorem, also known as the Taniyama–Shimura conjecture, asserts that every elliptic curve defined over the rational numbers is a modular form in disguise.

In 1985, starting with a fictitious solution to Fermat’s last theorem (the Frey curve), G. Frey showed that he could create an unusual elliptic curve which appeared not to be modular. If the curve were not modular, then this would show that if Fermat’s last theorem were false, then the Taniyama-Shimura conjecture would also be false. Furthermore, if the Taniyama-Shimura conjecture is true, then so is Fermat’s last theorem.

However, Frey did not actually prove that his curve was not modular. The conjecture that Frey’s curve was not modular came to be called the “epsilon conjecture,” and was quickly proved by Ribet (Ribet’s theorem) in 1986, establishing a very close link between two mathematical structures (the Taniyama-Shimura conjecture and Fermat’s last theorem) which appeared previously to be completely unrelated

By proving the semistable case of the conjecture, Andrew Wiles proved Fermat’s Last Theorem.

Some Elliptic curves:  