We establish asymptotic formulas for volumes of height balls in analytic varieties over local fields and in adelic points of algebraic varieties over number fields, relating the Mellin transforms of height functions to Igusa integrals and to global geometric invariants of the underlying variety. In the adelic setting, this involves the construction of general Tamagawa measures.
@article{CML_2010__2_3_351_0,
author = {Antoine Chambert-Loir and Yuri Tschinkel},
title = {Igusa integrals and volume asymptotics in analytic and adelic geometry},
journal = {Confluentes Mathematici},
pages = {351--429},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {2},
number = {3},
year = {2010},
doi = {10.1142/S1793744210000223},
language = {en},
url = {https://cml.centre-mersenne.org/articles/10.1142/S1793744210000223/}
}
TY - JOUR AU - Antoine Chambert-Loir AU - Yuri Tschinkel TI - Igusa integrals and volume asymptotics in analytic and adelic geometry JO - Confluentes Mathematici PY - 2010 SP - 351 EP - 429 VL - 2 IS - 3 PB - World Scientific Publishing Co Pte Ltd UR - https://cml.centre-mersenne.org/articles/10.1142/S1793744210000223/ DO - 10.1142/S1793744210000223 LA - en ID - CML_2010__2_3_351_0 ER -
%0 Journal Article %A Antoine Chambert-Loir %A Yuri Tschinkel %T Igusa integrals and volume asymptotics in analytic and adelic geometry %J Confluentes Mathematici %D 2010 %P 351-429 %V 2 %N 3 %I World Scientific Publishing Co Pte Ltd %U https://cml.centre-mersenne.org/articles/10.1142/S1793744210000223/ %R 10.1142/S1793744210000223 %G en %F CML_2010__2_3_351_0
Antoine Chambert-Loir; Yuri Tschinkel. Igusa integrals and volume asymptotics in analytic and adelic geometry. Confluentes Mathematici, Tome 2 (2010) no. 3, pp. 351-429. doi : 10.1142/S1793744210000223. https://cml.centre-mersenne.org/articles/10.1142/S1793744210000223/
[1] A. Baker , Transcendental Number Theory ( Cambridge Univ. Press , 1975 ) .
[2] V. V. Batyrev, New Trends in Algebraic Geometry (Cambridge Univ. Press, Warwick, 1996) pp. 1–11.
[3] V. V. Batyrev and Yu. I. Manin, Math. Ann. 286, 27 (1990), DOI: 10.1007/BF01453564.
[4] V. V. Batyrev and Yu. Tschinkel, Internat. Math. Res. Notices 12, 591 (1995).
[5] V. V. Batyrev and Yu. Tschinkel, J. Math. Sci. 82, 3220 (1996), DOI: 10.1007/BF02362469.
[6] B. J. Birch, Proc. London Math. Soc. A 265, 245 (1962), DOI: 10.1098/rspa.1962.0007.
[7] M. Borovoi and Z. Rudnick, Invent. Math. 119, 37 (1995), DOI: 10.1007/BF01245174.
[8] R. Brauer, Ann. Math. 48, 502 (1947), DOI: 10.2307/1969183.
[9] R. de la Bretèche, J. Number Th. 87, 315 (2001).
[10] M. Brion and S. Kumar , Frobenius Splitting Methods in Geometry and Representation Theory , Progress in Mathematics 231 ( Birkhäuser , 2005 ) .
[11] F. Bruhat and J. Tits, Publ. Math. Inst. Hautes Études Sci. 5 (1972).
[12] A. Chambert-Loir and Yu. Tschinkel, J. Number Th. 85, 172 (2000), DOI: 10.1006/jnth.2000.2539.
[13] A. Chambert-Loir and Yu. Tschinkel, Invent. Math. 148, 421 (2002), DOI: 10.1007/s002220100200.
[14] A. Chambert-Loir and Yu. Tschinkel, Integral points of bounded height on partial equivariant compactifications of vector groups , arXiv:0912.4751 .
[15] A. Chambert-Loir and Yu. Tschinkel, Integral points of bounded height on toric varieties , arXiv:1006.3345 .
[16] H. Clemens, Trans. Amer. Math. Soc. 136, 93 (1969), DOI: 10.1090/S0002-9947-1969-0233814-9.
