Fehm, Arno; Petersen, Sebastian On the rank of abelian varieties over ample fields. (English) Zbl 1193.14054 Int. J. Number Theory 6, No. 3, 579-586 (2010). A field \(F\) is said to be ample (or large) if any smooth curve over \(F\) has either infinitely many \(F\)-points or none. For example, an algebraically closed field is ample. It is conjectured here that if \(F\) is an ample field that is not algebraic over a finite field then any nontrivial abelian variety \(A\) over \(F\) has infinite Mordell-Weil rank. Note that if \(F\) is algebraic over a finite field then every abelian variety over \(F\) has rank zero.The main result here is that under these conditions \(A(F)\otimes{\mathbb Z}_{(p)}\) is not finitely generated as a \({\mathbb Z}_{(p)}\)-module if \(F\) has characteristic \(p\geq 0\). In particular the conjecture holds if \(F\) has characteristic zero. The proof uses the Mordell-Lang conjecture, established by Faltings and others. The weaker statement that \(A(F)\) is not finitely generated is given an alternative, elementary, proof. This is then used to improve on some earlier results by showing that if \(K\) is an infinite finitely generated field and \(A\) is defined over \(K\), then \(A\) has infinite rank over a large class of ample extensions of \(K\). Reviewer: G. K. Sankaran (Bath) Cited in 6 Documents MSC: 14K05 Algebraic theory of abelian varieties 12E30 Field arithmetic 14G05 Rational points Keywords:abelian variety; infinite rank; ample field PDFBibTeX XMLCite \textit{A. Fehm} and \textit{S. Petersen}, Int. J. Number Theory 6, No. 3, 579--586 (2010; Zbl 1193.14054) Full Text: DOI References: [1] Conrad B., Enseign. Math. (2) 52 pp 37– [2] G. Faltings, Barsotti Symposium in Algebraic Geometry, Perspectives in Mathematics 15 (Academic Press, 1994) pp. 175–182. [3] Frey G., Proc. London Math. Soc. 28 pp 112– [4] DOI: 10.1002/1522-2616(200203)236:1<119::AID-MANA119>3.0.CO;2-U · Zbl 1007.12003 · doi:10.1002/1522-2616(200203)236:1<119::AID-MANA119>3.0.CO;2-U [5] DOI: 10.1007/s00013-005-1492-x · Zbl 1140.12003 · doi:10.1007/s00013-005-1492-x [6] Hindry M., Astérisque 209 pp 39– [7] DOI: 10.1090/S0894-0347-96-00202-0 · Zbl 0864.03026 · doi:10.1090/S0894-0347-96-00202-0 [8] DOI: 10.4153/CJM-2006-032-4 · Zbl 1160.11031 · doi:10.4153/CJM-2006-032-4 [9] DOI: 10.1090/S0002-9947-07-04364-4 · Zbl 1193.11055 · doi:10.1090/S0002-9947-07-04364-4 [10] DOI: 10.4064/aa98-1-2 · Zbl 1124.14304 · doi:10.4064/aa98-1-2 [11] DOI: 10.1007/s00013-003-4610-7 · Zbl 1046.12003 · doi:10.1007/s00013-003-4610-7 [12] DOI: 10.4310/MRL.1999.v6.n6.a2 · Zbl 1016.11022 · doi:10.4310/MRL.1999.v6.n6.a2 [13] DOI: 10.3836/tjm/1170348168 · Zbl 1213.11119 · doi:10.3836/tjm/1170348168 [14] DOI: 10.2178/jsl/1190150142 · Zbl 1015.03041 · doi:10.2178/jsl/1190150142 [15] DOI: 10.1112/S0024609303002431 · Zbl 1140.11335 · doi:10.1112/S0024609303002431 [16] Milne J., Princeton Mathematical Series 33, in: Étale Cohomology (1980) [17] DOI: 10.1016/j.jnt.2005.12.006 · Zbl 1193.11059 · doi:10.1016/j.jnt.2005.12.006 [18] DOI: 10.1007/0-8176-4417-2_11 · Zbl 1200.11041 · doi:10.1007/0-8176-4417-2_11 [19] DOI: 10.4007/annals.2004.160.1099 · Zbl 1084.14026 · doi:10.4007/annals.2004.160.1099 [20] DOI: 10.2307/2118581 · Zbl 0862.12003 · doi:10.2307/2118581 [21] DOI: 10.1007/BFb0099956 · doi:10.1007/BFb0099956 [22] DOI: 10.1006/jnth.2001.2692 · Zbl 1001.11025 · doi:10.1006/jnth.2001.2692 [23] Shimura G., Abelian Varieties with Complex Multiplication and Modular Functions (1997) · Zbl 0908.11023 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.