class field theory - Intuition for Group Cohomology

See also this Math.SE post I wrote for some more motivation: http://math.stackexchange.com/a/270266/873. Recall that $mathrm H^*(G,M)=mathrm{Ext}^*(mathbb{Z},M)$.

After learning some more math, I've come across the following example of a use of group cohomology which sheds some light on its geometric meaning. (If you want to see a somewhat more concrete explanation of how group cohomology naturally arises, skip the next paragraph.)

We define an elliptic curve to be $E=mathbb{C}/L$ for a two-dimensional lattice $L$. Note that the first homology group of this elliptic curve is isomorphic to $L$ precisely because it is a quotient of the universal cover $mathbb{C}$ by $L$. A theta function is a section of a line bundle on an elliptic curve. Since any line bundle can be lifted to $mathbb{C}$, the universal cover, and any line bundle over a contractible space is trivial, the line bundle is a quotient of the trivial line bundle over $mathbb{C}$. We can define a function $j(omega,z):L times mathbb{C} to mathbb{C} setminus {0}$. Then we identify $(z,w) in mathbb{C}^2$ (i.e. the line bundle over $mathbb{C}$) with $(z+omega,j(omega,z)w)$. For this equivalence relation to give a well-defined bundle over $mathbb{C}/L$, we need the following: Suppose $omega_1,omega_2 in L$. Then $(z,w)$ is identified with $(z+omega_1+omega_2,j(omega_1+omega_2,z)w$. But $(z,w)$ is identified with $(z+omega_1,j(omega_1,z)w)$, which is identified with $(z+omega_1+omega_2,j(omega_2,z+omega_1)j(omega_1,z)w)$. In other words, this forces $j(omega_1+omega_2,z) = j(omega_2,z+omega_1)j(omega_1,z)$. This means that, if we view $j$ as a function from $L$ to the set of non-vanishing holomorphic functions $mathbb{C} to mathbb{C}$, with (right) L-action on this set defined by $(omega f)(z) mapsto f(z+omega)$, then $j$ is in fact a $1$-cocyle in the language of group cohomology. Thus $H^1(L,mathcal{O}(mathbb{C}))$, where $mathcal{O}(mathbb{C})$ denotes the (additive) $L$-module of holomorphic functions on $mathbb{C}$, classifies line bundles over $mathbb{C}/L$. What's more is that this set is also classified by the sheaf cohomology $H^1(E,mathcal{O}(E)^{times})$ (where $mathcal{O}(E)$ is the sheaf of holomorphic functions on $E$, and the $times$ indicates the group of units of the ring of holomorphic functions). That is, we can compute the sheaf cohomology of a space by considering the group cohomology of the action of the homology group on the universal cover! In addition, the $0$th group cohomology (this time of the meromorphic functions, not just the holomorphic ones) is the invariant elements under $L$, i.e. the elliptic functions, and similarly the $0$th sheaf cohomology is the global sections, again the elliptic functions.

More concretely, a theta function is a meromorphic function such that $theta(z+omega)=j(omega,z)theta(z)$ for all $z in mathbb{C}$, $omega in L$. (It is easy to see that $theta$ then gives a well-defined section of the line bundle on $E$ given by $j(omega,z)$ described above.) Then, note that $theta(z+omega_1+omega_1)=j(omega_1+omega_2,z)theta(z) = j(omega_2,z+omega_1)j(omega_1,z) theta(z)$, meaning that $j$ must satisfy the cocycle condition! More generally, if $X$ is a contractible Riemann surface, and $Gamma$ is a group which acts on $X$ under sufficiently nice conditions, consider meromorphic functions $f$ on $X$ such that $f(gamma z)=j(gamma,z)f(z)$ for $z in X$, $gamma in Gamma$, where $j: Gamma times X to mathbb{C}$ is holomorphic for fixed $gamma$. Then one can similarly check that for $f$ to be well-defined, $j$ must be a $1$-cocyle in $H^1(Gamma,mathcal{O}(X)^times)$! (I.e. with $Gamma$ acting by precomposition on $mathcal{O}(X)^times$, the group of units of the ring of holomorphic functions on $X$.) Thus the cocycle condition arises from a very simple and natural definition (that of a function which transforms according to a function $j$ under the action of a group). A basic example is a modular form such as $G_{2k}(z)$, which satisfies $G_{2k}(gamma z) = (cz+d)^{2k} G_{2k}(z)$, where $gamma = left(begin{array}{cc} a & b \ c & d end{array}right) in SL_2(mathbb{Z})$ acts as a fractional linear transformation. It follows automatically that something as simple as $(cz+d)^{2k}$ is a cocycle in group cohomology, since $G_{2k}$ is, for example, nonzero.

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