Elliptic Curves, Modular Forms, and Their L-functions (Student Mathematical Library) (Student Mathematical Library, 58)

Nice exposition but nothing original. Largely examples and treatment taken from other sources.

It's late at night, so I don't want to write a long review. I'm just here to support a great author and a great book. If you're looking for analytic number theory books, you should seriously consider buying this. I haven't found any other ANT book of this quality yet. In fact, this might be the best math book I've ever read.

A little complicated

It is always refreshing to find a book in mathematics that serves as a transition between very elementary treatments and books that emphasize formal developments. The discourse in the latter is rigorous, but usually fails to give the essential insight into the nature of the subject and why it is important. Transitional books, such as this one, give the motivation for the need for the more complicated formalism, and readers will be able to better appreciate the latter when they finally get to that level.Elliptic curves have to rank as the most beautiful objects in all of mathematics, and their applications are becoming so pervasive that even the practically-minded engineer or cryptographer has to understand their properties. The theory of elliptic curves is very rich, and much time and effort is needed to understand some of the important results in this theory, but this book serves as a useful tool in that regard.One of the main sticking points in the theory of elliptic curves is that there have been several "existence" theorems proven for their properties, but as typical in many areas of modern mathematics no "constructive" or "effective" procedures are given in the proofs for finding these properties explicitly. One well known example, and one that is discussed in the book, is Siegel's theorem, which shows that the collection of integral points of an elliptic curve over the rational numbers is finite. The proof of Siegel's theorem is not given in the book, but the author alludes to a result of the mathematician Alan Baker, who proves a bound on the size of this collection. Baker's proof is also omitted in the book.There is an analog of Siegel's theorem for elliptic curves defined over function fields, rather than number fields as done in this book. Such points are sometimes called 'S-integers', and it is an interesting fact that the proof of this analog is effective, dealing as it does with the theory of heights, which are briefly discussed in this book. As the author remarks, Siegel's theorem is based on Diophantine approximation and a theorem of the mathematician Klaus Roth, which gives a bound on the "approximation exponent" of a number field. The author with some justification does not want to go into the intricacies of Diophantine approximation, but readers who decide to go to more advanced studies of elliptic curves will find that Baker's bound boils down to finding explicit lower bounds for sums of logarithms, and these bounds are expressed in terms of logarithmic height functions.Of particular importance and given a succinct discussion in this book is the method of descent, which is used to find the points of infinite order on a rational elliptic curve. The author's explanation of descent revolves around showing that there is an (algebraic) equivalence, or homomorphism, between points on the elliptic curve and rational numbers modulo squares. This is proved in detail in the book, and in fact a connection is made to the prime divisor of the 'discriminant' of the elliptic curve. The weak Mordell-Weil theorem, which proves the finiteness of E(Q)/2E(Q), follows from all this analysis.One topic in the book that is given a very clear discussion is that of homogeneous spaces. The author motivates their use via explicit examples, and the reader can see clearly how homogeneous spaces are essentially auxiliary curves that can be used to find rational points of elliptic curves. One issue that immediately arises in the use of homogeneous spaces is that such spaces may have rational points in every completion of the number field over which the elliptic curve is defined, but still fail to have a rational point over the original number field. This is encapsulated in the statement of the "failure of the Hasse principle", and this failure is quantified by use of the Tate-Shafarevich group. The author defines this group and the related 2-Selmer group in the book. Both of these groups are homogeneous spaces and are used to measure the failure of the method of descent. The author treats the special case where the elliptic curve has four distinct rational points with torsion equal to two.Readers who decide to go on with their study of elliptic curves will discover that finding representations of elements of the Selmer group as explicit covering spaces of elliptic curves is of great value in the search for rational points. In particular, the computation of the n-Selmer group for general n occupies much space in the research literature, and is related to the topic of Galois cohomology. The elements of the Tate-Shafarevich group will be represented by principal homogeneous spaces or with points that are "everywhere locally". The failure of the Hasse principle is reflected in "nontrivial" elements of the Tate-Shatarevich group, namely those elements that are "everywhere locally" , but where there are no global points.

I got a different book. I don't know how the seller behaves like this. I wish to return it.