I'm a little confused by the definition of a period matrix given on p. 2 of this paper: https://link.springer.com/content/pdf/10.1007/BF02599319.pdf
The definition of a period matrix that I'm familiar with gives a $g \times 2g$ matrix and I'm not sure how this definition relates to it. In general, there seem to be a couple of different ways to write down the definition of a period matrix of an abelian variety depending on how a basis for the lattice is given.
What are possible ways to define a period matrix and how are they related to each other? For example, how would one write a period matrix for a product of elliptic curves using these defintions?
Edit: Using definition of period matrix provided here: https://books.google.com/books?id=ysvyCAAAQBAJ&pg=PA9&lpg=PA9&dq=%22period+matrix%22+birkenhake+lange&source=bl&ots=j7BnrEAfZh&sig=2xfB2eQm1lmI7u_HxszebLWof3A&hl=en&sa=X&ved=0ahUKEwiOzsTgjKnVAhWGjlQKHcfuBpwQ6AEIMjAC#v=onepage&q=%22period%20matrix%22%20birkenhake%20lange&f=false