Heights on groups and small multiplicative dependencies

Vaaler, Jeffrey

doi:10.1090/S0002-9947-2013-06029-1

Heights on groups and small multiplicative dependencies
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by Jeffrey D. Vaaler PDF

Trans. Amer. Math. Soc. 366 (2014), 3295-3323 Request permission

Abstract:

We generalize the absolute logarithmic Weil height from elements of the multiplicative group $\overline {\mathbb {Q}}^{\times }/\mathrm {Tor}\bigl (\overline {\mathbb {Q}}^{\times }\bigr )$ to finitely generated subgoups of $\overline {\mathbb {Q}}^{\times }/\mathrm {Tor}\bigl (\overline {\mathbb {Q}}^{\times }\bigr )$. The height of a finitely generated subgroup is shown to equal the volume of a certain naturally occurring, convex, symmetric subset of Euclidean space. This connection leads to a bound on the norm of integer vectors that give multiplicative dependencies among finite sets of algebraic numbers.

References

Daniel Allcock and Jeffrey D. Vaaler, A Banach space determined by the Weil height, Acta Arith. 136 (2009), no. 3, 279–298. MR 2475695, DOI 10.4064/aa136-3-6
Francesco Amoroso and Sinnou David, Le problème de Lehmer en dimension supérieure, J. Reine Angew. Math. 513 (1999), 145–179 (French, with English summary). MR 1713323, DOI 10.1515/crll.1999.058
Francesco Amoroso and Evelina Viada, Small points on rational subvarieties of tori, Comment. Math. Helv. 87 (2012), no. 2, 355–383. MR 2914852, DOI 10.4171/CMH/256
A. Baker, Linear forms in the logarithms of algebraic numbers. IV, Mathematika 15 (1968), 204–216. MR 258756, DOI 10.1112/S0025579300002588
Daniel Bertrand, Duality on tori and multiplicative dependence relations, J. Austral. Math. Soc. Ser. A 62 (1997), no. 2, 198–216. MR 1433209, DOI 10.1017/S1446788700000768
Ethan D. Bolker, A class of convex bodies, Trans. Amer. Math. Soc. 145 (1969), 323–345. MR 256265, DOI 10.1090/S0002-9947-1969-0256265-X
Enrico Bombieri and Walter Gubler, Heights in Diophantine geometry, New Mathematical Monographs, vol. 4, Cambridge University Press, Cambridge, 2006. MR 2216774, DOI 10.1017/CBO9780511542879
E. Bombieri and J. Vaaler, On Siegel’s lemma, Invent. Math. 73 (1983), no. 1, 11–32. MR 707346, DOI 10.1007/BF01393823
B. Brindza, On the generators of $S$-unit groups in algebraic number fields, Bull. Austral. Math. Soc. 43 (1991), no. 2, 325–329. MR 1102085, DOI 10.1017/S0004972700029129
Yann Bugeaud and Kálmán Győry, Bounds for the solutions of unit equations, Acta Arith. 74 (1996), no. 1, 67–80. MR 1367579, DOI 10.4064/aa-74-1-67-80
N. L. Carothers, A short course on Banach space theory, London Mathematical Society Student Texts, vol. 64, Cambridge University Press, Cambridge, 2005. MR 2124948
J. W. S. Cassels, An introduction to the geometry of numbers, Die Grundlehren der mathematischen Wissenschaften, Band 99, Springer-Verlag, Berlin-New York, 1971. Second printing, corrected. MR 0306130
L. Fuchs, Abelian groups, International Series of Monographs on Pure and Applied Mathematics, Pergamon Press, New York-Oxford-London-Paris, 1960. MR 0111783
L. Hajdu, A quantitative version of Dirichlet’s $S$-unit theorem in algebraic number fields, Publ. Math. Debrecen 42 (1993), no. 3-4, 239–246. MR 1229671
G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge U. Press, 1934.
L. Kronecker Zwei Sätz über Gleichungen mit ganzzahligen Coefficienten, J. Reine Angew. Math. 53 (1857), 173–175.
