Abstract
In this chapter we study Drinfeld modules defined over a finite field \(k= \mathbb {F}_{q^n}\). What distinguishes the theory of these Drinfeld modules from the general theory is that the Frobenius π := τ n commutes with every other element of \(k\!\left \{\tau \right \}\), hence A[π] is a subring of \( \operatorname {\mathrm {End}}_k(\phi )\) for any Drinfeld module ϕ; this simple observation is the starting point of the main results of this chapter.
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Notes
- 1.
If Δ is an algebra over a field L, Δ′⊂ Δ is a subalgebra, and L′∕L is a field extension, then Δ′⊗L L′ is a subalgebra of the L′-algebra Δ ⊗L L′ since L′ is a flat L-module; see [DF04, p. 400].
- 2.
Some assumptions are necessary since not every domain can be embedded into a division ring; see [Coh95, p. 9].
- 3.
\(F_{\mathfrak {l}}\) is flat over \(A_{\mathfrak {l}}\); cf. [DF04, p. 400].
- 4.
The key point here is that, up to conjugation, \(k\!\left \{\tau \right \}\) is the unique maximal \(\mathbb {F}_q[\pi ]\)-order in k(τ); see [Rei03, §21].
- 5.
It seems quite challenging to generalize Elkies’ method to higher rank Drinfeld modules. As far as I know, currently there are no known examples of Drinfeld modules ϕ of rank ≥ 3 over F with \( \operatorname {\mathrm {End}}(\phi )=A\) having infinitely many supersingular reductions.
References
Greg W. Anderson, t-motives, Duke Math. J. 53 (1986), no. 2, 457–502.
Bruno Anglès, On some characteristic polynomials attached to finite Drinfeld modules. Manuscripta Math., 93 (1997), no. 3, 369–379.
Alp Bassa and Peter Beelen, On the Deuring polynomial for Drinfeld modules in Legendre form, Acta Arith. 186 (2018), no. 2, 179–190.
P. Boyer, Mauvaise réduction des variétés de Drinfeld et correspondance de Langlands locale, Invent. Math. 138 (1999), no. 3, 573–629.
M. L. Brown, Singular moduli and supersingular moduli of Drinfeld modules, Invent. Math. 110 (1992), no. 2, 419–439.
P. M. Cohn, Skew fields, Encyclopedia of Mathematics and its Applications, vol. 57, Cambridge University Press, Cambridge, 1995, Theory of general division rings.
David S. Dummit and Richard M. Foote, Abstract algebra, third ed., John Wiley & Sons, Inc., Hoboken, NJ, 2004.
_________ , Elliptic modules. II, Mat. Sb. (N.S.) 102(144) (1977), no. 2, 182–194, 325.
Ernst-Ulrich Gekeler, Zur Arithmetik von Drinfeld-Moduln, Math. Ann. 262 (1983), no. 2, 167–182.
_________ , On finite Drinfeld modules, J. Algebra 141 (1991), no. 1, 187–203.
_________ , On the arithmetic of some division algebras, Comment. Math. Helv. 67 (1992), no. 2, 316–333.
_________ , Frobenius distributions of Drinfeld modules over finite fields, Trans. Amer. Math. Soc. 360 (2008), no. 4, 1695–1721.
Sumita Garai and Mihran Papikian, Endomorphism rings of reductions of Drinfeld modules, J. Number Theory 212 (2020), 18–39.
Everett W. Howe and Kiran S. Kedlaya, Every positive integer is the order of an ordinary abelian variety over \(\mathbb {F}_2\), Res. Number Theory. 7 (2021), no. 4, 59.
Liang-Chung Hsia and Jing Yu, On characteristic polynomials of geometric Frobenius associated to Drinfeld modules, Compositio Math. 122 (2000), no. 3, 261–280.
_________ , Algebra, third ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002.
Gérard Laumon, Cohomology of Drinfeld modular varieties. Part I, Cambridge Studies in Advanced Mathematics, vol. 41, Cambridge University Press, Cambridge, 1996, Geometry, counting of points and local harmonic analysis.
Richard Pink, The Galois representations associated to a Drinfeld module in special characteristic. II. Openness, J. Number Theory 116 (2006), no. 2, 348–372.
_________ , Drinfeld modules with no supersingular primes, Internat. Math. Res. Notices (1998), no. 3, 151–159.
I. Reiner, Maximal orders, London Mathematical Society Monographs. New Series, vol. 28, The Clarendon Press, Oxford University Press, Oxford, 2003, Corrected reprint of the 1975 original, With a foreword by M. J. Taylor.
_________ , Number theory in function fields, Graduate Texts in Mathematics, vol. 210, Springer-Verlag, New York, 2002.
Andreas Schweizer, On the Drinfeld modular polynomial ΦT(X, Y ), J. Number Theory 52 (1995), no. 1, 53–68.
_________ , On singular and supersingular invariants of Drinfeld modules, Ann. Fac. Sci. Toulouse Math. (6) 6 (1997), no. 2, 319–334.
_________ , The arithmetic of elliptic curves, second ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009.
Raymond van Bommel, Edgar Costa, Wanlin Li, Bjorn Poonen, and Alexander Smith, Abelian varieties of prescribed order over finite fields, arXiv 2106.13651 [math.NT] (2021).
Jiu-Kang Yu, Isogenies of Drinfeld modules over finite fields, J. Number Theory 54 (1995), no. 1, 161–171.
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Papikian, M. (2023). Drinfeld Modules over Finite Fields. In: Drinfeld Modules. Graduate Texts in Mathematics, vol 296. Springer, Cham. https://doi.org/10.1007/978-3-031-19707-9_4
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