• elliptic curves
• Picard group
• algebraic stacks
• W. Fulton and M.Olsson

The purpose of this thesis is to give an exposition of the computation of the Picard group of the moduli stack of elliptic curves over a base scheme S. Let M_1,1,S denote this stack.
In 1965 David Mumford proved that the Picard group of the moduli stack M_1,1,k is equal to Z/12, in case k is a field of characteristic not 2 or 3.
In 2010 William Fulton and Martin Olsson integrated the previous work of Mumford, with the article "The Picard group of M_1,1". They proved that if S is a connected Z[1/2]-scheme or a connected reduced scheme, then Pic(M_1,1,S) is isomorphic to Z/12 x Pic(A^{1}_S). The scheme A^{1}_S, which is the affine line over S, represents the coarse moduli space of M_1,1,S.
The proof of this Theorem is divided into many steps, in each of which the problem is analyzed by restricting oneself to a specific type of base scheme S for M_1,1,S, and in the thesis we study these steps in detail.