The most elusive and tantalizing problem in all of mathematics is the Riemann Hypothesis: I say this both as an opening salvo to start spats among the readers of this review - and even arguments, to be sure - but also because I think it’s a claim that can be defended (of course, I am a number theorist, so there may be some bias in this…).  In any case, its history certainly is spectacular, with Riemann revealing something utterly sublime, namely that in a profound sense the deep behavior of the primes, i.e. their distribution among the positive integers, at first sight an impenetrable business, is in fact encoded in what Riemann’s assertion claims: the nontrivial zeroes of the zeta function all have real part equal to ½.  For a quasi-popular account of this fact, see Du Satoy’s book, The Music of the Primes.  Ever since 1859 when Riemann phrased it, this assertion has captivated number theorists – as well as others: before he died, Atiyah was working on it.  However, lest we indulge ourselves (or I do so) and go off into a contemplation of this particular theme, let’s assume a somewhat different perspective relevant to the book under review, viz. the bigger picture, that of L-functions, of which the Riemann zeta function is the simplest qua form, and of course, the most famous one, given the promise it holds.

The zeta function made a spectacular early appearance in Euler’s proof of the infinitude of the primes, which fact reduces to the divergence of the harmonic series, or equivalently, to the singularity zeta, as a meromorphic complex function, has at 1.  This seminal proof also introduces the now-ubiquitous notion of the Euler product.  If we go to the generalization of zeta, i.e. to L-functions, these themes are still present, indeed central.  Over twenty years prior to Riemann’s claim, Dirichlet proved, using L-functions whose numerators are Dirichlet characters, that any arithmetic progression on coprime generators contains infinitely many primes.  What a beautiful result!

Thus, Euler, Dirichlet, Riemann: what a spectacular cast!  But 19^th century number theory, reaping Euler’s 18^th century pioneering work - think of the quadratic reciprocity law: Euler’s results can be combined to prove it, but the one who did the most with it was Gauss who gave six separate complete proofs (if memory serves) – grew prodigiously.  Gauss famously crowned number theory the queen of mathematics, and many 19^th century scholars set out to serve her: Eisenstein, Dirichlet, Riemann of course, Dedekind, Kummer, Kronecker, and as a figure spanning the late 19^th and early 20^th centuries, Hilbert.  It is fair to say that the thrust of 19^th century number theory was the theory of algebraic number fields, including in particular class field theory, one of the gems par excellence of Hilbert’s famous 1897 Zahlbericht which, like so much of what Hilbert did, defined much of the subject and what was to be done in it. 

And now we find ourselves face-to-face with what has, during the 20^th century, become a very well defined, if highly ramified (if you’ll excuse a pun), subject.  It is this marvelous business that forms the subject of the wonderful book under review.  Sorensen goes from Riemann and Dirichlet (as per the above themes) to Dedekind and then jumps (with class field theory in his pocket: he recommends Neukirch’s book, with which I am very much in agreement) to that foundation of modern algebraic number theory, Tate’s thesis, introducing harmonic analysis into number theory.  Written under E. Artin, Tate’s famous thesis presents the local vs. global perspectives (ideles and adeles, the latter supposedly named after an early girlfriend of Andre Weil), and all this forms the needed prelude to Hecke characters and Hecke L-functions, Galois characters, and Galois representations—all wonderful stuff!  I remember learning this material in my youth (1970’s, 1980’s) at UCLA and UCSD (where Sorensen holds forth), and how wonderful it all was.  Coincidentally, my transition to UCSD’s PhD program also saw the sharpening of my focus on what was all the rage that day, modular forms, and in his book, Sorensen’s coverage engenders a transition from the (highly evolved) Artin L-functions to modular forms and (certain) Galois representations.  Of course, these last-mentioned faunae should ring any number of bells: together with elliptic curves, they comprise the main players in Wiles and Taylor’s proof of Fermat’s Last Theorem.  (See the great documentary, The Proof, for a star-studded account.)  It bears mentioning explicitly that in the last chapter Sorensen trains his focus on what is probably the hottest theme in contemporary number theory, the Langlands program.  À propos, here is how Sorensen introduces this chapter: “In early January of 1967 Langlands wrote a 17 page letter to Weil, which laid out one of the most influential and exhilarating programs in mathematics,” and then Sorensen goes on to quote Langlands’ comment to Weil that “if you are willing to read it as pure speculation I would appreciate that; if not - I am sure you have a waste basket handy."  It is in this context, by the way, that what Sorensen means by “reciprocity” occurs: it’s the Langlands reciprocity conjecture, pitting a certain irreducible Galois representation against a cuspidal automorphic representation whose L-function coincides with a particular Artin L-function.  The more things change, the more they stay the same.

So, I really like this book and recommend it enthusiastically.  Three final points (including a quibble): 

First, the book is very well and carefully written, is full of very beautiful mathematics, and when studied carefully, will teach the diligent student a huge amount of very serious mathematics, overflowing the boundaries of strict number theory properly so-called.  But then, it’s always been this way.  Encore: le plus ça change ….

Second, it is a huge boon (and something difficult to do: Sorensen does a great job) to fit all of this material into a coherent historical framework, and do so in a balanced manner.  One way he achieves this, very effectively, is by means of his opening sections to the individual six chapters.  In addition to the blurb I quoted above, from Chapter 6, I recommend Sorensen’s introduction to Chapter 2, on Tate’s Thesis: it’s both compact and complete.  Read it first when you get your copy of the book.

Qua lay-out, it’s quite good: great proofs, good examples, lots of exercises.  But here’s my quibble: why is there no index?  There really should be one, I think, although sleuthing through the book, looking for something, does have its own pedagogical value.

And a personal note: it makes me happy, as someone who got his PhD at UCSD over forty years ago, that my alma mater sports a course like the one Sorensen teaches, and the university can boast of the book he’s written as an anabolic steroid enhanced outgrowth of his lecture notes: what a great course it must be.

Michael Berg is an emeritus professor at Loyola Marymount University, retired after three decades of teaching mathematics, some Thomist philosophy, and judo (eventually from his home dojo).  He continues his long-time research in the area of analytic methods in number theory, focusing in particular on higher reciprocity laws for number fields.