We give a survey of the recent developments in the understanding of the multizeta values for function fields.

/pdf/multizeta-values-for-function-fields-a-survey-2wbmnbrqdg.pdf

Multizeta values for function fields: A survey

We show that the module of Stark units associated to a sign-normalized rank one Drinfeld module can be obtained from Anderson's equivariant $A$-harmonic series. We apply this to obtain a class formula a la Taelman and to prove a several variable log-algebraicity theorem, generalizing Anderson's log-algebraicity theorem. We also give another proof of Anderson's log-algebraicity theorem using shtukas and obtain various results concerning the module of Stark units for Drinfeld modules of arbitrary rank.

Stark units in positive characteristic

Let K be a global function field over a finite field of characteristic p and let A be the ring of elements of K which are regular outside a fixed place of K. This report presents recent developments in the arithmetic of special L-values of Anderson A-modules. Provided that p does not divide the class number of K, we prove an "analytic class number formula" for Anderson A-modules with the help of a recent work of Debry. For tensor powers of the Carlitz module, we explain how to derive several log-algebraicity results from the class number formula for these Anderson modules.

/pdf/recent-developments-in-the-theory-of-anderson-modules-4uwsindr6g.pdf

Recent developments in the theory of anderson modules

Assuming everywhere good reduction we generalize the class number formula of Taelman to Drinfeld modules over arbitrary coefficient rings. In order to prove this formula we develop a theory of shtukas and their cohomology.

Shtuka cohomology and special values of Goss L-functions

We prove or conjecture several relations between the multizeta values for positive genus function fields of class number one, focusing on the zeta-like values, namely those whose ratio with the zeta value of the same weight is rational (or conjecturally equivalently algebraic). These are the first known relations between multizetas, which are not with prime field coefficients. We seem to have one universal family. We also find that interestingly the mechanism with which the relations work is quite different from the rational function field case, raising interesting questions about the expected motivic interpretation in higher genus. We provide some data in support of the guesses.

/pdf/zeta-like-multizeta-values-for-higher-genus-curves-54ae61r1op.pdf

Zeta-like Multizeta Values for higher genus curves

Let F be the function field of an irreducible, smooth, projective curve over a finite field. Let A be the ring of functions on the curve which are regular away from a fixed closed point ∞. Let F∞ be the completion of F at ∞. Consider an integral model φ of a Drinfeld A-module over a finite extension of F. We associate to such a model an element (the L-value of φ of F∞, mimicking the residue at 1 of the Dedekind zeta function of a number field, or the top coefficient at 1 of the L-function of an elliptic curve over Q. The class module of φ is an A-module of finite cardinality, which serves as an analogue of the class group of a number field, or the Tate-Shafarevich group of an elliptic curve. The regulator of φ is an invertible sub-A-module of F∞ and looks like the regulators of number fields and elliptic curves. The aim of this thesis is to present a conjectural formula which relates the L-value, the class module and the regulator of φ. It could be regarded as a function field analogue of the class number formula for number fields or the BSD conjecture for elliptic curves; hence the name “a class number formula for Drinfeld modules.” Our conjecture is a direct generalisation of a theorem by Taelman which states that the formula holds if A=Fq [t]. The current manuscript generalises his work, by introducing the analogues of his L-values, class modules and regulators for general A, by stating a conjectural formula, and by proving it for all A which are principal ideal domains. We also show how our techniques can be used to prove the class number formula in specific cases, for example, when the curve has genus zero.

/pdf/towards-a-class-number-formula-for-drinfeld-modules-1mzhmkd7sk.pdf

Towards a class number formula for Drinfeld modules

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Towards a class number formula for Drinfeld modules