By G. H. Hardy

ISBN-10: 0521092272

ISBN-13: 9780521092272

There may be few textbooks of arithmetic as recognized as Hardy's natural arithmetic. considering its e-book in 1908, it's been a vintage paintings to which successive generations of budding mathematicians have became at first in their undergraduate classes. In its pages, Hardy combines the passion of a missionary with the rigor of a purist in his exposition of the basic principles of the differential and indispensable calculus, of the houses of countless sequence and of different subject matters related to the thought of restrict.

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**Example text**

2/10 of [BGR], it is reduced since the fiber product of the affinoid B determined by 7rn,k(T) in Yn and YE is an affinoid subdomain of B. Hence all these exponents are at most 1. Since (by the dimension argument used above) every 7rtj (T) divides the restriction of one and only one of the 7r,,,k (T) we see that the fiber product of Wn and Y1 equals Wt. 3 and hence is the hypersurface cut out by a Fredholm series G(T). Moreover, for each n we know there exists an element Hn(T) E A(Y,,) such that G(T)HH(T) = F(T).

Note: We have jumped the gun in our terminology for the above definition does not immediately allow us to see that C, is a curve. In this paper we shall not define the eigencurve of general tame level, and only concern ourselves with tame level N = 1. Thus, in what follows, we will drop the tag "of tame level N = 1" and refer to the eigencurve of tame level 1 simply as the eigencurve. 5 implies that Cped is the union of its irreducible components. Here is a statement of some of the main results of this paper.

This follows directly from the construction of D in Chapter 7 (Prop. 2) together with the fact that D '-' Cpr d (Theorem. 1). It follows from Theorem C that Cp is a curve in the sense that it is an equidimensional rigid analytic space of dimension 1. 1 below) we have: Theorem D. If two classical modular points of tame level N, if, if, E (Xp x A') (C,,) lie on the same irreducible component of the reduced eigen- curve, then f - f modulo the maximal ideal of OcP in the sense that their Fourier expansions are congruent modulo that maximal ideal (and their associated residual representations have equivalent semi-simplifications).

### A course of pure mathematics by G. H. Hardy

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