Let p be a prime number. The Hasse invariant is a modular form modulo p that is often used to produce congruences between modular forms of different weights. We show how to produce such congruences between forms of weights 2 and p+1, in terms of group cohomology. We also show how our method works in the contexts of quadratic imaginary fields (where there is no Hasse invariant available) and Hilbert modular forms over totally real fields of even degree.
In 1985 Kazuya Kato formulated a fascinating framework of conjectures which generalizes the Hasse principle for the Brauer group of a global field to the so-called cohomological Hasse principle for an arithmetic scheme. In this paper we prove the prime-to-characteristic part of the cohomological Hasse principle. We also explain its implications on finiteness of motivic cohomology and special values of zeta functions.
Igusa noted that the Hasse invariant of the Legendre family of elliptic curves over a finite field of odd characteristic is a solution mod $p$ of a Gaussian hypergeometric equation. We show that any family of exponential sums over a finite field has a Hasse invariant which is a sum of products of mod $p$ solutions of $A$-hypergeometric systems.
Let $k$ be a finite field, a $p$-adic field or a number field. Let $K$ be a finite extension of the Laurent series field in $m$ variables $k((x_1,...,x_m))$ or, more generally, a finite extension of the field of rational functions $k((x_1,...,x_m))(y_1,...,y_n)$. When $r$ is an integer, we consider the Galois module $\mathbb{Q}/\mathbb{Z}(r)$ over $K$ and we prove several vanishing theorems for its cohomology. In the particular case when $K$ is a finite extension of the Laurent series field in two variables $k((x_1,x_2))$, we also prove exact sequences that play the role of the Brauer-Hasse-No…
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.
We show that the $p$-adic completion of any affine elliptic curve with ordinary reduction possesses Frobenius lifts whose "normalized" action on $1$-forms preserves mod $p$ the space of invariant $1$-forms. We next show that, after removing the $2$-torsion sections, the above situation can be "infinitesimally deformed" in the sense that the above mod $p$ result has a mod $p^2$ analogue. While the "eigenvalues" mod $p$ are given by the reciprocal of the Hasse polynomial, the "eigenvalues" mod $p^2$ are given by an appropriate $\d$-modular function whose reciprocal is a $p$-adic deformation of t…
Let $F$ be a totally real field of degree $g$, and let $p$ be a prime number. We construct $g$ partial Hasse invariants on the characteristic $p$ fiber of the Pappas-Rapoport splitting model of the Hilbert modular variety for $F$ with level prime to $p$, extending the usual partial Hasse invariants defined over the Rapoport locus. In particular, when $p$ ramifies in $F$, we solve the problem of lack of partial Hasse invariants. Using the stratification induced by these generalized partial Hasse invariants on the splitting model, we prove in complete generality the existence of Galois pseudo-re…
We construct a generalization of the Hasse invariant for any Shimura variety of PEL type A over a prime of good reduction, whose vanishing locus is the open and dense μ-ordinary locus.
If the Hasse invariant of a $p$-divisible group is small enough, then one can construct a canonical subgroup inside its $p$-torsion. We remark that, assuming the existence of a subgroup of adequate height in the $p$-torsion whose dual has small degree, the expected properties of the canonical subgroup can be easily proven. A fundamental relation is the equality between the Hasse invariant and the degree of the dual of the canonical subgroup. When one considers a $p$-divisible group $G$ with an action of the ring of integers of a (possibly ramified) finite extension of $\mathbb{Q}_p$, then much…
We construct a generalization of the Hasse invariant for certain unitary Shimura varieties of PEL type whose vanishing locus is the complement of the so-called μ-ordinary locus. We show that the μ-ordinary locus of those varieties is affine. As an application, we strengthen a special case of a theorem of one of us (W.G.) on the association of Galois representations to automorphic representations of unitary groups whose archimedean component is a holomorphic limit of discrete series.
We introduce a method for producing congruences between Hecke eigenclasses, possibly torsion, in the coherent cohomology of automorphic vector bundles on certain good reduction Shimura varieties. The congruences are produced using some "generalized Hasse invariants" adapted to the Ekedahl-Oort stratification of the special fiber.
It is shown that the number of irreducible quartic factors of the form $g(x) = x^4+ax^3+(11a+2)x^2-ax+1$ which divide the Hasse invariant of the Tate normal form $E_5$ in characteristic $l$ is a simple linear function of the class number $h(-5l)$ of the field $\mathbb{Q}(\sqrt{-5l})$, when $l \equiv 2,3$ modulo $5$. A similar result holds for irreducible quadratic factors of $g(x)$, when $l \equiv 1, 4$ modulo $5$. This implies a formula for the number of linear factors over $\mathbb{F}_p$ of the supersingular polynomial $ss_p^{(5*)}(x)$ corresponding to the Fricke group $Γ_0^*(5)$.
