Editing Modular form (section)

===Definition in terms of lattices or elliptic curves===
A modular form can equivalently be defined as a function ''F'' from the set of [[period lattice|lattice]]s in {{math|'''C'''}} to the set of [[complex number]]s which satisfies certain conditions:

# If we consider the lattice {{math|Λ {{=}} '''Z'''''α'' + '''Z'''''z''}} generated by a constant {{mvar|α}} and a variable {{mvar|z}}, then {{math|''F''(Λ)}} is an [[analytic function]] of {{mvar|z}}.
# If {{mvar|α}} is a non-zero complex number and {{math|''α''Λ}} is the lattice obtained by multiplying each element of {{math|Λ}} by {{mvar|α}}, then {{math|''F''(''α''Λ) {{=}} ''α''<sup>−''k''</sup>''F''(Λ)}} where {{mvar|k}} is a constant (typically a positive integer) called the '''weight''' of the form.
# The [[absolute value]] of {{math|''F''(Λ)}} remains bounded above as long as the absolute value of the smallest non-zero element in {{math|Λ}} is bounded away from 0.

The key idea in proving the equivalence of the two definitions is that such a function {{mvar|F}} is determined, because of the second condition, by its values on lattices of the form {{math|'''Z''' + '''Z'''''τ''}}, where {{math|''τ'' ∈ '''H'''}}.