Weil Conjectures Research Articles

Quadratic enrichment of the logarithmic derivative of the zeta function

We define an enrichment of the logarithmic derivative of the zeta function of a variety over a finite field to a power series with coefficients in the Grothendieck–Witt group. We show that this enrichment is related to the topology of the real points of a lift. For cellular schemes over a field, we prove a rationality result for this enriched logarithmic derivative of the zeta function as an analogue of part of the Weil conjectures. We also compute several examples, including toric varieties, and show that the enrichment is a motivic measure.

Frobenius Distributions of Low Dimensional Abelian Varieties Over Finite Fields

Abstract Given a $g$-dimensional abelian variety $A$ over a finite field $\mathbf{F}_{q}$, the Weil conjectures imply that the normalized Frobenius eigenvalues generate a multiplicative group of rank at most $g$. The Pontryagin dual of this group is a compact abelian Lie group that controls the distribution of high powers of the Frobenius endomorphism. This group, which we call the Serre–Frobenius group, encodes the possible multiplicative relations between the Frobenius eigenvalues. In this article, we classify all possible Serre–Frobenius groups that occur for $g \le 3$. We also give a partial classification for simple ordinary abelian varieties of prime dimension $g\geq 3$.

Birational anabelian Grothendieck Conjecture for curves over arbitrary cyclotomic extension fields of number fields

In anabelian geometry, various strong/desired forms of Grothendieck Conjecture-type results for hyperbolic curves over relatively small arithmetic fields — for instance, finite fields, number fields, or p-adic local fields — have been obtained by many researchers, especially by A. Tamagawa and S. Mochizuki. Let us recall that, in their proofs, the Weil Conjecture or p-adic Hodge theory plays an essential role. Therefore, to obtain such Grothendieck Conjecture-type results, it appears that the condition that the cyclotomic characters of the absolute Galois groups of the base fields are highly nontrivial is indispensable. On the other hand, in an author's recent joint work with Y. Hoshi and S. Mochizuki, we introduced the notion of TKND-AVKF-field [concerning the divisible subgroups of the groups of rational points of semi-abelian varieties] and obtained the semi-absolute version of the Grothendieck Conjecture for higher dimensional (≥2) configuration spaces associated to hyperbolic curves of genus 0 over TKND-AVKF-fields contained in the algebraic closure of the field of rational numbers. For instance, every [possibly, infinite] cyclotomic extension field of a number field is such a TKND-AVKF-field. In particular, this Grothendieck Conjecture-type result suggests that the condition that the cyclotomic character of the absolute Galois group of the base field under consideration is sufficiently nontrivial is, in fact, not indispensable for strong/desired form of anabelian phenomena. In the present paper, to pose another evidence for this observation, we prove the relative birational version of the Grothendieck Conjecture for smooth curves over TKND-AVKF-fields with a certain mild condition that every cyclotomic extension field of a number field satisfies. From the viewpoint of the condition on base fields, this result may be regarded as a partial generalization of F. Pop and S. Mochizuki's results on the birational version of the Grothendieck Conjecture for smooth curves.

Noncommutative Weil conjecture

In this article, following an insight of Kontsevich, we extend the Weil conjecture, as well as the strong form of the Tate conjecture, from the realm of algebraic geometry to the broad noncommutative setting of dg categories. Moreover, we establish a functional equation for the noncommutative Hasse-Weil zeta functions, compute the l-adic and p-adic absolute values of the eigenvalues of the cyclotomic Frobenius, and provide a complete description of the category of noncommutative numerical motives in terms of Weil q-numbers. As a first application, we prove the noncommutative Weil conjecture and the noncommutative strong form of the Tate conjecture in several cases: twisted schemes, Calabi-Yau dg categories associated to hypersurfaces, noncommutative gluings of schemes, root stacks, (twisted) global orbifolds, and finite-dimensional dg algebras. As a second application, we provide an alternative noncommutative proof of the Weil conjecture and of the strong form of the Tate conjecture in the particular cases of intersections of two quadrics and linear sections of determinantal varieties.

Almost all primes satisfy the Atkin–Serre conjecture and are not extremal

Let \(f(z)=\sum _{n=1}^{\infty } a_f(n)e^{2\pi i n z}\) be a non-CM holomorphic cuspidal newform of trivial nebentypus and even integral weight \(k\ge 2\). Deligne’s proof of the Weil conjectures shows that \(|a_f(p)|\le 2p^{\frac{k-1}{2}}\) for all primes p. We prove for 100% of primes p that \( 2p^{\frac{k-1}{2}}{\log \log p}/{\sqrt{\log p}}<|a_f(p)|<\lfloor 2p^{\frac{k-1}{2}}\rfloor .\) Our proof gives an effective upper bound for the size of the exceptional set. The lower bound shows that the Atkin–Serre conjecture is satisfied for 100% of primes, and the upper bound shows that \(|a_f(p)|\) is as large as possible (i.e., p is extremal for f) for 0% of primes. Our proofs use the effective form of the Sato–Tate conjecture proved by the second author, which relies on the recent proof of the automorphy of the symmetric powers of f due to Newton and Thorne.

