Numbers and functions. Introduction to Vojta s analogy Seminar talk by A. Eremenko, November 23, 1999, Purdue University. Absolute values. Let k be a field. An absolute value v is a function k R, x x v with the following roerties: 1. x v 0, and x v = 0 iff x = 0, 2. xy v = x v y v, 3. x + y v x v + y v 2 max{ x v, y v }. If 3 is relaced by the stronger roerty 3. x + y v max{ x v, y v }, then v us called non-archimedian. For an absolute value v we ut Then v : k R { }, v(x) = log x v. v(xy) = v(x) + v(y), and { min{v(x),v(y)} in non-archimedian case, and v(x + y) min{v(x), v(y)} log 2 in archimedian case. Examle 1. (Classical). For k = Q, x x is the usual, archimedian absolute value. For each rime we can write every rational number as x = s m/n, where m and n are not divisible by. Then x = s, v (x) = s log (1) is called -adic absolute value. U to roortionality of v-functions, these are the only absolute values in Q (A. Ostrowski s theorem). We have x v = 1 or v(x) = 0 for every x. v v This is called the Artin Whales Product Formula. For the filed Q it is equivalent to Euclid s theorem, that every rational number is a roduct of owers of rimes: x = ± s j j, x v = x v v 1 x v = x s j j = 1.
Examle 2. (Classical). For k = M(C), the field of meromorhic functions on the Riemann shere C, and f k, f = e (f), where (f) is the multilicity of zero of f at C (it is negative if is a ole; (0) = ). This time the Product Formula is (f) = 0. Each absolute value defines a metric and thus a toology on k. Equivalent toologies corresond to roortional functions v. One also considers comletion of a field with resect to an absolute value. Thus in Examle 1 we obtain R as the comletion of Q with resect to, and Q, the -adic numbers fields. In Examle 2 we obtain for each the field of formal Laurent series (with finitely many negative owers) at. Absolute values can be extended to algebraic extensions of a field. In Examle 2 one obtains the Puiseaux series in this way. Examle 3. (Vojta). Let f be a meromorhic function in z r with f(0) = 1. Then, for θ [ π,π) f r,θ = f(re iθ ) is an archimedian absolute value (well, almost; the second art of condition 1 is not satisfied), and for each C, < r: f r, = (f), vr, = (f) log r r is a non-archimedian absolute value (comare with (1)). To write the Product Formula, we have to average over the infinitely many archimedian absolute values: (f) log r + 1 π log f(re iθ ) dθ = 0. 2π π <r Another name for this is Jensen s formula. Heights in rojective saces. Let P n be the n-dimensional rojective sace over k, that is the set of non-zero vectors (x 0,...,x n ) k n+1 u to 2
roortionality. The (logarithmic) height is a function h : P n R defined for a oint P = (x 0 :... : x n ) P n by h(p) = v log max x j v = j v min j v(x j ), where the summation extends to a set of absolute values, satisfying the Product Formula. It follows from the Product formula that h is well defined. Examle 1, again. If k = Q, we use P = (m 0 :... : m n ), where m j are integers without a common factor, and obtain h(p) = log max{ m 0,..., m n } + log max{ m 0..., m n } = max{log m 0,...,log m n }, because max{ m 0,..., m n } = 1 for each rime, because our homogeneous coordinates m j have no common factor. Examle 2, again. If k = M(C), the field of rational functions, a oint P P n can be written as P = (f 0 :... : f n ), where f j are olynomials without a common factor, and again only the summand with = is different from zero: h(p) = log max{ f 0,..., f n } = max{deg f 0,...,deg f n }. In articular, h(x) for x k is the degree of a rational function x. Examle 3, again. Take k = C and consider a holomorhic curve f : B(r) P n, where B(r) = {z C : z r} and P n is comlex rojective sace. We write f = (f 0 :... : f n ), where f j are holomorhic functions without common zeros. Then the logarithmic height should be h f (r) = v log max{ f 0 v,..., f n v } = v S log max{ f 0 v,..., f n v } = 1 π log max{ f 0 (re iθ ),..., f n (re iθ ) }dθ, 2π π where S = {v = (r, θ) : θ π} is the set of archimedian absolute values. Another name for this is the Nevanlinna Cartan characteristic of a holomorhic curve, T f (r). When n = 1 one obtains the usual Nevanlinna or 3
