I am interested in $H¹$ right now and the cocycle condition $φ\_{jk} • φ\_{ij} = φ\_{ik}$ because of how it is said to relate to automorphic forms. I can't quite see the relationship between factors of automorphy and $H¹(G,M)$ that is mentioned [here](https://en.wikipedia.org/wiki/Automorphic_form).


Let $G$ be a group acting on a complex manifold $X$ and let $f$ be a holomorphic function from $X$ to $ℂ$. Recall that the factor of automorphy has
$$f(gx) = j\_g(x) f(x)$$, which already seems to bear some similarity to a cocycle condition.


The cohomology of a group $G$ with coefficients in $M$ can be calculated from the chain complex whose nth component is the $G$-module of functions from $Gⁿ$ to $M$. See [here](https://en.wikipedia.org/wiki/Group_cohomology#Cochain_complexes) for the formulas for the differential in group cohomology. I am interested in


$$H¹(G, M) = Z¹(G,M)/B¹(G,M)$$


The wikipedia article on automorphic forms mentions that "The formulation requires the general notion of factor of automorphy j for Γ, which is a type of 1-cocycle in the language of group cohomology."


Can someone spell this out? Is it possible to interpret the automorphy condition as a cocycle condition?



Modular forms and the cocycle condition in group cohomology
We have $j\_{gh}(x)f(x)=f((gh)x)=f(g(hx))=j\_g(hx)f(hx)=j\_g(hx)j\_h(x)f(x)$, and so $j\_{gh}(x)=j\_g(hx)j\_h(x)$, which is the one-cocycle condition.


