Modulo Torsion Research Articles

On the integral Hodge conjecture for real abelian threefolds

Abstract We prove the real integral Hodge conjecture for several classes of real abelian threefolds. For instance, we prove the property for real abelian threefolds whose real locus is connected, and for real abelian threefolds A which are the product A = B × E {A=B\times E} of an abelian surface B and an elliptic curve E with connected real locus E ⁢ ( ℝ ) {E(\mathbb{R})} . Moreover, we show that every real abelian threefold satisfies the real integral Hodge conjecture modulo torsion, and reduce the principally polarized case to the Jacobian case.

Realization of graded monomial ideal rings modulo torsion

Let $A$ be the quotient of a graded polynomial ring $\mathbb{Z}[x_1,\cdots,x_m]\otimes\Lambda[y_1,\cdots,y_n]$ by an ideal generated by monomials with leading coefficients 1. Then we constructed a space~$X_A$ such that $A$ is isomorphic to $H^*(X_A)$ modulo torsion elements.

On the coniveau of rationally connected threefolds

We prove that the integral cohomology modulo torsion of a rationally connected threefold comes from the integral cohomology of a smooth curve via the cylinder homomorphism associated to a family of $1$-cycles. Equivalently, it is of strong coniveau 1 in the sense of Benoist-Ottem.

Hyperbolic tessellations and generators of for imaginary quadratic fields

AbstractWe develop methods for constructing explicit generators, modulo torsion, of the$K_3$-groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic$3$-space or on direct calculations in suitable pre-Bloch groups and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite$K_3$-group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for$ K_3 $of any field, predict the precise power of$2$that should occur in the Lichtenbaum conjecture at$ -1 $and prove that this prediction is valid for all abelian number fields.

On infiniteness of integral overconvergent de Rham–Witt cohomology modulo torsion

In this article, we give examples of smooth varieties of positive characteristic whose first integral overconvergent de Rham–Witt cohomology modulo torsion is not finitely generated over the Witt ring of the base field.

Classification of Quantum Cellular Automata

There exists an index theory to classify strictly local quantum cellular automata in one dimension. We consider two classification questions. First, we study to what extent this index theory can be applied in higher dimensions via dimensional reduction, finding a classification by the first homology group of the manifold modulo torsion. Second, in two dimensions, we show that an extension of this index theory (including torsion) fully classifies quantum cellular automata, at least in the absence of fermionic degrees of freedom. This complete classification in one and two dimensions by index theory is not expected to extend to higher dimensions due to recent evidence of a nontrivial automaton in three dimensions. Finally, we discuss some group theoretical aspects of the classification of quantum cellular automata and consider these automata on higher dimensional real projective spaces.

Numerically trivial automorphisms of Enriques surfaces in characteristic 2

An automorphism of an algebraic surface $S$ is called cohomologically (numerically) trivial if it acts identically on the second cohomology group (this group modulo torsion subgroup). Extending the results of Mukai and Namikawa to arbitrary characteristic $p > 0$, we prove that the group of cohomologically trivial automorphisms $\operatorname{Aut}_{\operatorname{ct}}(S)$ of an Enriques surface $S$ is of order $\le 2$ if $S$ is not supersingular. If $p = 2$ and $S$ is supersingular, we show that $\mathrm{Aut}_{\operatorname{ct}}(S)$ is a cyclic group of odd order $n \in \{1,2,3,5,7,11\}$ or the quaternion group $Q_8$ of order 8 and we describe explicitly all the exceptional cases. If $K_S \neq 0$, we also prove that the group $\mathrm{Aut}_{\operatorname{nt}}(S)$ of numerically trivial automorphisms is a subgroup of a cyclic group of order $\le 4$ unless $p = 2$, where $\mathrm{Aut}_{\operatorname{nt}}(S)$ is a subgroup of a 2-elementary group of rank $\le 2$.