Ahlfors characteristic. f(0) = 1). Lang writes: (They differ by a bounded term, but coincide if The CR note where Cartan announced his Second Main Theorem for the case of hyerlanes was ublished in 1929, essentially at the same time as Weil s thesis in 1928, where he uses the height which today bears his name and which was defined simultaneously by Siegel [1929]. But no one at the time saw that Cartan s definition of this height was entirely analogous to the definition of the heights in algebraic number theory, and that both were based on the roduct formula. This ga in understanding is, to me, almost as striking as the ga in understanding between Artin and Hecke in Hamburg about the connection between nonabelian L-series and modular forms. One had to await 40 to 50 years for the connections to be made, conjecturally, by Langlands and Vojta resectively in these two cases. In both cases, some algebraic number theorist s failure to relate roerly to analysis (and conversely) contributed to that ga of understanding. Thue Siegel Roth Theorem. Usual formulation: for every algebraic number x, and all but finitely many rational numbers P = m/n we have or, using our notation, x y max{m,n} 2+ǫ, log x y (2 + ǫ)h(y), excet for finitely many y. The following generalization to -adic absolute values is due to Mahler: let k be a number field (= finite extension of Q), S a finite set of absolute values, and ǫ > 0. For each v S choose a v Q. Then the inequality v + (y a v ) (2 + ǫ)h(y) v S holds for all but finitely many y k. Notice that all but finitely many is the same as all but a set of bounded height. 4
Weak form of the Second Main Theorem of Nevanlinna. For every non-constant meromorhic function f in C, for every finite set of constants {a}, for every ǫ > 0 we have = a 1 2π a π π 1 2π π π log + f(re iθ ) a 1 dθ v + r,θ (f a)dθ (2 + ǫ)h f(r), (2) where r / E f (ǫ), an excetional set of finite length. Notice that a nonconstant meromorhic function is the same as a meromorhic function of bounded height. We call this form of the Second Main Theorem weak for two reasons. One is that one can write better error term, then ǫh f (r). But more imortant omission in (2) is the ramification term N 1,f (r), which counts the total multilicity of a-oints in B(r). In articular, the Second Main Theorem with the ramification term imlies: for three non-roortional entire functions f 0,f 1,f 2 without common zeros, and with f 0 + f 1 + f 2 = 0 1 2π π π log max{ f 0, f 1, f 2 }(re iθ )dθ (1 + o(1))n(r,f 0 f 1 f 2 ), r / E where N is the usual averaged counting function of different zeros, and E is a set of finite length. Examle 1, again. Let us denote by n(m) the number of different rimes which divide an integer m. The following statement is known as the abcconjecture of Masser and Oesterlé: for every ǫ > 0 there exists C(ǫ), such that for any non-zero relatively rime integers a, b, c with a + b + c = 0 max{ a, b, c } C(ǫ) (n(abc)) 1+ǫ. This easily imlies that each Fermat s equation has only finitely many solutions, but unlike the Fermat Theorem, abc is still a conjecture. Examle 2, again. Let us denote by n(f) the number of different zeros of a olynomial f. The following statement is known as Mason s theorem: for any relatively rime non-roortional olynomials a, b, c, with a + b + c = 0 Lang writes: max{deg a, deg b, deg c} n(abc) 1. 5
