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Distribution Theory of Algebraic Numbers

Pei-Chu Hu and Chung-Chu Yang
Walter De Gruyter
Publication Date: 
Number of Pages: 
De Gruyter Expositions in Mathematics 45
[Reviewed by
Michael Berg
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When some one comes to me with the hackneyed good news / bad news option, I always go for the bad news first, probably because I believe in happy endings. Thus as regards Hu and Yang’s Distribution Theory of Algebraic Numbers, let me go with the bad news first: in places, particularly in the Preface, it appears that de Gruyter’s assigned prose-editor went AWOL. With English being the lingua franca of modern mathematics (pace Paris, I fear), Hu and Chang are certainly right to write this important book in English, but dealing with a high density of grammatical errors already on pp. v–vi is off-putting. Also, in other places in the book, notably in their comments rather than in their development of mathematical arguments per se, articles tend to go missing, run-on sentences show up, and verb forms get abused. Happily I did not find any passages where the indicated meaning was obliterated, but, given the beauty and promise of the mathematics dealt with in these pages, faulty prose is a very unfortunate dissonant.

But now for the good news. The message of Distribution Theory of Algebraic Numbers is very exciting, indeed. In fact, it is irresistible to any number theorist with a taste for exploring the frontier guided by suggestive analogies and considerations of structure. Hu and Yang suggest as their main thesis that Diophantine questions, or, more specifically, Diophantine approximation, without doubt a theme at the very center of number theory, is comparable (at a deep level) to Nevanlinna theory.

Coming from analysis, Nevanlinna theory has to do with the distribution of the values of holomorphic and meromorphic functions (à la the Big Picard Theorem, and then some). Thus, there is the so-called deficiency index δf, which is characterized, for a suitable function f: CCP1, as a normalized way of counting points in f–1(a). We have δf ∊ [0,1] and if f-1(a) = Ø then δf(a) = 1 (cf. the online notes for more detail). Two of the principal results in value distribution theory state that if f is nonconstant holomorphic then δf = 0 almost everywhere and the sum of the δf’s for a ranging over all of CP1 is at most 2.

Now the very first point made in the Preface of Distribution Theory of Algebraic Numbers is that C. F. Osgood observed that this last-mentioned 2 of Nevanlinna theory is similar (?!) to the 2 in Roth’s Theorem. And what does Roth’s Theorem say? Well, it’s actually Thue-Siegel-Roth, to old(er)-timers like me, and it is the marvelous statement that if a is algebraic then, for all ε > 0, there are only finitely many fractions p/q within q–(2+ε) of a. Very, very evocative! Reading the book to find out why one 2 is so much like the other 2 is just the sort of thing some arithmeticians (like me) can’t resist. There’s a lot hiding in the shadows.

And things really take off from here. Historically speaking, Serge Lang (r.i.p.) soon entered the game, P. Vojta went on to compare the arithmetical theory of heights with the theory of characteristic functions à la Nevanlinna, and, in due course, the famous abc-conjecture appeared: cf. p. 306: “… influenced by Strothers-Mason’s theorem… Oesterlé and Messer formulated the conjecture as follows… Given ε > 0, there exists a number C(ε) [such that] [f]or all nonzero relatively prime integers a, b, c, such that a + b = c, we have that max{|a|, |b|, |c|} ≤ C(ε) r(abc)1+ε…” Here r is the radical function, defined as the product of the distinct primes dividing its input.

Tellingly, the function theoretic counterpart due to Strothers and Mason (cf. p. 304) states that if a(x), b(x), and c(x) are polynomials over C, with again a + b = c, and if deg* counts the number of distinct roots of its polynomial input, then max{deg(a), deg(b), deg(c)} ≤ deg*(abc) – 1. “Watson, the parallel is exact!”

Hu and Yand go on to quote Lang’s classic Algebra: “One of the most fruitful analogies is that between the integers Z and the ring of polynomials F[t] over a field F.” Truer words were never spoken.

The abc-conjecture is of course famous for being a sufficient hypothesis for a slew of hypersexy conclusions. It implies, for examples, Thue-Siegel-Roth, Fermat(-Wiles), and Mordell(-Faltings). Truly this is a giant among open problems! It doesn’t get any sexier than this, at least in number theory (no! — in all of mathematics!), this side of the Riemann Hypothesis.

There is a lot of important methodology, for lack of a better word, in the book under review, in preparation for the latter half — and rightly so. After a compact but thorough reprise of algebraic number fields, Hu and Yang hit algebraic geometry, to the extent it is needed for this kind of arithmetic. Varieties, linear systems, curves, sheaves, schemes, and Kobayashi hyperbolicity are dealt with, and then it’s on to height functions and Arakelov theory.

All of this sets the stage for the last five chapters of the book, which contain exquisitely beautiful material: abc, Thue-Siegel-Roth, Vojta’s conjectures, and (yes!) L-functions.

Obviously this is all too good to pass up. Factoring out by the aforementioned “bad news” (which, happily, does no mathematical damage) the authors present us with fine scholarship, developed cogently. I think that Distribution Theory of Algebraic Numbers promises to be an important and useful contribution to the literature, especially suited to ambitious youngsters with good taste and no small measure of daring.

Michael Berg is professor of mathematics at Loyola Marymount University in Los Angeles, CA.