This paper explores analogies between the Weil proof of the Riemann hypothesis for function fields and the geometry of the adeles class space, which is the noncommutative space underlying Connes' spectral realization of the zeros of the Riemann zeta function. We consider the cyclic homology of the cokernel (in the abelian category of cyclic modules) of the ``restriction map'' defined by the inclusion of the ideles class group of a global field in the noncommutative adeles class space. Weil's explicit formula can then be formulated as a Lefschetz trace formula for the induced action of the ideles class group on this cohomology. In this formulation the Riemann hypothesis becomes equivalent to the positivity of the relevant trace pairing. This result suggests a possible dictionary between the steps in the Weil proof and corresponding notions involving the noncommutative geometry of the adeles class space, with good working notions of correspondences, degree and codegree etc. In particular, we construct an analog for number fields of the algebraic points of the curve for function fields, realized here as classical points (low temperature KMS states) of quantum statistical mechanical systems naturally associated to the periodic orbits of the action of the ideles class group, that is, to the noncommutative spaces on which the geometric side of the trace formula is supported.
The ring of adèles of a global field and its group of units, the group of idèles, are fundamental objects in modern number theory. We discuss a formalization of their definitions in the Lean 3 theorem prover. As a prerequisite, we formalized adic valuations on Dedekind domains. We present some applications, including the statement of the main theorem of global class field theory and a proof that the ideal class group of a number field is isomorphic to an explicit quotient of its idèle class group.
Let K be a global field. Using natural spaces of functions on the adele ring and the idele class group of K, we construct a virtual representation of the idele class group of K whose character is equal to the Weil distribution that occurs in Andre Weil's Explicit Formula. Hence this representation encodes information about the distribution of the prime ideals of K, and it is a spectral interpretation for the poles and zeros of the L-function. It is closely related to the spectral interpretation of Alain Connes.
We prove the Hasse-Weil inequality for genus 2 curves given by an equation of the form y^2 = f(x) with f a polynomial of degree 5, using arguments that mimic the elementary proof of the genus 1 case obtained by Yu. I. Manin in 1956.
A well-known result of von zur Gathen asserts that a non-exceptional permutation polynomial of degree $n$ over $\mathbb{F}_{q}$ exists only if $q<n^{4}$. With the help of the Weil bound for the number of $\mathbb{F}_{q}$-points on an absolutely irreducible (possibly singular) affine plane curve, Chahal and Ghorpade improved von zur Gathen's proof to replace $n^{4}$ by a bound less than $n^{2}(n-2)^{2}$. Also based on the Weil bound, we further refine the upper bound for $q$ with respect to $n$, by a more concise and direct proof following Wan's arguments.
We present a proof of the Chevalley-Weil Theorem that is somewhat different from the proofs appearing in the literature and with somewhat weaker hypotheses, of purely topological type. We also provide a discussion of the assumptions, and an application to solutions of generalized Fermat equations, where our statement allows to simplify the original argument of Darmon and Granville.
We say that a function f defined on R or Qp has a well defined weak Mellin transform (or weak zeta integral) if there exists some function $M\_f(s)$ so that we have $Mell(ϕ\star f,s) = Mell(ϕ,s)M\_f(s)$ for all test functions $ϕ$ in $C\_c^\infty(R^*)$ or $C\_c^\infty(Q\_p^*)$. We show that if $f$ is a non degenerate second degree character on R or Qp, as defined by Weil, then the weak Mellin transform of $f$ satisfies a functional equation and cancels only for $\Re(s) = 1/2$. We then show that if $f$ is a non degenerate second degree character defined on the adele ring $A\_Q$, the same stateme…
In 1952 Peter Roquette gave an arithmetic proof of the Riemann hypothesis for algebraic function fields of a finite constants field, which was proved by André Weil in 1940. The construction of Weil's scalar product is essential in Roquette's proof. In this paper a scalar product for algebraic function fields over a number field is constructed which is the analogue of Weil's scalar product.
We give a proof that the Riemann hypothesis for hypersurfaces over finite fields implies the result for all smooth proper varieties, by a deformation argument which does not use the theory of Lefschetz pencils or the l-adic Fourier transform.
Let $F$ be a finite field, $μ$ be a fixed additive character and $s$ be an integer coprime to $|F^\times|$. For any $a\in F$, the corresponding Weil sum is defined to be $W_{F,s}(a)=\displaystyle\sum_{x \in F} μ(x^s-ax)$. The Weil spectrum counts distinct values of the Weil sum as $a$ runs through the invertible elements in the finite field. The value of these sums and the size of the Weil spectrum are of particular interest, as they link problems in coding and information theory to other areas of math such as number theory and arithmetic geometry. In the setting of Niho exponents, we examine …
We consider Weil sums of binomials of the form $W_{F,d}(a)=\sum_{x \in F} ψ(x^d-a x)$, where $F$ is a finite field, $ψ\colon F\to {\mathbb C}$ is the canonical additive character, $\gcd(d,|F^\times|)=1$, and $a \in F^\times$. If we fix $F$ and $d$ and examine the values of $W_{F,d}(a)$ as $a$ runs through $F^\times$, we always obtain at least three distinct values unless $d$ is degenerate (a power of the characteristic of $F$ modulo $|F^\times|$). Choices of $F$ and $d$ for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if $F$ is …
We give new proofs of two basic results in number theory: The law of quadratic reciprocity and the sign of the Gauss sum. We show that these results are encoded in the relation between the discrete Fourier transform and the action of the Weyl element in the Weil representation modulo p,q and pq.
In one of his papers, using arguments about l-adic representations, Taniyama expresses the zeta function of an abelian variety over a number field as an infinite product of modified Artin L-functions. The latter can be further decomposed as products of modified Dedekind zeta functions. After recalling Taniyama's work, we give a simple geometric proof of the resulting product formula for abelian and more general group schemes.
