This book is devoted to giving an account of the arithmetic theory of the moduli spaces of elliptic curves. The main emphasis is on understanding the behavior of these moduli spaces at primes dividing the "level" of the moduli problem being considered. Until recently, this seemed a very difficult problem, because one had no apriori construction of these spaces at the "bad" primes. One defined them as schemes over, say, Z[1/N] as the solution to some well-posed moduli problem which only made sense for elliptic curves over rings in which N was invertible, and then one used a process of normalization to extend them to schemes over Z, e.g., one took the "proj" of the graded subring of the ring of all modular forms of the type in question consisting of those with intgeral Fourier ("q-expansion") coefficients at the cusps. This procedure produced a scheme over Z, but one had no idea of what meduli interpretation this scheme had, nor afortiori did one have any idea of the modular interpretation of its reduction modulo p, for p a prime dividing the level.
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