The Weil Conjectures

The Weil Conjectures

The Weil Conjectures: On Math and the Pursuit of the Unknown. By KarenOlsson. Pp. 211, NY, Farrar, Straus and Giroux, 2019, $17.00.

The Heythrop JournalVolume 62, Issue 1 p. 186-187 Book Review The Weil Conjectures: On Math and the Pursuit of the Unknown. By Karen Olsson. Pp. 211, NY, Farrar, Straus and Giroux, 2019, $17.00. Patrick Madigan, Patrick Madigan Campion Hall, OxfordSearch for more papers by this author Patrick Madigan, Patrick Madigan Campion Hall, OxfordSearch for more papers by this author First published: 26 December 2020 https://doi.org/10.1111/heyj.13747Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat No abstract is available for this article. Volume62, Issue1January 2021Pages 186-187 RelatedInformation

The space of cubic surfaces equipped with a line

The Cayley--Salmon theorem implies the existence of a 27-sheeted covering space specifying lines contained in smooth cubic surfaces over $\mathbb{C}$. In this paper we compute the rational cohomology of the total space of this cover, using the spectral sequence in the method of simplicial resolution developed by Vassiliev. The covering map is an isomorphism in cohomology (in fact of mixed Hodge structures) and the cohomology ring is isomorphic to that of $PGL(4,\mathbb{C})$. We derive as a consequence of our theorem that over the finite field $\mathbb{F}_q$ the average number of lines on a cubic surface equals 1 (away from finitely many characteristics); this average is $1 + O(q^{-1/2})$ by a standard application of the Weil conjectures.

The Weil Conjectures: On Math and the Pursuit of the Unknown by Karen Olsson

The Weil Conjectures: On Math and the Pursuit of the Unknown by Karen Olsson

Noncommutative motives in positive characteristic and their applications

Let k be a base field of positive characteristic. Making use of topological periodic cyclic homology, we start by proving that the category of noncommutative numerical motives over k is abelian semi-simple, as conjectured by Kontsevich. Then, we establish a far-reaching noncommutative generalization of the Weil conjectures, originally proven by Dwork and Grothendieck. In the same vein, we establish a far-reaching noncommutative generalization of the cohomological interpretations of the Hasse-Weil zeta function, originally proven by Hesselholt. As a third main result, we prove that the numerical Grothendieck group of every smooth proper dg category is a finitely generated free abelian group, as claimed (without proof) by Kuznetsov. Then, we introduce the noncommutative motivic Galois (super-)groups and, following an insight of Kontsevich, relate them to their classical commutative counterparts. Finally, we explain how the motivic measure induced by Berthelot's rigid cohomology can be recovered from the theory of noncommutative motives.

Irrationality of generic cubic threefold via Weil's conjectures

An arithmetic method of proving the irrationality of smooth projective 3-folds is described, using reduction modulo [Formula: see text]. It is illustrated by an application to a cubic threefold, for which the hypothesis that its intermediate Jacobian is isomorphic to the Jacobian of a curve is contradicted by reducing modulo 3 and counting points over appropriate extensions of [Formula: see text]. As a spin-off, it is shown that the 5-dimensional Prym varieties arising as intermediate Jacobians of certain cubic 3-folds have the maximal number of points over [Formula: see text] which attains Perret's and Weil's upper bounds.

Lattice Models of Finite Fields

Finite fields form an important chapter in abstract algebra, and mathematics in general, yet the traditional expositions, part of Abstract Algebra courses, focus on the axiomatic presentation, while Ramification Theory in Algebraic Number Theory, making a suited topic for their applications, is usually a separated course. We aim to provide a geometric and intuitive model for finite fields, involving algebraic numbers, in order to make them accessible and interesting to a much larger audience, and bridging the above mentioned gap. Such lattice models of finite fields provide a good basis for later developing their study in a more concrete way, including decomposition of primes in number fields, Frobenius elements, and Frobenius lifts, allowing to approach more advanced topics, such as Artin reciprocity law and Weil Conjectures, while keeping the exposition to the concrete level of familiar number systems. Examples are provided, intended for an undergraduate audience in the first place.

Weil groups and the distribution of prime ideals

We generalize Chebotarev’s density theorem to Weil groups. Since the Artin–Weil conjecture on the integrality of the Artin–Hecke L-functions, constructed by A. Weil, has not been completely proved so far, we estimate the character sums both under and without the assumption of the validity of that conjecture.

