On the Zeta Function of Forms of Fermat Equations

The paper reviews Dwork's p-adic analytic methods used in the Weil Conjectures. The first two chapters review a version of his proof of the rationality conjecture. The rest of the paper is devoted to Dwork's original cohomological methods, along with some p-adic functional analysis. The last chapter applies what is developed to an example of Dwork's family of hypersurfaces.

Let $\mathbb{F}_q$ be a finite field with $q$ elements, $\psi$ a non-zero element of $\mathbb{F}_q$, and $n$ an integer $\geq 3$ prime to $q$. The aim of this article is to show that the zeta function of the projective variety over $\mathbb{F}_q$ defined by $X_\psi \colon x_1^n+...+x_n^n - n \psi x_1... x_n=0$ has, when $n$ is prime and $X_\psi$ is non singular (i.e. when $\psi^n \neq 1$), an explicit decomposition in factors coming from affine varieties of odd dimension $\le...

For a scheme $X$ whose $\mathbb F_q$-rational points are counted by a polynomial $N(q)=\sum a_iq^i$, the $\mathbb{F}_1$-zeta function is defined as $\zeta(s)=\prod(s-i)^{-a_i}$. Define $\chi=N(1)$. In this paper we show that if $X$ is a smooth projective scheme, then its $\mathbb{F}_1$-zeta function satisfies the functional equation $\zeta(n-s) = (-1)^\chi \zeta(s)$. We further show that the $\mathbb{F}_1$-zeta function $\zeta(s)$ of a split reductive group scheme $G$ of rank...

We give improvements of the deformation method for computing the zeta function of a generic projective hypersurface in characteristic~$p$ that either reduce the dependence on~$p$ of the time complexity to $\tilde{O}(p^{1/2})$ or that of the space complexity to $\tilde{O}(\log(p))$ while remaining polynomial in the other input parameters.

The purpose of this article is to give an explicit description, in terms of hypergeometric functions over finite fields, of zeta function of a certain type of smooth hypersurfaces that generalizes Dwork family. The point here is that we count the number of rational points employing both character sums and the theory of weights, which enables us to enlighten the calculation of the zeta function.

In this paper we study the Fermat equation $x^n+y^n=z^n$ over quadratic fields $\mathbb{Q}(\sqrt{d})$ for squarefree $d$ with $26 \leq d \leq 97$. By studying quadratic points on the modular curves $X_0(N)$, $d$-regular primes, and working with Hecke operators on spaces of Hilbert newforms, we extend work of Freitas and Siksek to show that for most squarefree $d$ in this range there are no non-trivial solutions to this equation for $n \geq 4$.

We consider a Fermat curve $F_n:x^n+y^n+z^n=1$ over an algebraically closed field $k$ of characteristic $p\geq0$ and study the action of the automorphism group $G=\left(\mathbb{Z}/n\mathbb{Z}\times\mathbb{Z}/n\mathbb{Z}\right)\rtimes S_3$ on the canonical ring $R=\bigoplus H^0(F_n,\Omega_{F_n}^{\otimes m})$ when $p>3$, $p\nmid n$ and $n-1$ is not a power of $p$. In particular, we explicitly determine the classes $[H^0(F_n,\Omega_{F_n}^{\otimes m})]$ in the Grothendieck group ...

We review a combinatoric approach to the Hodge Conjecture for Fermat Varieties and announce new cases where the conjecture is true.

We show that even dimensional Fermat cubic hypersurfaces are rational over any field of characteristic different from three. As a byproduct of our rationality constructions we get estimates on the number of their rational points over a number field, and a class of quadro-cubic Cremona correspondences of even dimensional projective spaces.

In this paper, we compute the number of real forms of Fermat hypersurfaces for degree $d \ge 3$ except the degree 4 surface case, and give explicit descriptions of them.