Reference for existence of quotients of modular Jacobians?

Question

In the construction of Galois representations attached to modular forms of weight 2, the representation spaces can be found in Tate modules of certain quotients of Jacobians of modular curves by ideals in the appropriate Hecke algebra. Is the existence\construction of such quotients treated in any of the modern references on abelian varieties? Does one just regard the ideal as a group acting as automorphisms on the Jacobian and then invoke some general theorem about quotients of abelian varieties by group actions, or is it more specialized than that? Maybe since the Hecke action on the modular Jacobian can be defined via Picard functoriality, one looks at a certain quotient of this functor and then proves it's representable by an abelian variety? Any pointers to theorems that encompass this type of quotient would be greatly appreciated. Thanks.

Answer 1

If $R$ is a finite type $\mathbb Z$-algebra acting on an abelian variety $A$, and $M$ is any finitely generated $R$-module, then $M\otimes_R A$ is well-defined as an abelian variety. (This construction includes the case where $M$ is a quotient of $R$, which is the case you are asking about.)

The construction is as follows: let $R^a \to R^b \to M \to 0$ be a presentation of $M$, and define $M\otimes_R A$ to be the cokernel of the induced map $A^a \to A^b$.

You can check that $M\otimes_R A$ is well-defined independently of the choice of presentation, e.g. by finding a functor that it represents (on the category of abelian varieties).

My memory is that I learned this from Serre's article on CM in Cassels and Frolich. It's probably written up in other places too.