Como Grothendieck simplificou a geometria algébrica

Alexandre Grothendieck (1922-2014) foi um dos maiores matemáticos do século 20 e um dos mais atípicos. Nascido na Alemanha a um pai anarquista de origem russa, sua infância foi marcada pela militância política dos seus pais, assim passando por revoluções, guerras e sobrevivência. Descoberto por sua precocidade matemática por Henri Cartan, Grothendieck fez seu doutorado sob orientação de Laurent Schwartz e Jean Dieudonné. As principais contribuições dele são na área da topologia e na geometria algébrica, assim como na teoria das categorias. No final dos anos de 1960, ele se dedicou à militância política e ecológica, organizando a revista Survivre durante três anos. Em 1986, publicou um manuscrito autobiográfico de 1000 páginas, Récoltes et semailles, em que ele descreve sua experiência e sua prática da matemática, assim suas contribuições à comunidade matemática francesa. Pouco comentado na filosofia, as implicações dos seus descobrimentos fora mais recentemente discutidas por Alain Badiou na sua "fenômeno-lógica", em Logiques des mondes (2016) e Arkady Plonitsky, Mathgematics, Science and postclassical Theory (1997), pesquisa trata da semelhança entre os aspectos formais da filosofia de Gilles Deleuze e da topologia de Grothendieck.

The Riemann hypothesis over finite fields: from Weil to the present day

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. It is my contribution to the book “The legacy of Bernhard Riemann after one hundred and fifty years”, edited by S.T. Yau et al.

Newman's conjecture in function fields

TextDe Bruijn and Newman introduced a deformation of the completed Riemann zeta function ζ, and proved there is a real constant Λ which encodes the movement of the nontrivial zeros of ζ under the deformation. The Riemann hypothesis is equivalent to the assertion that Λ≤0. Newman, however, conjectured that Λ≥0, remarking, “the new conjecture is a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so”. Andrade, Chang and Miller extended the machinery developed by Newman and Pólya to L-functions for function fields. In this setting we must consider a modified Newman's conjecture: supf∈F⁡Λf≥0, for F a family of L-functions.We extend their results by proving this modified Newman's conjecture for several families of L-functions. In contrast with previous work, we are able to exhibit specific L-functions for which ΛD=0, and thereby prove a stronger statement: maxL∈F⁡ΛL=0. Using geometric techniques, we show a certain deformed L-function must have a double root, which implies Λ=0. For a different family, we construct particular elliptic curves with p+1 points over Fp. By the Weil conjectures, this has either the maximum or minimum possible number of points over Fp2n. The fact that #E(Fp2n) attains the bound tells us that the associated L-function satisfies Λ=0. VideoFor a video summary of this paper, please visit http://youtu.be/hM6-pjq7Gi0.

Hypersurfaces and the Weil Conjectures

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.

Rational Points on Elliptic Curves $y^{2}=x^{3}+a^{3}$ in ${\bf F}_p$ where $p\equiv 1\pmod6$ is Prime

In this work, we consider the rational points on elliptic curves over finite fields F_{p}. We give results concerning the number of points on the elliptic curve y^2{\equiv}x^3+a^3(mod p)where p is a prime congruent to 1 modulo 6. Also some results are given on the sum of abscissae of these points. We give the number of solutions to y^2{\equiv}x^3+a^3(modp), also given in ([1], p.174), this time by means of the quadratic residue character, in a different way, by using the cubic residue character. Using the Weil conjecture, one can generalize the results concerning the number of points in F_{p} to F_{p^{r}}.

Fourier transforms and $p$-adic ‘Weil II’

We give a purity theorem in the manner of Deligne's ‘Weil II’ theorem for rigid cohomology with coefficients in an overconvergent $F$ -isocrystal; the proof mostly follows Laumon's Fourier-theoretic approach, transposed into the setting of arithmetic $\mathcal{D}$ -modules. This yields in particular a complete, purely $p$ -adic proof of the Weil conjectures when combined with recent results on $p$ -adic differential equations by Andre, Christol, Crew, Kedlaya, Matsuda, Mebkhout and Tsuzuki.

Deforming 𝑙-adic representations of the fundamental group of a smooth variety

There has long been a philosophy that every deformation problem in characteristic zero should be governed by a differential graded Lie algebra (DGLA). This paper develops the theory of Simplicial Deformation Complexes (SDCs) as an alternative to DGLAs. These work in all characteristics, and for many problems can be constructed canonically. This theory is applied to study the deformation functor for representations of the étale fundamental group of a variety X X . We are chiefly concerned with establishing an algebraic analogue of a result proved by Goldman and Millson for compact Kähler manifolds. By applying the Weil Conjectures instead of Hodge theory, we see that if X X is a smooth proper variety defined over a finite field, and we consider deformations of certain continuous l l -adic representations of the algebraic fundamental group, then the hull of the deformation functor will be defined by quadratic equations. Moreover, if X X is merely smooth, then the hull will be defined by equations of degree at most four.