In our previous paper [math.NT/0408050], we established a correspondence between vector-valued holomorphic Siegel modular forms and cohomology with local coefficients for local symmetric spaces $X$ attached to real orthogonal groups of type $(p,q)$. This correspondence is realized using theta functions associated to explicitly constructed "special" Schwartz forms. Furthermore, the theta functions give rise to generating series of certain "special cycles" in $X$ with coefficients.
In this paper, we study the boundary behaviour of these theta functions in the non-compact case and show that the theta functions extend to the Borel-Sere compactification $\bar{X}$ of $X$. However, for the $\Q$-split case for signature $(p,p)$, we have to construct and consider a slightly larger compactification, the "big" Borel-Serre compactification. The restriction to each face of $\bar{X}$ is again a theta series as in [math.NT/0408050], now for a smaller orthogonal group and a larger coefficient system.
As application we establish the cohomological nonvanishing of the special (co)cycles when passing to an appropriate finite cover of $X$. In particular, the (co)homology groups in question do not vanish.
We give a direct description of the category of sheaves on Lichtenbaum's Weil-étale site of a number ring. Then we apply this result to define a spectral sequence relating Weil-étale cohomology to Artin-Verdier étale cohomology. Finally we construct complexes of étale sheaves computing the expected Weil-étale cohomology.
We study Weil-etale cohomology, introduced by Lichtenbaum for varieties over finite fields. In the first half of the paper we give an explicit description of the base change from Weil-etale cohomology to etale cohomology. As a consequence, we get a long exact sequence relating Weil-etale cohomology to etale cohomology, show that for finite coefficients the cohomology theories agree, and with rational coefficients a Weil-etale cohomology group is the direct sum of two etale cohomology groups. In the second half of the paper we restrict ourselves to Weil-etale cohomology of the motivic complex. …
We establish duality results for the cohomology of the Weil group of a $p$-adic field, analogous to, but more general than, results from Galois cohomology. We prove a duality theorem for discrete Weil modules, which implies Tate-Nakayama Duality. We define Weil-smooth cohomology for varieties over local fields, and prove a duality theorem for the cohomology of $\G_m$ on a smooth, proper curve with a rational point. This last theorem is analogous to, and implies, a classical duality theorem for such curves.
In this article, we first briefly introduce the history of the Weil-étale cohomology theory of arithmetic schemes and review some important results established by Lichtenbaum, Flach and Morin. Next we generalize the Weil-etale cohomology to $S$-integers and compute the cohomology for constant sheaves $\mathbb{Z}$ or $\mathbb{R}$. We also define a Weil-étale cohomology with compact support $H_c(Y_W, -)$ for $Y=Spec \mathcal{O}_{F,S}$ where $F$ is a number field, and computed them. We verify that these cohomology groups satisfy the axioms state by Lichtenbaum. As an application, we derive a cano…
We calculate the total derived functor for the map from the Weil-etale site introduced by Lichtenbaum to the etale site for varieties over finite fields. In particular, there is a long exact sequence relating Weil-etale cohomology and etale cohomology. In the second half of the paper, we apply this to study the Weil-etale cohomology of the motivic complex for smooth and projective varieties. These groups are expected to be finitely generated, to give an integral model for l-adic cohomology, and to be related to special values of the zeta function. We give necessary and sufficient conditions fo…
We generalize a theorem of Tate and show that the second cohomology of the Weil group of a global or local field with coefficients in $\C^*$ (or more generally, with coefficients in the complex points of a tori over $\C$) vanish, where the cohomology groups are defined using measurable cochains in the sense of Moore. We recover a theorem of Labesse proving that admissible homomorphisms of a Weil group to the Langlands dual of a reductive group can be lifted to an extension of the Langlands dual group by a tori.
We construct the Weil-étale cohomology and Euler characteristics for a subclass of the class of $\mathbb{Z}$-constructible sheaves on the spectrum of the ring of integers of a totally imaginary number field. Then we show that the special value of an Artin L-function of toric type at zero is given by the Weil-étale Euler characteristic of an appropriate $\mathbb{Z}$-constructible sheaf up to signs. As applications of our result, we will derive some classical formulas of special values of L-functions of tori and their class numbers.
We construct the Weil-étale cohomology and Euler characteristics for a subclass of the class of $\mathbb{Z}$-constructible sheaves on an open subscheme of the spectrum of the ring of integers of a number field. Then we show that the special value of an Artin L-function of toric type at zero is given by the Weil-étale Euler characteristic of an appropriate $\mathbb{Z}$-constructible sheaf up to signs. As applications of our result, we will prove a formula for the special value of the L-function of an algebraic torus at zero which is similar to Ono's Tamagawa Number Formula.
