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International audience; We provide a theoretical study of Algebraic Geometry codes constructed from abelian surfaces defined over finite fields. We give a general bound on their minimum distance and we investigate how this estimation can be sharpened under the assumption that the abelian surface does not contain low genus curves. This approach naturally leads us to consider Weil restrictions of elliptic curves and abelian surfaces which do not admit a principal polarization.

The Dickson–Guralnick–Zieve curve, briefly DGZ curve, defined over the finite field F q arises naturally from the classical Dickson invariant of the projective linear group P G L ( 3 , F q ) . The DGZ curve is an (absolutely irreducible, singular) plane curve of degree q 3 − q 2 and genus 1 2 q ( q − 1 ) ( q 3 − 2 q − 2 ) + 1 . In this paper we show that the DGZ curve has several remarkable features, those appearing most interesting are: the DGZ curve has a large automorphism group compared to its genus albeit its Hasse–Witt invariant is positive; the Fermat curve of degree q − 1 is a quotient curve of the DGZ curve; among the plane curves with the same degree and genus of the DGZ curve and defined over F q 3 , the DGZ curve is optimal with respect the number of its F q 3 -rational points.

Let n be a positive integer, q = 2 n , and let F q be the finite field with q elements. For each positive integer m, let D m ( X ) be the Dickson polynomial of the first kind of degree m with parameter 1. Assume that m > 1 is a divisor of q + 1 . We study the existence of α ∈ F q ⁎ such that D m ( α ) = D m ( α − 1 ) = 0 . We also explore the connections of this question to an open question by Wiedemann and a game called “Button Madness”.

Let t be an integer ≥ 3 such that t ≡ 1 mod 4. The absolute irreducibility of the polynomial $\phi _{t}(x, y) = \frac {x^{t} + y^{t} + 1 + (x + y + 1)^{t}}{(x + y)(x + 1)(y + 1)}$ (over $\mathbb {F}_{2}$ ) plays an important role in the study of APN functions. We prove that this polynomial is absolutely irreducible under the assumptions that the largest odd integer which divides t − 1 is large enough and can not be written in a specific form.

This paper contains three new results. {\bf 1}.We introduce new notions of projective crystalline representations and twisted periodic Higgs-de Rham flows. These new notions generalize crystalline representations of \'etale fundamental groups introduced in [7,10] and periodic Higgs-de Rham flows introduced in [19]. We establish an equivalence between the categories of projective crystalline representations and twisted periodic Higgs-de Rham flows via the category of twisted Fontaine-Faltings module which is also introduced in this paper. {\bf 2.}We study the base change of these objects over very ramified valuation rings and show that a stable periodic Higgs bundle gives rise to a geometrically absolutely irreducible crystalline representation. {\bf 3.} We investigate the dynamic of self-maps induced by the Higgs-de Rham flow on the moduli spaces of rank-2 stable Higgs bundles of degree 1 on $\mathbb{P}^1$ with logarithmic structure on marked points $D:=\{x_1,\,...,x_n\}$ for $n\geq 4$ and construct infinitely many geometrically absolutely irreducible $\mathrm{PGL_2}(\mathbb Z_p^{\mathrm{ur}})$-crystalline representations of $\pi_1^\text{et}(\mathbb{P}^1_{{\mathbb{Q}}_p^\text{ur}}\setminus D)$. We find an explicit formula of the self-map for the case $\{0,\,1,\,\infty,\,\lambda\}$ and conjecture that a Higgs bundle is periodic if and only if the zero of the Higgs field is the image of a torsion point in the associated elliptic curve $\mathcal{C}_\lambda$ defined by $ y^2=x(x-1)(x-\lambda)$ with the order coprime to $p$., Comment: 84 pages

International audience; We give a construction of singular curves with many rational points over finite fields. This construction enables us to prove some results on the maximum number of rational points on an absolutely irreducible projective algebraic curve defined over Fq of geometric genus g and arithmetic genus π.

