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Finite group subschemes of abelian varieties over finite fields. (English) Zbl 1326.14106

The finite group subschemes in question are finite (étale) group schemes of abelian varieties belonging to an isogeny class of the varieties over finite fields. The purpose of this paper is to classify of group schemes \(B(l),\) where \(B\) runs through the \(k\)-isogeny class of an abelian variety \(A\) over a finite field \(k,\) in terms of certain Newton polygon associated to the Weil polynomial \(f_A\) (the characteristic polynomial of the Frobenius endomorphism \(F\) of \(A\) which acts on the Tate module \(T_{l}(A)\) ). Here prime number \(l \neq \mathrm{char} \; k\).
Earlier the author [Cent. Eur. J. Math. 8, No. 2, 282–288 (2010; Zbl 1198.14043)] studied the classification of groups of \(k\)-points \(A(k).\)
The main results of the paper under review are Theorem 3.1 and Theorem 3.2.
Consider the situation: (*) \(S\) is the ring of integers of an unramified extension of \({\mathbb Q}_l \), \(T\) is “a finitely generated free \(S\)-module \(T\) endowed with an \(S\)-linear injective endomorphism \(E\) which induces on \(T/lT\) an nilpotent endomorphism \(N\)”. \(Q(t) = \det (t - E)\). “There is a basis of \(T/lT\) such that the matrix of \(N\) is a sum of Jordan cells of dimensions \(m_1, \ldots, m_r.\)” “We associate to \(N\) the Young polygon \(\mathrm{Yp}(N)\) given by the sequence \(m_1,\dots ,m_r.\) We also denote this Young polygon by \(\mathrm{Yp}(E|T)\).”
Theorem 3.1 states that under conditions (*) the Newton polygon \(\mathrm{Np}(Q)\) lies on or above Young polygon \(\mathrm{Yp}(E|T).\) If furthermore, \(R = S[t]/Q(t)S[t], V= R\otimes_{{\mathbb Z}_l} {\mathbb Q}_l,\) \(Y\) is a Young polygon such that \(\mathrm{Np}(Q)\) lies on or above \(Y\), then there exists an \(R-\)lattice \(T\) in \(V\) such that \(\mathrm{Yp}(x|T) = Y\) (Theorem 3.2.).
(Example 3.3 contains a misprint: Fig. 2 is not the drawing of the Newton polygon of the polynomial \(t^2 - lt - l\)).
The paper under review exploits next fundamental results, the following two without reference to primary sources: (i) an abelian variety is an algebraic group that is a complete algebraic variety [Abelian varieties, B. B. Venkov, N. P. Parshin, in: M. Hazewinkel (ed.), Encyclopaedia of mathematics. Volume 1: A-B. An updated and annotated translation of the Soviet ‘Mathematical Encyclopaedia’. Dordrecht (Netherlands) etc.: D. Reidel Publishing Company (1988; Zbl 0625.00029)]; (ii) over finite fields the possibility of the \(p-\)adic representation of the endomorphism ring of an abelian variety on its Tate module follows from results by J.-P. Serre [Am. J. Math. 80, 715–739 (1958; Zbl 0099.16201)] and by Yu. I. Manin [Russ. Math. Surv. 18, No. 6, 1–83 (1963); translation from Usp. Mat. Nauk 18, No. 6(114), 3–90 (1963; Zbl 0128.15603)]; (iii) the fact that the action of \(F\) on \(V_{l} = T_{l}(A)\otimes_{{\mathbb Z}_l}{\mathbb Q}_l \) is semisimple follows from the semisimplicity theorem by P. Deligne [Publ. Math., Inst. Hautes Étud. Sci. 52, 137–252 (1980; Zbl 0456.14014)].
In section 4 ‘a relationship between Young polygons for the Frobenius actions on an abelian variety‘ is given.
Next the author proves ‘that (generalized) matrix factorizations correspond to Tate modules.‘
The paper concludes with the application of some previous results to the classification of zeta functions of Kummer surfaces.
The techniques used are computational. This is a good place to see the interplay between nilpotent matrices over finite fields and their lifting to rings of integers of unramified extensions of the fields of \(l-\)adic numbers with respect to characteristic polynomials of semisimple linear transforms.

MSC:

14K15 Arithmetic ground fields for abelian varieties
14L15 Group schemes
14G05 Rational points
14G15 Finite ground fields in algebraic geometry
11G07 Elliptic curves over local fields
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References:

[1] Banaszak, G.; Gajda, W.; Krason, P., On the image of l-adic Galois representations for abelian varieties of type I and II, Doc. Math. Extra, Coats, 35-75 (2006) · Zbl 1186.11028
[2] Demazure, M., Lectures on \(p\)-Divisible Groups, Lect. Notes Math., vol. 302 (1972), Springer · Zbl 0247.14010
[3] Grothendieck, A., Éléments de géométrie algébrique IV. Étude locale des śchemas et des morphismes de schémas, Quatrième partie, Publ. Math. IHÉS, 32, 5-361 (1967) · Zbl 0153.22301
[4] Hartshorne, R., Algebraic Geometry, Grad. Texts Math., vol. 52 (1977), Springer-Verlag: Springer-Verlag New York, Heidelberg · Zbl 0367.14001
[5] Maisner, D.; Nart, E., Abelian surfaces over finite fields as Jacobians, Exp. Math., 11, 3, 321-337 (2002), with an appendix by Everett W. Howe · Zbl 1101.14056
[6] Milne, J., Abelian varieties (2008) · Zbl 0604.14028
[7] Mumford, D., Abelian Varieties, Tata Inst. Fund. Res. Stud. Math., vol. 5 (1970), Oxford University Press: Oxford University Press London · Zbl 0198.25801
[8] Pohst, M.; Zassenhaus, H., Algorithmic Algebraic Number Theory, Encycl. Math. Appl., vol. 30 (1997), Cambridge University Press: Cambridge University Press Cambridge, revised reprint of the 1989 original · Zbl 0685.12001
[9] Rybakov, S., The groups of points on abelian varieties over finite fields, Cent. Eur. J. Math., 8, 2, 282-288 (2010) · Zbl 1198.14043
[10] Tate, J., Endomorphisms of abelian varieties over finite fields, Invent. Math., 2, 2, 134-144 (1966) · Zbl 0147.20303
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