[17] R. Cluckers, G. Comte and F. Loeser, Local metric properties and regular stratifications of p-adic definable sets , arXiv:0910.0799 .
[18] J.-L. Colliot-Thélène and J.-J. Sansuc, Duke Math. J. 54, 375 (1987), DOI: 10.1215/S0012-7094-87-05420-2.
[19] C. De Concini and C. Procesi, Invariant Theory, Lecture Notes in Math 996 (Springer, Montecatini, 1982) pp. 1–44.
[20] P. Deligne, Publ. Math. Inst. Hautes Études Sci. 43, 273 (1974), DOI: 10.1007/BF02684373.
[21] J. Denef, Amer. J. Math. 109, 991 (1987), DOI: 10.2307/2374583.
[22] W. Duke, Z. Rudnick and P. Sarnak, Duke Math. J. 71, 143 (1993), DOI: 10.1215/S0012-7094-93-07107-4.
[23] A. Eskin and C. McMullen, Duke Math. J. 71, 181 (1993), DOI: 10.1215/S0012-7094-93-07108-6.
[24] A. Eskin, S. Mozes and N. Shah, Ann. Math. 143, 253 (1996), DOI: 10.2307/2118644.
[25] J. Franke, Yu. I. Manin and Yu. Tschinkel, Invent. Math. 95, 421 (1989), DOI: 10.1007/BF01393904.
[26] G. L. Gordon, Trans. Amer. Math. Soc. 261, 93 (1980), DOI: 10.1090/S0002-9947-1980-0576865-1.
[27] A. Gorodnik, F. Maucourant and H. Oh, Ann. Sci. École Norm. Sup. 41, 385 (2008).
[28] A. Gorodnik, H. Oh and N. Shah, Amer. J. Math. 131, 1 (2009).
[29] B. Hassett and Yu. Tschinkel, Duke Math. J. 120, 577 (2003).
[30] M. Hindry and J. H. Silverman , Diophantine Geometry. An Introduction , Graduate Texts in Mathematics 201 ( Springer , 2000 ) .
[31] J.-I. Igusa, J. Reine Angew. Math 269 (1974) pp. 110–130.
[32] J.-I. Igusa, J. Reine Angew. Math. 279, 307 (1975).
[33] F. Knop, The Luna-Vust theory of spherical embeddings, Proc. of the Hyderabad Conference on Algebraic Groups (Manoj Prakashan, 1989) pp. 225–249.
[34] S. Lang , Fundamentals of Diophantine Geometry ( Springer , 1983 ) .
[35] S. Lang and A. Weil, Amer. J. Math. 76, 819 (1954), DOI: 10.2307/2372655.
[36] F. Maucourant, Duke Math. J. 136, 357 (2007), DOI: 10.1215/S0012-7094-07-13626-3.
[37] J. S. Milne , Étale Cohomology , Math. Notes ( Princeton Univ. Press , 1980 ) .
[38] E. Peyre, Duke Math. J. 79, 101 (1995), DOI: 10.1215/S0012-7094-95-07904-6.
[39] E. Peyre (ed.) , Nombre et Répartition des Points de Hauteur Bornée ( Astérisque , 1998 ) .
[40] E. Peyre, Nombre et Répartition des Points de Hauteur Bornée, ed. () pp. 259–298.
[41] P. Salberger, Nombre et Répartition des Points de Hauteur Bornée () pp. 91–258.
[42] J.-P. Serre, Inst. Hautes Études Sci. Publ. Math. 323 (1981).
[43] J.-P. Serre , Lectures on the Mordell–Weil Theorem , 3rd edn. , Aspects of Mathematics , ed. ( Friedr. Vieweg & Sohn , 1997 ) .
[44] N. A. Shah, Sankhyā Ser. A 62, 386 (2000).
[45] J. Shalika, R. Takloo-Bighash and Yu. Tschinkel, J. Amer. Math. Soc. 20, 1135 (2007), DOI: 10.1090/S0894-0347-07-00572-3.
[46] A. Skorobogatov , Torsors and Rational Points , Cambridge Tracts in Mathematics 144 ( Cambridge Univ. Press , 2001 ) .
[47] A. Weil , Adeles and Algebraic Groups , Progr. Math. ( Birkhäuser , 1982 ) .
[48] A. Weil , Basic Number Theory ( Springer , 1995 ) .
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