Mason started a trend of thought by discovering an entirely new relation among olynomials, in a very original work as follows... HW Exercise: Derive Mason s theorem from the Riemann Hurwitz formula. Aendix (not included in the seminar talk) A1. The following generalization of the Thue Siegel Roth Theorem to several dimensions was obtained by Schmidt. For a hyerlane H P n given by a 0 x 0 +... + a n x n and an absolute value v we define the Weil function λ v,h (P) = log a 0x 0 +... + a n x n v max{ x 0 v,..., x n v }, P = (x 0 :... : x n ). Let k be a number field, S a finite set of absolute values, H 1,...,H q, q n+2 hyerlanes in general osition on P n, and ǫ > 0. Then q λ v,hj (P) (n + 1 + ǫ)h(p), j=1 v S excet those P lying in a finite collection of hyerlanes (deending on {H j } and ǫ.) The corresonding statement for holomorhic curves is called Cartan s Second Main Theorem (in an imroved form, due to Vojta): Let H 1,...,H q, q n + 2 be hyerlanes in general osition on P n, and ǫ > 0. Then q q 1 π m(r,h j,f) := λ (r,θ),hj (f)dθ (n + 1 + ǫ)h f (r), j=1 j=1 2π π for all non-constant holomorhic curves f : C P n, excet those which lie in a finite collection of hyerlanes (deending on {H j } and ǫ.) A2. A common use of the Second Main Theorem(s) is to rove Picardtye theorems, that certain equations do not have meromorhic solutions, or have very few of them. Similarly, theorems on diohantine aroximation can be used to rove that certain diohantine equations have few solutions. Thus the arithmetic counterart of the classical Picard s Theorem is the 6
theorem of Thue and Siegel, about integral oints on an affine curve of genus zero. Another examle is Borel s Theorem: if f 1,...,f q are zero-free entire functions, and f 1 +... + f q = 0, then some of these functions are roortional. The arithmetic counterart is due to Schlickewei. References [1] H. Cartan, Mathematica (Cluj) 7 (1933), 5-31. [2] A. Eremenko, Russian Math. Surveys 37:4 (1982) 61-95. [3] S. Lang, Algebra, 1965. [4] S. Lang, Fundamentals of Diohantine Geometry, Sringer, 1983. [5] S. Lang, Introduction to Comlex Hyerbolic Saces, Sringer, 1986. [6] S. Lang, Diohantine Geometry, in: S. Lang (ed.), Number Theory III. Encycl. Math. Sci., vol. 60, Sringer, 1991. [7] M. Ru, P. Vojta, Inv. math. 127 (1997) 51-65. [8] M. Ru, J. reine angew. Math. 442 (1993), 163-176. [9] H. Schlickewei, Acta Arithm. 33 (1977) 183-185. [10] H. Selberg, Math. Z. 31 (1930) 709-728. [11] P. Vojta, Diohantine aroximations and Value Distribution Theory, LNM, 1239, Sringer, 1987. [12] E. Wirsing, Proc. Sym. Pure Math. 20 (1968), 213-247. 7
A3. We summarize what was said (and a art of what was not) in the following table, artially taken from Vojta s book. Vojta s Analogy Function Theory Number Theory f : C P 1 {y} k r y θ [ π,π) a finite set S log f(re iθ ) a 1 v(y a), v S (f), C v(y), v / S Jensen s Formula Artin Whales Product Formula Proximity function Weil s function Nevanlinna s characteristic Height on k Cartan s characteristic Height on rojective sace Selberg Valiron characteristic Absolute height Nevanlinna s SMT Thue Siegel Roth Theorem Cartan s SMT Schmidt s Theorem and its imrovement by Vojta and its imrovement by Vojta Borel s Theorem van der Poorten s Theorem Selberg Valiron SMT Wirsing s Theorem Picard s Theorem Thue Siegel s Theorem Another Picard s Theorem Faltings Theorem Borel s Theorem Schlickewei s Theorem Precise error term in SMT Lang s conjecture SMT with the ramification term??? its corollary for 3 functions abc-conjecture?? (some arguments of Osgood?) Roth s roof?? Ahlfors roof of Cartan s SMT Schmidt s roof Cartan s and Nevanlinna s roofs??? (Lemma on log derivative)??? Curvature??? moving target generalizations even better analogy? 8