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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.59 Modular Forms

Kevin Buzzard

1 A Lattice in the Complex Numbers

When one first learns about the complex numbers, one is taught to think of them as a two-dimensional space, with one real and one imaginary dimension: a complex number z = x + iy has real part x and imaginary part y, where i is a square root of - 1.

Now let us consider what the complex numbers that have integers for their real and imaginary parts look like. These complex numbers, such as 3 + 4i or -23i, form a “lattice” in the complex plane (see figure 1).

By definition, every element of this lattice is of the form m + ni for some pair of integers m and n. We say that the lattice is generated by 1 and i, and use the notation + i for it. Note that this lattice can be generated ...

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