Let K be a global function field of positive characteristic p and let M be a (commutative) finite and flat K-group scheme. We show that the kernel of the canonical localization map H^{1}(K,M)\to\prod_{all v}H^{1}(K_{v},M) in flat (fppf) cohomology can be computed solely in terms of Galois cohomology. We then give applications to the case where M is the kernel of multiplication by p^{m} on an abelian variety defined over K.
We give a simple and natural proof for the compatibility of the Hasse invariant with duality. We then study a $p$-divisible group with an action of the ring of integers of a finite ramified extension of $\mathbb{Q}_p$. We suppose that it satisfies the Pappas-Rapoport condition ; in that case the Hasse invariant is a product of partial Hasse invariants, each of which can be expressed in terms of primitive Hasse invariants. We then show that the dual of the $p$-divisible group naturally satisfies a Pappas-Rapoport condition, and prove the compatibility with the duality for the partial and primit…
In this article, we construct in a purely local way partial (Hasse) invariants for $p$-divisible groups with given endomorphisms, using crystalline cohomology. Theses invariants generalises the classical Hasse invariant, and allow us to study families of such groups. We also study a few geometric properties of theses invariants. Used in the context of Shimura varieties, for example, theses invariants are detecting some Newton strata, including the $μ$-ordinary locus.
Let $G$ be a $p$-divisible group over the ring of integers of $\mathbb{C}_p$, and assume that it is endowed with an action of the ring of integers of a finite unramified extension $F$ of $\mathbb{Q}_p$. Let us fix the type $μ$ of this action on the sheaf of differentials $ω_G$. V. Hernandez, following a construction of Goldring and Nicole, defined partial Hasse invariants for $G$. The product of these invariants is the $μ$-ordinary Hasse invariant, and it is non-zero if and only if the $p$-divisible group is $μ$-ordinary (i.e. the Newton polygon is minimal given the type of the action). \\ We …
A formula is proved for the number of linear factors over $\mathbb{F}_l$ of the Hasse invariant of the Tate normal form $E_5(b)$ for a point of order $5$, as a polynomial in the parameter $b$, in terms of the class number of the imaginary quadratic field $K=\mathbb{Q}(\sqrt{-l})$, proving a conjecture of the author from 2005. A similar theorem is proved for quadratic factors with constant term $-1$, and a theorem is stated for the number of quartic factors of a specific form in terms of the class number of $\mathbb{Q}(\sqrt{-5l})$. These results are shown to imply a recent conjecture of Nakaya…
We generalize the Hasse invariant of local class field theory to the tame Brauer group of a higher dimensional local field, and use it to study the arithmetic of central simple algebras over such fields, which are given {\it a priori} as tensor products of standard cyclic algebras. We also compute the tame Brauer dimension (or {\it period-index bound}) and the cyclic length of a general henselian-valued field of finite rank and finite residue field.
Let $(X, D_{X})$ be an arbitrary pointed stable curve of topological type $(g_{X}, n_{X})$ over an algebraically closed field of characteristic $p>0$. We prove that the generalized Hasse-Witt invariants of prime-to-$p$ cyclic admissible coverings of $(X, D_{X})$ attain maximum. As applications, we obtain an anabelian formula for $(g_{X}, n_{X})$, and prove that the field structures associated to inertia subgroups of marked points can be reconstructed group-theoretically from open continuous homomorphisms of admissible fundamental groups. Moreover, the formula for maximum generalized Hasse-Witt…
Let X be an irreducible symplectic variety defined over a number field K. Assume either that X has Picard number at least two or that X has even second Betti number. We prove that there exist a finite algebraic field extension L/K and a density 1 set S of non-archimedean places of L such that the reduction of X at any place in S has nonzero Hasse-Witt invariant.
