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I remember watching the BBC Horizon documentary on Fermat's Last Theorem many years ago and it was very inspiring. Though I have always wondered, starting from 25 minutes and 53 seconds of the documentary available here, what are the symbols that Andrew Wiles is writing on the board? I can see that he writes

$$\#H^1_f(\Bbb{Q}_{\Sigma}/\Bbb{Q}, \textrm{Sym}...\text{(the video cuts out)}$$

as well as

$$\textrm{Sym}^2 \rho \otimes \Bbb{Q}_p/\Bbb{Z}_p.$$

Out of curiousity, what do these mean? Is the first the order of some first cohomology group? I can't tell what the rest is because the video cuts out. As for the second, is this some representation theoretic thing, tensoring the second symmetric power of $\rho$ - which I presume from the video is a representation $\rho : \textrm{Gal}(\overline{\Bbb{Q}}/\Bbb{Q}) \to \textrm{GL}_2(\Bbb{Z}_p)$ - with the quotient $\Bbb{Q}_p/\Bbb{Z}_p$? What is the quotient group in question. Also, what is $\Bbb{Q}_p$?

As a learner of Algebraic number theory, I am interested in what these symbols mean.

Thanks.

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2 Answers 2

up vote 12 down vote accepted

First of all: $\rho$ is usually used for a Galois representation; in the context, I'm pretty sure it's supposed to be the Galois representation given by the Tate module of an elliptic curve over $\mathbf{Q}$, which is a continuous homomorphism $\operatorname{Gal}(\overline{\mathbf{Q}} / \mathbf{Q}) \to \operatorname{GL}_2(\mathbf{Z}_p)$ for some prime $p$. $\operatorname{Sym}^2 \rho$ is the symmetric square Galois representation into $\operatorname{GL}_3(\mathbf{Z}_p)$.

In the $H^1$ bit: $\mathbf{Q}_\Sigma$ is the maximal extension of $\mathbf{Q}$ unramified outside a finite set $\Sigma$ of primes, and $H^1(\mathbf{Q}_\Sigma / \mathbf{Q}, -)$ is shorthand for the group cohomology $H^1(\operatorname{Gal}(\mathbf{Q}_\Sigma / \mathbf{Q}), -)$ (more strictly, continuous gropu cohomology, which respects the Krull topology on the Galois group). We can plug in for the "$-$" any Galois representation factoring through the canonical map $$\operatorname{Gal}(\overline{\mathbf{Q}} / \mathbf{Q}) \to \operatorname{Gal}(\mathbf{Q}_\Sigma / \mathbf{Q}),$$ i.e. any Galois representation unramified outside $\Sigma$; the Galois representation attached to an elliptic curve (or its symmetric square) satisfies this as long as $\Sigma$ contains $p$ and all primes dividing the discriminant of the elliptic curve.

As for $\mathbf{Q}_p / \mathbf{Z}_p$: this is the "Prufer group", an infinite torsion group isomorphic to the direct limit of cyclic groups of order $p^n$ over all $n$. It's useful here because taking homs into $\mathbf{Q}_p / \mathbf{Z}_p$ gives a well-behaved duality theory for topological modules (one form of Pontryagin duality).

The only thing I haven't explained yet is the subscript $f$ in $H^1_f(...)$. This is perhaps the most delicate thing here: it's the "finite part" of the Galois cohomology group $H^1(...)$, a certain canonical submodule defined in a famous 1990 paper of Bloch and Kato. $H^1_f(\mathbf{Q}_\Sigma / \mathbf{Q}, \rho)$ is closely related to the Selmer group of the elliptic curve, so the $H^1_f$ functor is a sort of "Selmer group of an arbitrary Galois representation". The size of the Bloch--Kato Selmer group of $\operatorname{Sym}^2 \rho$ is important here because it determines how the deformation theory of $\rho$ behaves.

A good place to learn more about these things would be the book by Cornell, Silverman and Stevens, which user33240 has already linked to.

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Thanks for your answer. I believe $\Bbb{Z}_p$ denotes the $p$ - adics, what about $\Bbb{Q}_p$? Thanks. –  user38268 Dec 30 '12 at 10:30

I recommend you to take a look at Wiles' original paper "Modular elliptic curves and Fermat's Last Theorem" on JSTOR instead. I remember according to my ANT professor it used heavy machinary in Galois representation and automorphic forms. There is a book by Silverman on this topic called Moduar forms and Fermat's last theorem. See here. This is one of the usual reference.

Other real experts on this site should be able to answer this more precisely.

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6  
"Wile's original paper"? Who is this chap Wile? –  David Loeffler Dec 28 '12 at 18:22

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