We investigate the action of the Weil group on the compactly supported l-adic cohomology groups of rigid spaces over local fields. We prove that every eigenvalue of the action is a Weil number when either a rigid space is smooth or the characteristic of the base field is equal to 0. Since a smooth rigid space is locally isomorphic to the Raynaud generic fiber of an algebraizable formal scheme, we can use the techniques of alterations and monodromy spectral sequences in the smooth case. In the general case, we use the continuity theorem of Huber, which requires the restriction on the characteri…
We construct an intermediate cohmology between motivic cohomology and Weil-etale cohomology. Using this, the Bass conjecture on finite generation of motivic cohomology, and the Beilinson-Tate on the finite generation of Weil-etale cohomology are related.
We define the fundamental group underlying to Lichtenbaum's Weil-étale cohomology for number rings. To this aim, we define the Weil-étale topos as a refinement of the Weil-étale sites introduced in \cite{Lichtenbaum}. We show that the (small) Weil-étale topos of a smooth projective curve defined in this paper is equivalent to the natural definition given in \cite{Lichtenbaum-finite-field}. Then we compute the Weil-étale fundamental group of an open subscheme of the spectrum of a number ring. Our fundamental group is a projective system of locally compact topological groups, which represents fi…
We define and study a Weil-étale topos for any regular, proper scheme $X$ over $\Spec(Z)$ which has some of the properties suggested by Lichtenbaum for such a topos. In particular, the cohomology with $R$-coefficients has the expected relation to $ζ(X,s)$ at $s=0$ if the Hasse-Weil L-functions $L(h^i(X_Q),s)$ have the expected meromorphic continuation and functional equation. If $\X$ has characteristic $p$ the cohomology with $Z$-coefficients also has the expected relation to $ζ(X,s)$ and our cohomology groups recover those previously studied by Lichtenbaum and Geisser.
We give a conjectural description of the vanishing order and leading Taylor coefficient of the Zeta function of a proper, regular arithmetic scheme $\mathcal{X}$ at any integer $n$ in terms of Weil-étale cohomology complexes. This extends work of Lichtenbaum \cite{Lichtenbaum05} and Geisser \cite{Geisser04b} for $\mathcal{X}$ of characteristic $p$, of Lichtenbaum \cite{li04} for $\mathcal{X}=\mathrm{Spec}(\mathcal{O}_F)$ and $n=0$ where $F$ is a number field, and of the second author for arbitrary $\mathcal{X}$ and $n=0$ \cite{Morin14}. We show that our conjecture is compatible with the Tamaga…
We consider the Tate cohomology of the circle group acting on the topological Hochschild homology of schemes. We show that in the case of a scheme smooth and proper over a finite field, this cohomology theory naturally gives rise to the cohomological interpretation of the Hasse-Weil zeta function by regularized determinants envisioned by Deninger. In this case, the periodicity of the zeta function is reflected by the periodicity of said cohomology theory, whereas neither is periodic in general.
In the setting of generalized class field theory, one has a field K with a hyper-class formation C, a complex of G_K-modules that plays the role of a class formation. Letting C(L) denote H^0(G_L, C), there is a distinguished cohomology class in H^2(Gal(L/K), C(L)), the hyper-fundamental class. In this paper, we prove that a group extension of Gal(L/K) by C(L) whose corresponding cohomology class is the hyper-fundamental class satisfies the classical axioms of a Weil group.
We give a new formula for the special value at s=2 of the Hasse-Weil zeta function for smooth projective fourfolds under some assumptions (the Tate and Beilinson conjecture, finiteness of some cohomology groups, etc.). Our formula may be considered as a generalization of the Artin-Tate(-Milne) formula for smooth surfaces, and expresses the special zeta value almost exclusively in terms of inner geometric invariants such as higher Chow groups (motivic cohomology groups). Moreover we compare our formula with Geisser's formula for the same zeta value in terms of Weil-étale motivic cohomology grou…
We would like to construct a new Grothendieck topology for arithmetic schemes, whose cohomology groups associated with motivic complexes of sheaves are finitely generated and whose Euler characteristics are related to special values of zeta-functions. In this paper we construct this topology for rings of algebraic integers and show that the cohomology of the constant sheaf Z is related to the behavior of zeta-functions at s = 0.
