I'm currently reading The 1-2-3 of Modular Forms: Lectures at a Summer School in Nordfjordeid and I got stuck (on the fourth page) on the following paragraph:
... the quotient space $\Gamma_1$ \ $\mathfrak{H}$ are moduli (=parameters) for the isomorphism classes of elliptic curves over $\mathbb{C}$. To each point $z \in \mathfrak{H}$ one can associate the lattice $\Lambda_z = \mathbb{Z}.z + \mathbb{Z}.1 \subset \mathbb{C}$ ...
What does it mean to have $\mathbb{Z}.z$ and $\mathbb{Z}.1 $? What kind of product is it? I haven't read much about lattices, so I'd appreciate any help to get started.
Edit: $\mathfrak{H} = \left\{ \tau \in \mathbb{C} \, | \, \Im(\tau) > 0 \right\}$ the upper half complex plane.
When you see a set involved in an arithmetic expression, the semantics is usually that it is the set of results when the expression is applied to all elements of the set(s). In particular,
$$\Bbb Z\cdot z+\Bbb Z\cdot 1 = z\Bbb Z +\Bbb Z= \{az+b\in\Bbb C\mid a,b\in\Bbb Z\} $$
Breaking it up one level, we have that
$$z\Bbb Z = \{z\cdot k\mid k\in\Bbb Z\}$$
And the "sum" of tho sets are all possible sums of their elements:
$$A+B=\{a+b\mid a\in A, b\in B\}$$
Notice that this notation can lead to counter-intuitive results like $ 2\Bbb Z \neq \Bbb Z + \Bbb Z = \Bbb Z$ or $ \Bbb Z = \Bbb Z - \Bbb Z \neq 0\Bbb Z = \{0\}$.
Also notice that sometimes, "−" (minus sign) is used to denote set-theoretic difference of two sets, and then
$$A-B := A\setminus B$$
where $A-B$ does not denote all possible differences of elements from $A$ and $B$.
And presumably, the notations look more like "$\Bbb Z\cdot z$" than "$\Bbb Z.z$".