Arithmetic invariants from Sato-Tate moments. (Invariants arithmétiques provenant des moments de Sato-Tate.) (English. French summary) Zbl 1444.11127

Summary: We give some arithmetic-geometric interpretations of the moments \(\operatorname{M}_2 [a_1]\), \(\operatorname{M}_1 [a_2]\), and \(\operatorname{M}_1 [s_2]\) of the Sato-Tate group of an abelian variety \(A\) defined over a number field by relating them to the ranks of the endomorphism ring and Néron-Severi group of \(A\).


11G10 Abelian varieties of dimension \(> 1\)
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14K15 Arithmetic ground fields for abelian varieties
11M50 Relations with random matrices
Full Text: DOI arXiv


[1] Banaszak, G.; Kedlaya, K. S., An algebraic Sato-Tate group and Sato-Tate conjecture, Indiana Univ. Math. J., 64, 245-274 (2015) · Zbl 1392.11041
[2] Cantoral Farfán, V.; Commelin, J., The Mumford-Tate conjecture implies the algebraic Sato-Tate conjecture of Banaszak and Kedlaya · Zbl 1506.14094
[3] Faltings, G., Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math., 73, 349-366 (1983) · Zbl 0588.14026
[4] Fité, F.; Kedlaya, K. S.; Sutherland, A. V.; Rotger, V., Sato-Tate distributions and Galois endomorphism modules in genus 2, Compos. Math., 148, 1390-1442 (2012) · Zbl 1269.11094
[5] Kim, S., The Sato-Tate conjecture and Nagao’s conjecture · Zbl 1529.11077
[6] Mumford, D., Abelian Varieties (1970), Tata Institute of Fundamental Research, Bombay, Oxford University Press · Zbl 0198.25801
[7] Serre, J.-P., Linear Representations of Finite Groups (1977), Springer-Verlag: Springer-Verlag New York · Zbl 0355.20006
[8] Serre, J.-P., Lectures on \(N_X(p) (2012)\), CRC Press: CRC Press Boca Raton, FL, USA · Zbl 1238.11001
[9] Tate, J., Algebraic cycles and poles of zeta functions, (Arithmetical Algebraic Geometry, Proceedings of a Conference Held at Purdue University, IN, USA. Arithmetical Algebraic Geometry, Proceedings of a Conference Held at Purdue University, IN, USA, 5-7 December 1963 (1965), Harper & Row: Harper & Row New York), 93-110 · Zbl 0213.22804
[10] Tate, J., Endomorphisms of abelian varieties over finite fields, Invent. Math., 2, 134-144 (1966) · Zbl 0147.20303
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