We study $p$-divisible groups $G$ endowed with an action of the ring of integers of a finite (possibly ramified) extension of $\mathbb{Q}_p$ over a scheme of characteristic $p$. We suppose moreover that the $p$-divisible group $G$ satisfies the Pappas-Rapoport condition for a certain datum $μ$ ; this condition consists in a filtration on the sheaf of differentials $ω_G$ satisfying certain properties. Over a perfect field, we define the Hodge and Newton polygons for such $p$-divisible groups, normalized with the action. We show that the Newton polygon lies above the Hodge polygon, itself lying …
We present a novel randomized algorithm to factor polynomials over a finite field $\F_q$ of odd characteristic using rank $2$ Drinfeld modules with complex multiplication. The main idea is to compute a lift of the Hasse invariant (modulo the polynomial $f \in \F_q[x]$ to be factored) with respect to a random Drinfeld module $ϕ$ with complex multiplication. Factors of $f$ supported on prime ideals with supersingular reduction at $ϕ$ have vanishing Hasse invariant and can be separated from the rest. Incorporating a Drinfeld module analogue of Deligne's congruence, we devise an algorithm to compu…
School of Mathematics and Physics, University of Science and Technology Beijing（北京科技大学数理学院）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）；Instituto de Matemática Pura e Aplicada, IMPA（巴西纯粹与应用数学研究所）
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 construct group-theoretical generalizations of the Hasse invariant on strata closures of the stacks $G$-Zip$^μ$. Restricting to zip data of Hodge type, we obtain a group-theoretical Hasse invariant on every Ekedahl-Oort stratum closure of a general Hodge-type Shimura variety. A key tool is the construction of a stack of zip flags $G$-ZipFlag$^μ$, fibered in flag varieties over $G$-Zip$^μ$. It provides a simultaneous generalization of the "classical case" homogeneous complex manifolds studied by Griffiths-Schmid and the "flag space" for Siegel varieties studied by Ekedahl-van der Geer. Four …
Let $Y$ be a scheme in which 2 is invertible and let $V$ be a rank $n$ vector bundle on $Y$ endowed with a non-degenerate symmetric bilinear form $q$. The orthogonal group ${\bf O}(q)$ of the form $q$ is a group scheme over $Y$ whose cohomology ring $H^*(B_{{\bf O}(q)},{\bf Z}/2{\bf Z})\simeq A_Y[HW_1(q),..., HW_n(q)]$ is a polynomial algebra over the étale cohomology ring $A_Y:=H^*(Y_{et},{\bf Z}/2{\bf Z})$ of the scheme $Y$. Here the $HW_i(q)$'s are Jardine's universal Hasse-Witt invariants and $B_{{\bf O}(q)}$ is the classifying topos of ${\bf O}(q)$ as defined by Grothendieck and Giraud. T…
The concept of weights on the cohomology of algebraic varieties was initiated by fundamental ideas and work of A. Grothendieck and P. Deligne. It is deeply connected with the concept of motives and appeared first on the singular cohomology as the weights of (possibly mixed) Hodge structures and on the etale cohomology as the weights of eigenvalues of Frobenius. But weights also appear on algebraic fundamental groups and in p-adic Hodge theory, where they become only visible after applying the comparison functors of Fontaine. After rehearsing various versions of weights, we explain some more re…
In this paper we study weight homology of singular schemes. Weight homology is an invariant of a singular scheme defined in terms of hypercoverings of resolution of singularities. Our main result is McKay principle for weight homology of quotient singularities, i.e. we describe weight homology of a quotient scheme in terms of weight homology of an equivariant scheme. Our method is to reduce the geometric McKay principle for weight homology to Kato's cohomological Hasse principle for arithmetic schemes. The McKay principle for weight homology implies McKay principle for the homotopy type of the…
A general structure theorem on higher order invariants is proven. For an arithmetic group, the structure of the corresponding Hecke module is determined. It is shown that the module does not contain any irreducible submodule. This explains the fact that L-functions of higher order forms have no Euler-product. Higher order cohomology is introduced, classical results of Borel are generalized and a higher order version of Borel's conjecture is stated.
In our previous paper "Strata Hasse invariants, Hecke algebras and Galois representations", initially motivated by questions about the Hodge line bundle of a Hodge-type Shimura variety, we singled out a generalization of the notion of {\em minuscule character} which we termed {\em quasi-constant}. Here we prove that the character of the Hodge line bundle is always quasi-constant. Furthermore, we classify the quasi-constant characters of an arbitrary connected, reductive group over an arbitrary field. As an application, we observe that, if $μ$ is a quasi-constant cocharacter of an ${\mathbf F}_…
We present novel algorithms to factor polynomials over a finite field $\F_q$ of odd characteristic using rank $2$ Drinfeld modules with complex multiplication. The main idea is to compute a lift of the Hasse invariant (modulo the polynomial $f(x) \in \F_q[x]$ to be factored) with respect to a Drinfeld module $ϕ$ with complex multiplication. Factors of $f(x)$ supported on prime ideals with supersingular reduction at $ϕ$ have vanishing Hasse invariant and can be separated from the rest. A Drinfeld module analogue of Deligne's congruence plays a key role in computing the Hasse invariant lift. We …
In this article, we prove results about the cohomology of compact unitary group Shimura varieties at split places. In nonendoscopic cases, we are able to give a full description of the cohomology, after restricting to integral Hecke operators at p on the automorphic side. We allow arbitrary ramification at p; even the PEL data may be ramified. This gives a description of the semisimple local Hasse-Weil zeta function in these cases. We also treat cases of nontrivial endoscopy. For this purpose, we give a general stabilization of the expression given in previous work, following the stabilization…
The relation of the Weierstrass semigroup with several invariants of a curve is studied. For Galois covers of curves with group $G$ we introduce a new filtration of the group decomposition subgroup of $G$. The relation to the ramification filtration is investigated in the case of cyclic covers. We relate our results to invariants defined by Boseck and we study the one point ramification case. We also give applications to Hasse-Witt invariant and symmetric semigroups.
We study the integral Hasse principle for affine varieties of the form ax^2+y^2+z^2-xyz=m ,using Brauer-Manin obstruction, and we produce examples whose Brauer groups include 4-torsion elements .We will construct their explicit representatives and compute local invariants in special places.
