Quantum arithmetic of Drinfeld modules

Year 2026, Volume: 9 Issue: 1, 39 - 46, 06.03.2026

Igor Nikolaev

https://doi.org/10.33205/cma.1868040

https://izlik.org/JA37XJ97NE

Abstract

We study quantum invariants of projective varieties over number fields. Namely, an explicit formula for a functor $\mathscr{Q}$ on such varieties is proved. The case of abelian varieties with complex multiplication is treated in detail.

Keywords

Drinfeld modules , noncommutative tori , projective variety

References

• B. Blackadar: K-Theory for Operator Algebras, MSRI Publications, Springer (1986).
• Z. I. Borevich, I. R. Shafarevich: Number Theory, Acad. Press, New Y ork (1966).
• E. G. Effros: Dimensions and C∗-Algebras, CBMS Regional Conference Series in Mathematics, 46, Amer. Math. Soc., Rhode Island (1981).
• D. Handelman: Positive matrices and dimension groups affiliated to C∗-algebras and topological Markov chains, J. Operator Theory, 6 (1981), 55–74.
• S. Lang: Complex Multiplication, Springer Verlag, New York (1983).
• X. Li: Semigroup C∗-algebras, In: Carlsen, T.M., Larsen, N.S., Neshveyev, S., Skau, C. (eds) Operator Algebras and Applications. Abel Symposia, Springer, 12 (2016).
• M. Namba: On branched coverings of projective manifolds, Proc. Japan Acad. Ser. A, 61 (4) (1985), 121–124.
• I. V. Nikolaev: On algebraic values of function exp(2πix + log log y), Ramanujan J., 47 (2018), 417–425.
• I. V. Nikolaev: Noncommutative Geometry, 2nd Edition, De Gruyter Studies Math., 66, De Gruyter, Berlin (2022).
• I. V. Nikolaev: Non-abelian class field theory and higher dimensional noncommutative tori, Algebr. Represent. Theory, 28 (2025), 1041–1053.
• I. V. Nikolaev: Quantum arithmetic, (2024), arXiv:2412.09148
• M. A. Rieffel: Non-commutative tori – a case study of non-commutative differentiable manifolds, Contemp. Math., 105 (1990), 191–211.
• M. Rosen: Number Theory in Function Flelds, Graduate Text Math., 210, Springer, New York (2002).
• J. T. Stafford, M. van den Bergh: Noncommutative curves and noncommutative surfaces, Bull. Amer. Math. Soc., 38 (2001), 171–216.