Multiples zeta values and alternating multiple zeta values in positive characteristic were introduced by Thakur and Harada as analogues of classical multiple zeta values of Euler and Euler sums. In this paper, we determine all linear relations between alternating multiple zeta values and settle the main goals of these theories. As a consequence, we completely establish Zagier–Hoffman’s conjectures in positive characteristic formulated by Todd and Thakur which predict the dimension and an explicit basis of the span of multiple zeta values of Thakur of fixed weight.

In 2007 Chang and Yu determined all the algebraic relations among Goss’s zeta values for $A=\mathbb F_q[\theta ]$, also known as the Carlitz zeta values. Goss raised the problem of determining all algebraic relations among Goss’s zeta values at positive integers for a general base ring A, but very little is known. In this paper, we develop a general method, and we determine all algebraic relations among Goss’s zeta values for the base ring A which is the coordinate ring of an elliptic curve defined over $\mathbb F_q$. To our knowledge, this is the first work tackling Goss’s problem when the base ring has class number strictly greater than 1.

We extend our previous work in collaboration with Ngô Bao Châu and give a fixed point formula for the elliptic part of moduli spaces of $G$-shtukas with arbitrary modifications. Our formula is similar to the fixed point formula of Kottwitz for certain Shimura varieties. Our method is inspired by that of Kottwitz and simpler than that of Lafforgue for the fixed point formula of the moduli space of Drinfeld $\text{GL} (r)$-shtukas.

Nous proposons une façon simple de compter le nombre de $G$-chtoucas de Drinfeld avec modifications arbitraires, inspirée par les travaux de Kottwitz sur les variétés de Shimura.

We propose a simple method to count the number of Drinfeld $G$-shtukas with arbitrary modifications which is inspired by works by Kottwitz on Shimura varieties.

Cameron–Praeger designs with parameters $t-(v,k,\lambda)$ are studied. Cameron and Praeger showed that in such designs, $t=2$ or 3. In 1989, Delandtsheer and Doyen proved that if $t=2$ then \[ v \leqslant \left(\left({k \atop 2}\right)-1\right)^2. \] In 2000, Mann and Tuan improved this equality and showed that if $t=3$ then \[ v \leqslant \left({k \atop 2}\right)+1. \] Three infinite families of Cameron–Praeger 3-designs for which this bound is met have been constructed by Mann and Tuan and by Sebille. The paper constructs infinitely many infinite families of such Cameron–Praeger 3-designs via a study of a divisibility problem for polynomials. Further, the construction generalizes the previous constructions.