THE HEIGHT OF ALGEBRAIC UNITS IN LOCAL FIELDS*

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1 THE HEIGHT OF ALGEBRAIC UNITS IN LOCAL FIELDS* CLAYTON PETSCHE Abstract. Given a number field k and a non-archimedean place v of k, we give a quantitative lower bound on the height of non-torsion algebraic units α Q, all of whose conjugates over k lie in the completion k v of k at v. As an application, we resolve Lehmer s problem for those algebraic units α for which a small prime splits completely in Q(α). 1. Introduction Let k be a number field, let v be a place of k, and let α Q be a nonzero algebraic number with monic minimal polynomial f(x) k[x] over k. We say that α is totally v-adic if f(x) splits completely over the completion k v of k at v. If v is a finite place corresponding to a prime ideal P O k, then an equivalent condition is that P splits completely into [k(α) : k] distinct prime factors in the ring of integers O k(α). When k = Q and v =, the archimedean place of Q, one would usually say that α is totally real. Let h : Q [ 0, ) denote the logarithmic Weil height, and for each place v of k, define σ(k, v) = inf h(α), where the infimum is taken over all non-root of unity, totally v-adic α Q. Define the similar quantity τ(k, v) = inf h(α), this time taking the infimum over all non-root of unity, totally v-adic algebraic units α Q. Thus 0 σ τ. If v is a complex place, then σ(k, v) = τ(k, v) = 0, since k v C is algebraically closed. In fact the converse of this last statement is true; that is (1) σ(k, v) > 0 for any non-complex place v. The first result in this direction was the sharp result of Schinzel [6], stating in our notation that (2) σ(q, ) = τ(q, ) = 1 ( ) log. 2 *This is a short paper I wrote up in I never got around to submitting it for publication, and anyway it was quickly thereafter superseded by Dubickas-Mossinghoff, Auxiliary polynomials for some problems regarding Mahler s measure, Acta Arith. 119 (2005), no. 1,

2 2 CLAYTON PETSCHE In other words, the golden number φ = is the smallest nonzero, nonroot of unity totally real algebraic number. The positivity of σ(k, v), for noncomplex places v, was first established for finite places p of Q by Bombieri- Zannier [3], and in full generality in Baker-Hsia [1], via their generalizations of Bilu s equidistribution theorem [2]. One drawback of these results, with the exception of Schinzel s, is that the positivity of σ is established purely qualitatively, and that no general explicit lower bounds seem to be available by present techniques. In this paper we give a quantitative lower bound on the (potentially) larger number τ(k, v) for finite places v. Theorem 1. Let k be a number field. Then for finite places v of k we have (3) τ(k, v) [k : Q] 1 log{2 [k:q] q v }, q v 1 where q v denotes the absolute norm of v. Notice that the lower bound (3) is only nontrivial when q v > 2 [k:q]. Of course for fixed k this accounts for all but finitely many of the nonarchimedean places of k. However, in the case k = Q, a special argument at the prime p = 2 gives a nontrivial lower bound for all finite places of Q. Theorem 2. For all rational primes p we have { log(p/2) (4) τ(q, p) p 1 if p 2 log 2 if p = 2. The lower bound (4) is asymptotically best possible as p, as the following upper bound on τ shows. Theorem 3. For all primes p 3 we have (5) τ(q, p) log {( p + p ) /2 } p 1 In particular, combining (4) and (5) we obtain the asymptotic formula τ(q, p) log p p 1 as p. These results have an application to Lehmer s problem [5], the statement of which we now recall. For a nonzero polynomial d d (6) f(x) = a j x j = a d (x α j ) C[x], j=0 j=1 we define its logarithmic Mahler measure m(f) by (7) m(f) = 1 0 log f(e 2πit ) dt = log a d +. d log + α j. j=1