Complex-valued periodic sequences, u, constructed by Goran Bjorck, are analyzed with regard to the behavior of their discrete periodic narrow-band ambiguity functions A_p(u). The Bjorck sequences, which are defined on Z/pZ for p>2 prime, are unimodular and have zero autocorrelation on (Z/pZ)\{0}. These two properties give rise to the acronym, CAZAC, to refer to constant amplitude zero autocorrelation sequences. The bound proven is |A_p(u)| \leq 2/\sqrt{p} + 4/p outside of (0,0), and this is of optimal magnitude given the constraint that u is a CAZAC sequence. The proof requires the full power …
The statement of the Riemann hypothesis makes sense for all global fields, not just the rational numbers. For function fields, it has a natural restatement in terms of the associated curve. Weil's work on the Riemann hypothesis for curves over finite fields led him to state his famous "Weil conjectures", which drove much of the progress in algebraic and arithmetic geometry in the following decades. In this article, I describe Weil's work and some of the ensuing progress: Weil cohomology (etale, crystalline); Grothendieck's standard conjectures; motives; Deligne's proof; Hasse-Weil zeta functio…
Ideles and adeles can be viewed as a generalization of Minkowski theory, in which embedding of a number field to the Cartesian product of its completions at the archimedean valuation is generalized to an embedding of the Cartesian product of all its completions with some restriction. This paper introduces the basic notions of point-set topology and builds the real numbers from the rational numbers. Then we review concepts from local fields that will lead to the product formula and the approximation theorem. We, then, construct adeles and ideles. The ideles modulo $k^*$ maps surjectively to the…
Let $V$ be a finite abelian group of odd order, equipped with a non-degenerate, alternating form $ω\colon V\times V \to \mathbb{Z}/m\mathbb{Z}$. We give closed formulas for the character values of the Weil representation associated with $(V,ω)$. These formulas generalize the ones given by S. Gurevich and R. Hadani (2007) and by T. Thomas (2008, 2013) from finite vector spaces to abelian groups. Our formulas do not involve the choice of a Lagrangian subgroup of $V$. Our proofs are based on an elementary algebraic approach introduced by H. N. Ward (1972, 2017) for finite vector spaces over field…
We proof a Garrett-Böcherer decomposition of a vector valued Siegel Eisenstein series $E_{l,0}^2$ of genus 2 transforming with the Weil representation of $\text{Sp}_2(\mathbb{Z})$ on the group ring $\mathbb{C}[(L'/L)^2]$. We show that the standard zeta function associated to a vector valued common eigenform $f$ for the Weil representation can be meromorphically continued to the whole $s$-plane and that it satisfies a functional equation. The proof is based on an integral representation of this zeta function in terms of $f$ and $E_{l,0}^2$.
If E is a non-isotrivial elliptic curve over a global function field F of odd characteristic we show that certain Mordell-Weil groups of E have 1-dimensional eigenspace relative to a fixed complex ring class character provided that the projection onto this eigenspace of a suitable Drinfeld-Heegner point is nonzero. This represents the analogue in the function field setting of a theorem for rational elliptic curves due to Bertolini and Darmon, and at the same time is a generalization of the main result proved by Brown in his monograph on Heegner modules. As in the number field case, our proof e…
Let $X$ be a smooth cubic hypersurface of dimension $n \ge 1$ over the rationals. It is well-known that new rational points may be obtained from old ones by secant and tangent constructions. In view of the Mordell--Weil theorem for $n=1$, Manin (1968) asked if there exists a finite set $S$ from which all other rational points can be thus obtained. We give an affirmative answer for $n \ge 48$, showing in fact that we can take the generating set $S$ to consist of just one point. Our proof makes use of a weak approximation theorem due to Skinner, a theorem of Browning, Dietmann and Heath-Brown on…
We push further the classical proof of Weil upper bound for the number of rational points of an absolutely irreducible smooth projective curve $X$ over a finite field in term of euclidean relationships between the Neron Severi classes in $X\times X$ of the graphs of iterations of the Frobenius morphism. This allows us to recover Ihara's bound, which can be seen as a {\em second order} Weil upper bound, to establish a new {\em third order} Weil upper bound, and using {\tt magma} to produce numerical tables for {\em higher order} Weil upper bounds. We also give some interpretation for the defect…
One of the driving motivations to develop $\F_1$-geometry is the hope to translate Weil's proof of the Riemann hypothesis from positive characteristics to number fields, which might result in a proof of the classical Riemann hypothesis. The underlying idea is that the spectrum of $\Z$ should find an interpretation as a curve over $\F_1$, which has a completion $\bar{\Spec\Z}$ analogous to a curve over a finite field. The hope is that intersection theory for divisors on the arithmetic surface $\bar{\Spec\Z} \times \bar{\Spec\Z}$ will allow to mimic Weil's proof. It turns out that it is possible…
We show that many quadratic binomial functions on a finite field of characteristic 2 are not APN infinitely often. This is of interest in the light of recent discoveries of new families of quadratic binomial APN functions. The proof uses the Weil bound from algebraic geometry.
The local Langlands correspondence for GL(n) of a non-Archimedean local field $F$ parametrizes irreducible admissible representations of $GL(n,F)$ in terms of representations of the Weil-Deligne group $WD_F$ of $F$. The correspondence, whose existence for $p$-adic fields was proved in joint work of the author with R. Taylor, and then more simply by G. Henniart, is characterized by its preservation of salient properties of the two classes of representations. The article reviews the strategies of the two proofs. Both the author's proof with Taylor and Henniart's proof are global and rely ultimat…