The number of rank two $\overline{\Bbb{Q}}_\ell$-local systems, or $\overline{\Bbb{Q}}_\ell$-smooth sheaves, on $(X-\{u\})\otimes_{\Bbb{F}_q}\Bbb{F}$, where $X$ is a smooth projective absolutely irreducible curve over $\Bbb{F}_q$, $\Bbb{F}$ an algebraic closure of $\Bbb{F}_q$ and $u$ is a closed point of $X$, with principal unipotent monodromy at $u$, and fixed by ${\rm Gal}(\Bbb{F}/\Bbb{F}_q)$, is computed. It is expressed as the trace of the Frobenius on the virtual $\overline{\Bbb{Q}}_\ell$-smooth sheaf found in the author's work with Deligne on the moduli stack of curves with \'etale divisors of degree $M\ge 1$. This completes the work with Deligne in rank two. This number is the same as that of representations of the fundamental group $\pi_1((X-\{u\})\otimes_{\Bbb{F}_q}\Bbb{F})$ invariant under the Frobenius ${\rm Fr}_q$ with principal unipotent monodromy at $u$, or cuspidal representations of ${\rm GL}(2)$ over the function field $F=\Bbb{F}_q(X)$ of $X$ over $\Bbb{F}_q$ with Steinberg component twisted by an unramified character at $u$ and unramified elsewhere, trivial at the fixed id\`ele $\alpha$ of degree 1. This number is computed in Theorem 4.1 using the trace formula evaluated at $f_u\prod_{v\not=u}\chi_{K_v}$, with an Iwahori component $f_u=\chi_{I_u}/|I_u|$, hence also the pseudo-coefficient $\chi_{I_u}/|I_u|-2\chi_{K_u}$ of the Steinberg representation twisted by any unramified character, at $u$. Theorem 2.1 records the trace formula for ${\rm GL}(2)$ over the function field $F$. The proof of the trace formula of Theorem 2.1 recently appeared elsewhere. Theorem 3.1 computes, following Drinfeld, the number of $\overline{\Bbb{Q}}_\ell$-local systems, or $\overline{\Bbb{Q}}_\ell$-smooth sheaves, on $X\otimes_{\Bbb{F}_q}\Bbb{F}$, fixed by ${\rm Fr}_q$, namely $\overline{\Bbb{Q}}_\ell$-representations of the absolute fundamental group $\pi(X\otimes_{\Bbb{F}_q}\Bbb{F})$ invariant under the Frobenius, by counting the nowhere ramified cuspidal representations of ${\rm GL}(2)$ trivial at a fixed id\`ele $\alpha$ of degree 1. This number is expressed as the trace of the Frobenius of a virtual $\overline{\Bbb{Q}}_\ell$-smooth sheaf on a moduli stack. This number is obtained on evaluating the trace formula at the characteristic function $\prod_v\chi_{K_v}$ of the maximal compact subgroup, with volume normalized by $|K_v|=1$. Section 5, based on a letter of P. Deligne to the author dated August 8, 2012, computes the number of such objects with any unipotent monodromy, principal or trivial, in our rank two case. Surprisingly, this number depends only on $X$ and ${\rm deg}(S)$, and not on the degrees of the points in $S_1$.

Let $t$ be an integer $\ge 5$. The absolute irreducibility of the polynomial $\phi_t(x, y) = \frac{x^t + y^t + 1 + (x + y + 1)^t}{(x + y)(x + 1)(y + 1)}$ (over $\mathbb{F}_2$) plays an important role in the study of APN functions. If $t \equiv 5 \bmod{8}$, we give a criterion that ensures that $\phi_t(x, y)$ is absolutely irreducible. We prove that if $\phi_t(x, y)$ is not absolutely irreducible, then it is divisible by $\phi_{13}(x, y)$. We also exhibit an infinite family of integers $t$ such that $\phi_t(x, y)$ is not absolutely irreducible.