We introduce a cohomology theory for a class of projective varieties over a finite field coming from the canonical trace on a C*-algebra attached to the variety. Using the cohomology, we prove the rationality, functional equation and the Betti numbers conjectures for the zeta function of the variety.
Suppose that $M$ is an even lattice with dual $M^{*}$ and level $N$. Then the group $Mp_{2}(\mathbb{Z})$, which is the unique non-trivial double cover of $SL_{2}(\mathbb{Z})$, admits a representation $ρ_{M}$, called the Weil representation, on the space $\mathbb{C}[M^{*}/M]$. The main aim of this paper is to show how the formulae for the $ρ_{M}$-action of a general element of $Mp_{2}(\mathbb{Z})$ can be obtained by a direct evaluation which does not depend on ``external objects'' such as theta functions. We decompose the Weil representation $ρ_{M}$ into $p$-parts, in which each $p$-part can be…
We develop a general classification theory for Brumer's dihedral quintic polynomials by means of Kummer theory arising from certain elliptic curves. We also give a similar theory for cubic polynomials.
We give a reformulation of the Birch and Swinnerton-Dyer conjecture over global function fields in terms of Weil-etale cohomology of the curve with coefficients in the Neron model, and show that it holds under the assumption of finiteness of the Tate-Shafarevich group.
We prove the plectic conjecture of Nekovář-Scholl over global function fields $Q$. For example, when the cocharacter is defined over $Q$ and the structure group is a Weil restriction from a geometric degree $d$ separable extension $F/Q$, consider the complex computing $\ell$-adic intersection cohomology with compact support of the associated moduli space of shtukas over $Q_I$. We endow this with the structure of a complex of $\DeclareMathOperator{\Weil}{Weil}(\Weil(F)^d\rtimes\mathfrak{S}_d)^I$-modules, which extends its structure as a complex of $\Weil(Q)^I$-modules constructed by Arinkin-Gai…
The purpose of this paper is to give a formula for the leading coefficient at $s=1$ of the $L$-function of one-motives over function fields in terms of Weil-étale cohomology, generalizing the Weil-étale version of the Birch and Swinnerton-Dyer conjecture in the authors' previous work. As a consequence we express the Tamagawa number of a torus introduced by Ono-Oesterlé in terms of Weil-étale cohomology, and reprove their Tamagawa number formula.
We prove, for all reductive groups, that the cohomology sheaves with compact support of stacks of shtukas are ind-smooth over $(X - N)^I$ and that their geometric generic fibers are equiped with an action of $\text{Weil}(X - N, \overlineη)^I$. Our method does not use any compactification of stacks of shtukas.
Lichtenbaum has conjectured the existence of a Grothendieck topology for an arithmetic scheme $X$ such that the Euler characteristic of the cohomology groups of the constant sheaf $\mathbb{Z}$ with compact support at infinity gives, up to sign, the leading term of the zeta-function $ζ_X(s)$ at $s=0$. In this paper we consider the category of sheaves $\bar{X}_L$ on this conjectural site for $X=Spec(\mathcal{O}_F)$ the spectrum of a number ring. We show that $\bar{X}_L$ has, under natural topological assumptions, a well defined fundamental group whose abelianization is isomorphic, as a topologic…
Emil Artin defined a zeta function for algebraic curves over finite fields and made a conjecture about them analogous to the famous Riemann hypothesis. This and other conjectures about these zeta functions would come to be called the Weil conjectures, which were proved by Weil for curves and later, by Deligne for varieties over finite fields. Much work was done in the search for a proof of these conjectures, including the development in algebraic geometry of a Weil cohomology theory for these varieties, which uses the Frobenius operator on a finite field. The zeta function is then expressed as…
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…
Let $E/\bbq$ be an elliptic curve defined over $\bbq$ with conductor $N$ and $\gq$ the absolute Galois group of an algebraic closure $\bar{\bbq}$ of $\bbq$. We prove that for every $σ\in \gq$, the Mordell-Weil group $E(\oqs)$ of $E$ over the fixed subfield of $\bar{\bbq}$ under $σ$ has infinite rank. Our approach uses the modularity of $E/\bbq$ and a collection of algebraic points on $E$ -- the so-called {\em Heegner points} -- arising from the theory of complex multiplication.
We prove that the triviality of the Galois action on the suitably twisted odd-dimensional étale cohomogy group of a smooth projective varietiy with finite coefficients implies the existence of certain primitive roots of unity in the field of definition of the variety. This text was inspired by an exercise in Serre's Lectures on the Mordell--Weil theorem.