Polarized endomorphisms of normal projective threefolds in arbitrary characteristic

Let $X$ be a projective variety over an algebraically closed field $k$ of arbitrary characteristic $p \ge 0$. A surjective endomorphism $f$ of $X$ is $q$-polarized if $f^\ast H \sim qH$ for some ample Cartier divisor $H$ and integer $q > 1$. Suppose $f$ is separable and $X$ is $\mathbb{Q}$-Gorenstein and normal. We show that the anti-canonical divisor $-K_X$ is numerically equivalent to an effective $\mathbb{Q}$-Cartier divisor, strengthening slightly the conclusion of Boucksom, de Fernex and Favre (Theorem C) and also covering singular varieties over an algebraically closed field of arbitrary characteristic. Suppose $f$ is separable and $X$ is normal. We show that the Albanese morphism of $X$ is an algebraic fibre space and $f$ induces polarized endomorphisms on the Albanese and also the Picard variety of $X$, and $K_X$ being pseudo-effective and $\mathbb{Q}$-Cartier means being a torsion $\mathbb{Q}$-divisor. Let $f^{Gal}:\overline{X}\to X$ be the Galois closure of $f$. We show that if $p>5$ and co-prime to $deg\, f^{Gal}$ then one can run the minimal model program (MMP) $f$-equivariantly, after replacing $f$ by a positive power, for a mildly singular threefold $X$ and reach a variety $Y$ with torsion canonical divisor (and also with $Y$ being a quasi-\'etale quotient of an abelian variety when $\dim(Y)\le 2$). Along the way, we show that a power of $f$ acts as a scalar multiplication on the Neron-Severi group of $X$ (modulo torsion) when $X$ is a smooth and rationally chain connected projective variety of dimension at most three.

Stein fillings and SU(2) representations

We recently defined invariants of contact 3-manifolds using a version of instanton Floer homology for sutured manifolds. In this paper, we prove that if several contact structures on a 3-manifold are induced by Stein structures on a single 4-manifold with distinct Chern classes modulo torsion then their contact invariants in sutured instanton homology are linearly independent. As a corollary, we show that if a 3-manifold bounds a Stein domain that is not an integer homology ball then its fundamental group admits a nontrivial homomorphism to SU(2). We give several new applications of these results, proving the existence of nontrivial and irreducible SU(2) representations for a variety of 3-manifold groups.

A Weil Banach Algebra for Multiplicative Algebraic Numbers

We present a generalization of the construction of Allcock and Vaaler in [AV09] which established that the Weil height provides non-zero algebraic numbers modulo torsion with the structure of a normed vector space and construct its completion as L1 of a locally compact measure space. In the present work we show that this space is naturally a Banach algebra and describe its structure and a method for computing the Banach multiplication explicitly using the theory of harmonic analysis on compact groups.

Representation theory and the cycle class map of a classifying space

We compute the Chern subgroup of the 4-th integral cohomology group of a certain classifying space and show that it is a proper subgroup. Such a classifying space gives us new counterexamples for the integral Hodge and Tate conjectures modulo torsion.

Iwasawa theory of quadratic twists of X 0(49)

The field \(K = \mathbb{Q}\left( {\sqrt { - 7} } \right)\) is the only imaginary quadratic field with class number 1, in which the prime 2 splits, and we fix one of the primes p of K lying above 2. The modular elliptic curve X0(49) has complex multiplication by the maximal order O of K, and we let E be the twist of X0(49) by the quadratic extension \(KK(\sqrt M )/K\), where M is any square free element of O with M ≡ 1 mod 4 and (M,7) = 1. In the present note, we use surprisingly simple algebraic arguments to prove a sharp estimate for the rank of the Mordell-Weil group modulo torsion of E over the field F∞ = K(Ep∞), where Ep∞ denotes the group of p∞-division points on E. Moreover, writing B for the twist of X0(49) by \(K(\sqrt[4]{{ - 7}})/K\), our Iwasawa-theoretic arguments also show that the weak form of the conjecture of Birch and Swinnerton-Dyer implies the non-vanishing at s = 1 of the complex L-series of B over every finite layer of the unique Z2-extension of K unramified outside p. We hope to give a proof of this last non-vanishing assertion in a subsequent paper.

Modifications of torsion-free coherent analytic sheaves

We study the transformation of torsion-free coherent analytic sheaves under proper modifications. More precisely, we study direct images of inverse image sheaves, and torsion-free preimages of direct image sheaves. Under some conditions, it is shown that torsion-free coherent sheaves can be realized as the direct image of locally free sheaves under modifications. Thus, it is possible to study coherent sheaves modulo torsion by reducing the problem to study vector bundles on manifolds. We apply this to reduced ideal sheaves and to the Grauert-Riemenschneider canonical sheaf of holomorphic n-forms.

