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 prove that there is a positive proportion of L-functions associated to cubic characters over $\mathbb F_q[T]$ that do not vanish at the critical point $s=1/2$. This is achieved by computing the first mollified moment using techniques previously developed by the authors in their work on the first moment of cubic L-functions, and by obtaining a sharp upper bound for the second mollified moment, building on work of Lester and Radziwiłł, which in turn develops further ideas from the work of Soundararajan, Harper and Radziwiłł. We work in the non-Kummer setting when $q\equiv 2 \,(\mathrm {mod}\,3)$, but our results could be translated into the Kummer setting when $q\equiv 1\,(\mathrm {mod}\,3)$ as well as into the number-field case (assuming the generalised Riemann hypothesis). Our positive proportion of nonvanishing is explicit, but extremely small, due to the fact that the implied constant in the upper bound for the mollified second moment is very large.

Let $\mathbb {F}_q$ be the finite field of q elements. In this paper, we study the vanishing behavior of multizeta values over $\mathbb {F}_q[t]$ at negative integers. These values are analogs of the classical multizeta values. At negative integers, they are series of products of power sums $S_d(k)$ which are polynomials in t. By studying the t-valuation of $S_d(s)$ for $s < 0$ , we show that multizeta values at negative integers vanish only at trivial zeros. The proof is inspired by the idea of Sheats in the proof of a statement of “greedy element” by Carlitz.

We compute the complete set of candidates for the zeta function of a K$3$ surface over $\mathbb{F}_{2}$ consistent with the Weil and Tate conjectures, as well as the complete set of zeta functions of smooth quartic surfaces over $\mathbb{F}_{2}$. These sets differ substantially, but we do identify natural subsets which coincide. This gives some numerical evidence towards a Honda–Tate theorem for transcendental zeta functions of K$3$ surfaces; such a result would refine a recent theorem of Taelman, in which one must allow an uncontrolled base field extension.

Let $C/\mathbf{Q}$ be a curve of genus three, given as a double cover of a plane conic. Such a curve is hyperelliptic over the algebraic closure of $\mathbf{Q}$, but may not have a hyperelliptic model of the usual form over $\mathbf{Q}$. We describe an algorithm that computes the local zeta functions of $C$ at all odd primes of good reduction up to a prescribed bound $N$. The algorithm relies on an adaptation of the ‘accumulating remainder tree’ to matrices with entries in a quadratic field. We report on an implementation and compare its performance to previous algorithms for the ordinary hyperelliptic case.

We present an efficient algorithm to compute the Hasse–Witt matrix of a hyperelliptic curve $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}C/\mathbb{Q}$ modulo all primes of good reduction up to a given bound $N$, based on the average polynomial-time algorithm recently proposed by the first author. An implementation for hyperelliptic curves of genus 2 and 3 is more than an order of magnitude faster than alternative methods for $N = 2^{26}$.

Using the cohomology theory of Dwork, as developed by Adolphson and Sperber, we exhibit a deterministic algorithm to compute the zeta function of a nondegenerate hypersurface defined over a finite field. This algorithm is particularly well suited to work with polynomials in small characteristic that have few monomials (relative to their dimension). Our method covers toric, affine, and projective hypersurfaces, and also can be used to compute the L-function of an exponential sum.

Let $\chi(n)$ be a quadratic character modulo a prime $p$. For a fixed integer $s\ne 0$, we estimate certain exponential sums with truncated $L$-functions \[L_{s,p}(n) = \sum_{j=1}^n \frac{\chi(\,j)}{j^s}\qquad (n =1, 2, \ldots)\]. Our estimate implies certain uniformly of distribution properties of reductions of $L_{s,p}(n)$ in the residue classes modulo $p$.

We present a new approach to the problem of computing the zeta function of a hypersurface over a finite field. For a hypersurface defined by a polynomial of degree $d$ in $n$ variables over the field of $q$ elements, one desires an algorithm whose running time is a polynomial function of $d^n \log(q)$. (Here we assume $d \geq 2$, for otherwise the problem is easy.) The case $n = 1$ is related to univariate polynomial factorisation and is comparatively straightforward. When $n = 2$ one is counting points on curves, and the method of Schoof and Pila yields a complexity of $\log(q)^{C_d}$, where the function $C_d$ depends exponentially on $d$. For arbitrary $n$, the theorem of the author and Wan gives a complexity which is a polynomial function of $(pd^n \log(q))^n$, where $p$ is the characteristic of the field. A complexity estimate of this form can also be achieved for smooth hypersurfaces using the method of Kedlaya, although this has only been worked out in full for curves. The new approach we present should yield a complexity which is a small polynomial function of $pd^n\log(q)$. In this paper, we work this out in full for Artin–Schreier hypersurfaces defined by equations of the form $Z^p - Z = f$, where the polynomial $f$ has a diagonal leading form. The method utilises a relative $p$-adic cohomology theory for families of hypersurfaces, due in essence to Dwork. As a corollary of our main theorem, we obtain the following curious result. Let $f$ be a multivariate polynomial with integer coefficients whose leading form is diagonal. There exists an explicit deterministic algorithm which takes as input a prime $p$, outputs the number of solutions to the congruence equation $f = 0 \bmod{p}$, and runs in ${\mathcal O}(p^{2 + \varepsilon})$ bit operations, for any $\varepsilon > 0$. This improves upon the elementary estimate of ${\mathcal O}(p^{n-1 + \varepsilon})$ bit operations, where $n$ is the number of variables, which can be achieved using Berlekamp's root counting algorithm.