On the Zeta Function of Forms of Fermat Equations
نویسنده
چکیده
We study “forms of the Fermat equation” over an arbitrary field k, i.e. homogenous equations of degree m in n unknowns that can be transformed into the Fermat equation Xm 1 +. . .+Xm n by a suitable linear change of variables over an algebraic closure of k. Using the method of Galois descent, we classify all such forms. In the case that k is a finite field of characteristic greater than m that contains the m-th roots of unity, we compute the Galois representation on l-adic cohomology (and so in particular the zeta function) of the hypersurface associated to an arbitrary form of the Fermat equation.
منابع مشابه
A more accurate half-discrete Hardy-Hilbert-type inequality with the best possible constant factor related to the extended Riemann-Zeta function
By the method of weight coefficients, techniques of real analysis and Hermite-Hadamard's inequality, a half-discrete Hardy-Hilbert-type inequality related to the kernel of the hyperbolic cosecant function with the best possible constant factor expressed in terms of the extended Riemann-zeta function is proved. The more accurate equivalent forms, the operator expressions with the norm, the rever...
متن کاملOn some mathematical connections between Fermat’s Last Theorem, Modular Functions, Modular Elliptic Curves and some sector of String Theory
This paper is fundamentally a review, a thesis, of principal results obtained in some sectors of Number Theory and String Theory of various authoritative theoretical physicists and mathematicians. Precisely, we have described some mathematical results regarding the Fermat’s Last Theorem, the Mellin transform, the Riemann zeta function, the Ramanujan’s modular equations, how primes and adeles ar...
متن کاملThe Zeta-function of Monomial Deformations of Fermat Hypersurfaces
This paper intends to give a mathematical explanation for results on the zeta-function of some families of varieties recently obtained in the context of Mirror Symmetry [4], [9]. In doing so, we obtain concrete and explicit examples for some results recently used in algorithms to count points on smooth hypersurfaces in Pn. In particular, we extend the monomial-motive correspondence of Kadir and...
متن کاملOn group equations
Suppose $f$ is a map from a non-empty finite set $X$ to a finite group $G$. Define the map $zeta^f_G: Glongrightarrow mathbb{N}cup {0}$ by $gmapsto |f^{-1}(g)|$. In this article, we show that for a suitable choice of $f$, the map $zeta^f_G$ is a character. We use our results to show that the solution function for the word equation $w(t_1,t_2,dots,t_n)=g$ ($gin G$) is a character, where $w(t_1,...
متن کاملQuantum Computing and Zeroes of Zeta Functions
A possible connection between quantum computing and Zeta functions of finite field equations is described. Inspired by the spectral approach to the Riemann conjecture, the assumption is that the zeroes of such Zeta functions correspond to the eigenvalues of finite dimensional unitary operators of natural quantum mechanical systems. The notion of universal, efficient quantum computation is used ...
متن کامل