New invariants ofG2–structures

We define a Z/48-valued homotopy invariant nu of a G_2-structure on the tangent bundle of a closed 7-manifold in terms of the signature and Euler characteristic of a coboundary with a Spin(7)-structure. For manifolds of holonomy G_2 obtained by the twisted connected sum construction, the associated torsion-free G_2-structure always has nu = 24. Some holonomy G_2 examples constructed by Joyce by desingularising orbifolds have odd nu. We define a further homotopy invariant xi of G_2-structures such that if M is 2-connected then the pair (nu, xi) determines a G_2-structure up to homotopy and diffeomorphism. The class of a G_2-structure is determined by nu on its own when the greatest divisor of p_1(M) modulo torsion divides 224; this sufficient condition holds for many twisted connected sum G_2-manifolds. We also prove that the parametric h-principle holds for coclosed G_2-structures.

Small Points and Free Abelian Groups

Let $F$ be an algebraic extension of the rational numbers and $E$ an elliptic curve defined over some number field contained in $F$. The absolute logarithmic Weil height, respectively the N\'eron-Tate height, induces a norm on $F^*$ modulo torsion, respectively on $E(F)$ modulo torsion. The groups $F^*$ and $E(F)$ are free abelian modulo torsion if the height function does not attain arbitrarily small positive values. In this paper we prove the failure of the converse to this statement by explicitly constructing counterexamples.

Homotopy colimits of classifying spaces of abelian subgroups of a finite group

The classifying space BG of a topological group $G$ can be filtered by a sequence of subspaces $B(q,G)$, using the descending central series of free groups. If $G$ is finite, describing them as homotopy colimits is convenient when applying homotopy theoretic methods. In this paper we introduce natural subspaces $B(q,G)_p$ of $B(q,G)$ defined for a fixed prime $p$. We show that $B(q,G)$ is stably homotopy equivalent to a wedge of $B(q,G)_p$ as $p$ runs over the primes dividing the order of $G$. Colimits of abelian groups play an important role in understanding the homotopy type of these spaces. Extraspecial $2$-groups are key examples, for which these colimits turn out to be finite. We prove that for extraspecial 2-groups, $B(2,G)$ does not have the homotopy type of a $K(\pi,1)$ space. For a finite group $G$, we compute the complex K-theory of $B(2,G)$ modulo torsion.

On the algebraic K-theory of truncated polynomial algebras in several variables

AbstractWe consider the algebraic K-theory of a truncated polynomial algebra in several commuting variables, . This naturally leads to a new generalization of the big Witt vectors. If k is a perfect field of positive characteristic we describe the K-theory computation in terms of a cube of these Witt vectors on ℕn. If the characteristic of k does not divide any of the ai we compute the K-groups explicitly. We also compute the K-groups modulo torsion for k = ℤ.To understand this K-theory spectrum we use the cyclotomic trace map to topological cyclic homology, and write as the iterated homotopy cofiber of an n-cube of spectra, each of which is easier to understand.

A generalization of Dirichlet's unit theorem

We generalize Dirichlet’s S-unit theorem from the usual group of S-units of a number field K to the infinite rank group of all algebraic numbers having nontrivial valuations only on places lying over S. Specifically, we demonstrate that the group of algebraic S-units modulo torsion is a Q-vector space which, when normed by the Weil height, spans a hyperplane determined by the product formula, and that the elements of this vector space which are linearly independent over Q retain their linear independence over R.

Orthogonal decomposition of the space of algebraic numbers and Lehmerʼs problem

Building on work of Dubickas and Smyth regarding the metric Mahler measure and the authors regarding extremal norms associated to the Mahler measure, the authors introduce a new set of norms associated to the Mahler measure of algebraic numbers which allow for an equivalent reformulation of problems like the Lehmer problem and the Schinzel–Zassenhaus conjecture on a single spectrum. We present several new geometric results regarding the space of algebraic numbers modulo torsion using the L p Weil height introduced by Allcock and Vaaler, including a canonical decomposition of an algebraic number into an orthogonal series with respect to the L 2 height.

Computing Borel's regulator

Abstract We present an infinite series formula based on the Karoubi–Hamida integral, for the universal Borel class evaluated on H 2n+1(GL(ℂ)). For a cyclotomic field F we define a canonical set of elements in K 3(F) and present a novel approach (based on a free differential calculus) to constructing them. Indeed, we are able to explicitly construct their images in H 3(GL(ℂ)) under the Hurewicz map. Applying our formula to these images yields a value V 1(F), which coincides with the Borel regulator R 1(F) when our set is a basis of K 3(F) modulo torsion. For F = ℚ(e 2πi/3) a computation of V 1(F) has been made based on our techniques.