This note aims at providing evidence for the section conjecture of anabelian geometry by establishing its behaviour under Weil restriction of scalars. In particular, the etale fundamental group of the Weil restriction is determined by means of a Shapiro Lemma for non-abelian group cohomology.
Lichtenbaum conjectured the existence of a Weil-étale cohomology in order to describe the vanishing order and the special value of the Zeta function of an arithmetic scheme $\mathcal{X}$ at $s=0$ in terms of Euler-Poincaré characteristics. Assuming the (conjectured) finite generation of some étale motivic cohomology groups we construct such a cohomology theory for regular schemes proper over $\mathrm{Spec}(\mathbb{Z})$. In particular, we obtain (unconditionally) the right Weil-étale cohomology for geometrically cellular schemes over number rings. We state a conjecture expressing the vanishing …
We give a general formula relating self cup products in cohomology to connecting maps in nonabelian cohomology, and apply it to obtain a formula for the self cup product associated to the Weil pairing.
We construct a cohomology theory with compact support H^i_c(X_ar,Z(n))$ for separated schemes of finite type over a finite field, which should play a role analog to Lichtenbaum's Weil-etale cohomology groups for smooth and projective schemes. In particular, if Tate's conjecture holds and rational and numerical equivalence agree up to torsion, then the groups H^i_c(X_ar,Z(n)) are finitely generated, form an integral version of l-adic cohomology with compact support, and admit a formula for the special values of the zeta-function of X.
In recent years Lichtenbaum has conjectured a description for the special values of Hasse--Weil zeta functions in terms of ``Weil-étale cohomology''. In earlier papers we studied a class of foliated dynamical systems which had some similarities with arithmetic schemes. Assuming that certain zeta regularized determinants exist we now prove an analogue of Lichtenbaum's conjectures for a particularly simple class of such dynamical systems: Using results of Álvarez López and Kordyukov we express the leading coefficient $ζ^*_R (0)$ of the Ruelle zeta function $ζ_R (s)$ at $s = 0$ in terms of analyt…
We study the family of elliptic curves $y^2=x(x-a^2)(x-b^2)$ parametrized by Pythagorean triples $(a,b,c)$. We prove that for a generic triple the lower bound of the rank of the Mordell-Weil group over $\mathbb{Q}$ is 1, and for some explicitly given infinite family the rank is 2. To each family we attach an elliptic surface fibered over the projective line. We show that the lower bounds for the rank are optimal, in the sense that for each generic fiber of such an elliptic surface its corresponding Mordell-Weil group over the function field $\mathbb{Q}(t)$ has rank 1 or 2, respectively. In ord…
We investigate the action of the Weil group on the compactly supported l-adic etale cohomology groups of rigid spaces over a local field. We prove that the alternating sum of the traces of the action is an integer and is independent of l when either the rigid space is smooth or the characteristic of the base field is equal to 0. We modify the argument of T. Saito to prove a result on l-independence for nearby cycle cohomology, which leads to our l-independence result for smooth rigid spaces. In the general case, we use the finiteness theorem of R. Huber, which requires the restriction on the c…
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 idel…
For varieties over a perfect field of characteristic p, etale cohomology with Q_l-coefficients is a Weil cohomology theory only when l is not equal to p; the corresponding role for l = p is played by Berthelot's rigid cohomology. In that theory, the coefficient objects analogous to lisse l-adic sheaves are the overconvergent F-isocrystals. This expository article is a brief user's guide for these objects, including some features shared with l-adic cohomology (purity, weights) and some features exclusive to the p-adic case (Newton polygons, convergence and overconvergence). The relationship bet…
To a pair of elliptic curves, one can naturally attach two K3 surfaces: the Kummer surface of their product and a double cover of it, called the Inose surface. They have prominently featured in many interesting constructions in algebraic geometry and number theory. There are several more associated elliptic K3 surfaces, obtained through base change of the Inose surface; these have been previously studied by Kuwata. We give an explicit description of the geometric Mordell-Weil groups of each of these elliptic surfaces in the generic case (when the elliptic curves are non-isogenous). In the non-…
In this paper, we prove that for suitably chosen Dwork motives, the local Galois representation arising from middle cohomology of fibers over a point with $p$-adic valuation $<0$ on the base is regular and ordinary. The result will be crucial in the forthcoming work Potential Automorphy for $GL_n$. The proof consists of the construction of a semistable blowup and a use of Hyodo-Kato's log crystalline cohomology theory.
