Drinfeld modules are analogs of elliptic curves for the finite characteristic case, and Anderson t-motives – their high-dimensional generalizations – are analogs of abelian varieties. The theory of Drinfeld modules and Anderson t-motives is as rich as the theory of abelian varieties. It offers many research problems of all levels, from the relatively simple ones, until the levels of the top of contemporary mathematics.
Drinfeld modules can be used in cryptography, like elliptic curves.

선수과목(권장): Math301 Modern Algebra I

References:
[1] Grishkov A., Logachev, D. Introduction to Anderson t-motives: a survey. 2020. https://arxiv.org/pdf/2008.10657.pdf
[2] Goss, D. Basic structures of function field arithmetic. Springer-Verlag, Berlin, 1996. (I have an electronic version of this book kindly sent by the author).
[3] Papikian, M. Drinfeld modules. Grad. Texts in Math., 296. Springer, 2023 (I have an electronic version of this book kindly sent by the author).
[4] Deligne, P.; Husemoller, D. Survey of Drinfelʹd modules. Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), 25–91. Contemp. Math., 67.

DRINFELD MODULES - SYLLABUS

1. A brief survey on elliptic curves over Q and C, in order to understand why we shall need Drinfeld modules: they are their
characteristic p analogs. The Weierstrass P-functions. Elliptic curves and lattices in C. The Tate module of an elliptic curve.
Galois action on the Tate module. Elliptic curves with complex multiplication.

2. F_q[\theta] - the ring of polynomials over a finite field: a characteristic p analog of the ring Z. F_q(\theta) - the field
of rational functions over a finite field: a characteristic p analog of the field Q. F_q((\theta^{-1})) - the field of Laurent series
over a finite field: a characteristic p analog of the field R. C_p - the completion of its algebraic closure: a characteristic p
analog of the field C.

3. Topology in C_p. Lattices in C_p. Additive polynomials and the ring C_p[\tau].

4. The Anderson ring C_p[t,\tau]. Anderson t-motives, examples. The Carlitz module. Drinfeld modules.
Basis of a Drinfeld module over C_p[t]. The matrix of multiplication by \tau.

5. t-module E(M) corresponding to a t-motive M. The Tate module of a t-motive. E(M) as a homomorphism group.

6. Exponential map for a Drinfeld module. Its existence and convergence.

7. Periodic maps associated to lattices in C_p - analogs of the Weierstrass P-functions.

8. 1 -- 1 correspondence between Drinfeld modules and lattices in C_p.

9. Endomorphism rings of Drinfeld modules - analogs of elliptic curves with complex multiplication.

10. Homomorphisms and tensor products of Anderson t-motives. Examples: homomorphisms and tensor products of the Carlitz module,
of Drinfeld modules. Nilpotent operator N. Duals of Anderson t-motives.

11. Exponential map for an Anderson t-motive M. Definitions of the Lie(M) and of H_1(M)=L(M) - the lattice of M.

12. Three ways to represent the principal exact sequence: from L(M) to Lie(M) and from Lie(M) to E(M).

13. Definition of H^1(M). Calculation of H^1(M) for the Carlitz module. Pairing between H_1(M) and H^1(M).

14. Drinfeld upper half plane \Omega. Its covering by the spaces D_{(n,x)}. First definition of the Bruhat-Tits building of \Omega.

15. Lattices in F_q((\theta^{-1}))^2. Second definition of the Bruhat-Tits building of \Omega.

16. Drinfeld modular forms.

Topics:
The Anderson ring. Drinfeld modules and Anderson t-motives are modules over the Anderson ring. 1 – 1 correspondence between Drinfeld modules and lattices. Lattices of Anderson t-motives, the exponential map. H^1 and H_1 of Anderson t-motives. Tensor products and Hom’s of Anderson t-motives. Endomorphisms of Drinfeld modules. Tate module of Drinfeld modules, Galois action on it. Drinfeld upper half plane, its Bruhat – Tits building. Drinfeld modular forms, Drinfeld modular curves. If there will be time: F-sheaves, shtukas of Drinfeld.

Prerequisite: Graduate Algebra.

This course consists of lectures and students’participation.
Students’participation include: attendance and interaction in class, provide (more detailed) proofs of some results whose proofs missed (or sketched) in lectures.