3 THE HEIGHT OF ALGEBRAIC UNITS IN LOCAL FIELDS 3 Here log + t = max(0, log t), and the second equality in (7) is Jensen s formula. Lehmer s problem is to decide whether or not there exists an absolute constant λ > 0, such that m(f) λ for all nonzero, non-cyclotomic irreducible polynomials f(x) Z[x] \ {±x}. The following result resolves Lehmer s problem for a certain special class of polynomials. Theorem 4. There exists an absolute constant c > 0 with the following property. If f(x) Z[x] \ {±x} is an irreducible, non-cyclotomic polynomial of degree d that splits completely over Q p, if T 1, and if (8) p T d log d, then m(f) ct 1. This improves upon a result of Silverman [7], who obtained by different methods an absolute lower bound on m(f) with the stronger condition p d log d in place of (8). See also [10]. An argument similar to that used in the proof of Theorem 1 occurs also in [4]. The author would like to thank Jeff Vaaler, who proposed this problem, and who simplified and sharpened the upper bound (5). 2. Absolute Values and the Weil Height Let k be a number field, and at each place v of k, let k v denote the completion of k at v. Then v p [k v : Q v ] = [k : Q] for all places p of Q. We will use two normalized absolute values v and v on k, which we now define. If v, then v restricted to Q is the usual archimedean absolute value; and if v p for a rational prime number p, then v restricted to Q is the usual p-adic absulute value. We then set v = [kv:qv]/[k:q] v. If v is a finite place of k dividing the rational prime p, we let O v = {x k v x v 1}, and M v = {x k v x v < 1} denote the ring of integers of k v and maximal ideal of O v, respectively. The absolute norm q v and residual degree f v are defined by q v = [O v : M v ] = p fv. Thus q v is the cardinality of the finite residue field O v /M v. We let π v M v denote a uniformizer; that is, an element such that M v = π v O v, and therefore π v v = qv 1/[k:Q]. With these normalizations, we have the product formula: if x k, then x v = 1, where the product is taken over all places v of k. v

4 4 CLAYTON PETSCHE The logarithmic Weil height h : Q [0, ) is now defined as follows. Given α Q, select any number field k containing α, and let (9) h(α) = v log + α v, the sum being taken over all places v of k. The height does not depend on the choice of k containing α. We point out that if α is an algebraic unit, then α v = 1 for all finite places v, and therefore the sum (9) can be taken over the infinite places only. We write H = exp(h) for the multiplicative Weil height. Finally, let α Q have degree d = [Q(α) : Q] over Q, and minimal polynomial f(x) Z[x] over Z (so f(x) vanishes at α, is irreducible over Q, and has relatively prime integer coefficients). Then we have (10) h(α) = 1 d m(f), where m(f) is the Mahler measure of f(x), defined as in (7). This is well known; see for example [8], Lemma Proofs of the Main Results Proof. Proof of Theorem 1 Fix a finite place v of k, and let α be a non-root of unity, totally v-adic algebraic unit, with monic minimal polynomial f(x) over k. Put L = k(α), and consider the number If w is an infinite place of L, we have and hence = α qv 1 1 L. w = α qv 1 1 w 2 max{1, α w } qv 1, w = 2 [Lw:R]/[L:Q] max{1, α w } qv 1. Applying this estimate to all of the infinite places w of L, we obtain w 2 [Lw:R]/[L:Q] max{1, α w } qv 1 (11) w w = 2H(α) qv 1, using the fact that α is a unit, and that the sum of the local degrees is the global degree. Now consider the places w of L dividing v. Since f(x) splits completely in k v, it follows that the place v splits completely in L. That is [L w : k v ] = 1 for each of the [L : k] places w v of L. We then have the equality f w = f v of residual degrees and q w = q v of absolute norms, and therefore q v 1 is the

5 THE HEIGHT OF ALGEBRAIC UNITS IN LOCAL FIELDS 5 order of the multiplicative group of the residue field O w /M w. Since α is a unit, we have α / M w, and therefore α qv 1 1 (mod M w ). It follows that w = α qv 1 1 w π w w = q 1/[L:Q] w = q 1/[L:Q] v, and combining all of the places w v of L, we obtain w qv [L:k]/[L:Q] (12) w v = qv 1/[k:Q]. Since is integral, w 1 at any finite place. Therefore, combining (11) and (12) with the product formula, we have 1 = w w 2q 1/[k:Q] v H(α) qv 1. Taking logarithms, we have the promised lower bound on h(α). Proof of Theorem 2. If p is odd, this is exactly (3). The special lower bound on τ(q, 2) is proved as follows. Let α be a nonzero, non-root of unity, totally 2-adic algebraic unit, and let L = Q(α). In this case we put = α 2 1 L. Arguing similarly to the proof of (11), we have w w 2H(α) 2, the product being over all infinite places of L. Since α is totally 2-adic, 2 splits completely in L. It follows that f w = 1 for each place w 2 of L, meaning q w = [O w : M w ] = 2 for these places. Thus α ± 1 0 (mod M w ), and therefore we have w = α 2 1 w = α + 1 w α 1 w π w 2 w = { q 1/[L:Q] w = 4 1/[L:Q]. Taking the product over all [L : Q] places w 2 of L, we have w 4 1. w 2 } 2