D. H. Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. (2) 34 (1933), no. 3, 461–479. MR 1503118, DOI 10.2307/1968172
Thomas Loher and David Masser, Uniformly counting points of bounded height, Acta Arith. 111 (2004), no. 3, 277–297. MR 2039627, DOI 10.4064/aa111-3-5
M. A. Lopez and S. Reisner, A special case of Mahler’s conjecture, Discrete Comput. Geom. 20 (1998), no. 2, 163–177. MR 1637864, DOI 10.1007/PL00000076
J. H. Loxton and A. J. van der Poorten, Multiplicative dependence in number fields, Acta Arith. 42 (1983), no. 3, 291–302. MR 729738, DOI 10.4064/aa-42-3-291-302
K. Mahler, Ein Minimalproblem für konvexe Polygone, Mathematica (Zutphen) B7 (1939), 118–127.
Kurt Mahler, Ein Übertragungsprinzip für konvexe Körper, Časopis Pěst. Mat. Fys. 68 (1939), 93–102 (German). MR 0001242
E. M. Matveev, Linear and multiplicative relations, Mat. Sb. 184 (1993), no. 4, 23–40 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math. 78 (1994), no. 2, 411–425. MR 1225976, DOI 10.1070/SM1994v078n02ABEH003477
E. M. Matveev, On the index of multiplicative groups of algebraic numbers, Mat. Sb. 196 (2005), no. 9, 59–70 (Russian, with Russian summary); English transl., Sb. Math. 196 (2005), no. 9-10, 1307–1318. MR 2195707, DOI 10.1070/SM2005v196n09ABEH003643
Peter McMullen, Volumes of projections of unit cubes, Bull. London Math. Soc. 16 (1984), no. 3, 278–280. MR 738519, DOI 10.1112/blms/16.3.278
Władysław Narkiewicz, Elementary and analytic theory of algebraic numbers, Monografie Matematyczne, Tom 57, PWN—Polish Scientific Publishers, Warsaw, 1974. MR 0347767
D. G. Northcott, An inequality in the theory of arithmetic on algebraic varieties, Proc. Cambridge Philos. Soc. 45 (1949), 502–509. MR 33094, DOI 10.1017/s0305004100025202
Shlomo Reisner, Random polytopes and the volume-product of symmetric convex bodies, Math. Scand. 57 (1985), no. 2, 386–392. MR 832364, DOI 10.7146/math.scand.a-12124
Shlomo Reisner, Zonoids with minimal volume-product, Math. Z. 192 (1986), no. 3, 339–346. MR 845207, DOI 10.1007/BF01164009
Y. Gordon, M. Meyer, and S. Reisner, Zonoids with minimal volume-product—a new proof, Proc. Amer. Math. Soc. 104 (1988), no. 1, 273–276. MR 958082, DOI 10.1090/S0002-9939-1988-0958082-9
Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
L. A. Santaló, An affine invariant for convex bodies of $n$-dimensional space, Portugal. Math. 8 (1949), 155–161 (Spanish). MR 39293
Rolf Schneider and Wolfgang Weil, Zonoids and related topics, Convexity and its applications, Birkhäuser, Basel, 1983, pp. 296–317. MR 731116
H. M. Stark, Further advances in the theory of linear forms in logarithms, Diophantine approximation and its applications (Proc. Conf., Washington, D.C., 1972) Academic Press, New York, 1973, pp. 255–293. MR 0352016
A. C. Thompson, Minkowski geometry, Encyclopedia of Mathematics and its Applications, vol. 63, Cambridge University Press, Cambridge, 1996. MR 1406315, DOI 10.1017/CBO9781107325845
A. J. Van der Poorten and J. H. Loxton, Computing the effectively computable bound in Baker’s inequality for linear forms in logarithms, Bull. Austral. Math. Soc. 15 (1976), no. 1, 33–57. MR 417068, DOI 10.1017/S0004972700036741
A. J. Van der Poorten and J. H. Loxton, Multiplicative relations in number fields, Bull. Austral. Math. Soc. 16 (1977), no. 1, 83–98. MR 491537, DOI 10.1017/S0004972700023042