6 6 CLAYTON PETSCHE Since w 1 for any finite place, combining our estimates for all places of L, we have 1 = w w 4 1 2H(α) 2 = 2 1 H(α) 2. Taking logarithms, we have h(α) log 2. Proof of Theorem 3. Factor the polynomial f(x) = x p 1 px (p 1)/2 1 = i g i (x) into its irreducible factors g i (x) in the ring Z[x]. By Gauss Lemma, the roots of f(x) are algebraic units; and by Hensel s Lemma, f(x) splits completely over Q p. Therefore, if f(x) vanished at a root of unity, it must be a (p 1)- st, which is visibly impossible. From these considerations it follows that each g i (x) is the minimal polynomial over Z of a nonzero, non-root of unity, totally p-adic algebraic unit, say α i. As such, by (10) we have m(f) = i = i m(g i ) h(α i ) deg(g i ) τ(q, p) i deg(g i ) = τ(q, p)(p 1), and therefore τ(q, p) m(f)/(p 1). We now invoke the fact that Mahler measure is invariant under the change of variables x x n for any integer n. Therefore, using (7), we have ( p + p m(f) = m(x 2 px 1) = log 2 ) + 4, 2 and the upper bound follows. Proof of Theorem 4. Assume that f(x) is given by (6). If a d a 0 ±1, then m(f) log 2, so we may therefore assume that a d a 0 = ±1. Thus f(x) is the minimal polynomial over Z of a non-root of unity algebraic unit α Q.

7 THE HEIGHT OF ALGEBRAIC UNITS IN LOCAL FIELDS 7 Now by assumption f(x) splits completely over Q p, and p T d log d. The function t log(t/2) t 1 is positive and decreasing on [7, + ). If p < 7, then m(f) = dh(α) min{τ(q, 2), τ(q, 3), τ(q, 5)} 1 T 1. On the other hand, if p 7, then completing the proof. m(f) = dh(α) dτ(k, p) d log(p/2) p 1 log(t d log d/2) d T d log d 1 T 1, References [1] M. Baker and L.-C. Hsia. Canonical heights, transfinite diameters, and polynomial dynamics. To appear in J. Reine Angew. Math. [2] Y. Bilu. Limit distribution of small points on algebraic tori. Duke Math. J., 89: , [3] E. Bombieri and U. Zannier. A note on heights in certain infinite extensions of Q. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 12: 5 14, [4] T. Callahan, M. Newman, and M. Sheingorn. Fields with large Kronecker constants. J. Number Theory, 9: , [5] D. H. Lehmer. Factorization of certain cyclotomic functions. Ann. of Math., 34: , [6] A. Schinzel. On the product of the conjugates out side the unit circle of an algebraic number. Acta Arith., 24: , [7] J. H. Silverman. Lehmer s conjecture and primes of small norm. unpublished manuscript, [8] M. Waldschmidt. Diophantine approximation on linear algebraic groups. Grundlehren der Mathematischen Wissenschaften, 326. Springer-Verlag, Berlin, [9] A. Weil. Basic Number Theory. Springer-Verlag, Berlin-Heidelberg-New York, [10] F. F. Zheludevich. Local estimates in the Lehmer problem. (Russian) Acta Arith. 57: , Department of Mathematics; The University of Texas at Austin; Austin, TX 78701

Auxiliary polynomials for some problems regarding Mahler s measure

ACTA ARITHMETICA 119.1 (2005) Auxiliary polynomials for some problems regarding Mahler s measure by Artūras Dubickas (Vilnius) and Michael J. Mossinghoff (Davidson NC) 1. Introduction. In this paper we