Elementary abelian group
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TASI Lectures on Abelian and Discrete Symmetries in F-theory Open
In F-theory compactifications, the abelian gauge sector is encoded in global structures of the internal geometry. These structures lie at the intersection of algebraic and arithmetic description of elliptic fibrations: While the Mordell–We…
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Cohomology of abelian arrangements Open
An abelian arrangement is a finite set of codimension one abelian subvarieties (possibly translated) in a complex abelian variety. In this paper, we study the cohomology of the complement of an abelian arrangement. For unimodular abelian a…
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Categories of abelian varieties over finite fields, I: Abelian varieties over 𝔽<sub><i>p</i></sub> Open
We assign functorially a [math] -lattice with semisimple Frobenius action to each abelian variety over [math] . This establishes an equivalence of categories that describes abelian varieties over [math] avoiding [math] as an eigenvalue of …
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Chevalley-Warning type results on abelian groups Open
We develop a notion of degree for functions between two abelian groups that allows us to generalize the Chevalley-Warning Theorems from fields to noncommutative rings or abelian groups of prime power order.
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Distinguished cycles on varieties with motive of abelian type and the Section Property Open
A remarkable result of Peter O’Sullivan asserts that the algebra epimorphism from the rational Chow ring of an abelian variety to its rational Chow ring modulo numerical equivalence admits a (canonical) section. Motivated by Beauville’s sp…
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On Transitive and Fully Transitive Abelian P-Groups Open
It is a long standing problem of A.L.S. Corner as to whether there exists a non-transitive fully transitive Abelian p-group with finite Ulm subgroups. The principal result of this paper shows that no such group exists for a special class o…
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On the -adic Galois representations attached to nonsimple abelian varieties Open
We study Galois representations attached to nonsimple abelian varieties over finitely generated fields of arbitrary characteristic. We give sufficient conditions for such representations to decompose as a product, and apply them to prove a…
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The Betti map associated to a section of an abelian scheme Open
Given a point ξ on a complex abelian variety A, its abelian logarithm can be expressed as a linear combination of the periods of A with real coefficients, the Betti coordinates of ξ. When (A, ξ) varies in an algebraic family, these coordin…
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On strongly sequenceable abelian groups Open
A group is strongly sequenceable if every connected Cayley digraph on the group admits an orthogonal directed cycle or an orthogonal directed path. This paper deals with the problem of whether finite abelian groups are strongly sequenceabl…
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THE CHARACTERISTIC POLYNOMIALS OF ABELIAN VARIETIES OF DIMENSIONS 3 OVER FINITE FIELDS Open
We describe the set of characteristic polynomials of abelian varieties of\ndimension 3 over finite fields.\n
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On superspecial abelian surfaces over finite fields Open
In this paper we establish a new lattice description for superspecial abelian varieties over a finite field \mathbb F_q of q=p^{a} elements. Our description depends on the parity of the exponent a of q . When q is an odd power of the prime…
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Morita equivalence classes of blocks with elementary abelian defect groups of order 16 Open
We classify the Morita equivalence classes of blocks with elementary abelian defect groups of order $16$ with respect to a complete discrete valuation ring with algebraically closed residue field of characteristic two. As a consequence, bl…
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Tensor ideals and varieties for modules of quantum elementary abelian groups Open
In a previous paper we constructed rank and support variety theories for "quantum elementary abelian groups," that is, tensor products of copies of Taft algebras. In this paper we use both variety theories to classify the thick tensor idea…
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Locality of percolation for Abelian Cayley graphs Open
We prove that the value of the critical probability for percolation on an abelian Cayley graph is determined by its local structure. This is a partial positive answer to a conjecture of Schramm: the function pc defined on the set of Cayley…
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ON GRADED -ALGEBRAS Open
We show that every topological grading of a $C^{\ast }$ -algebra by a discrete abelian group is implemented by an action of the compact dual group.
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The basepoint-freeness threshold and syzygies of abelian varieties Open
We show how a natural constant introduced by Jiang and Pareschi for a\npolarized abelian variety encodes information about the syzygies of the section\nring of the polarization. As a particular case this gives a quick and\ncharacteristic-f…
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A characterization of ordinary abelian varieties by the Frobenius push-forward of the structure sheaf Open
For an ordinary abelian variety X, $$F^e_*\mathcal {O}_X$$ is decomposed into line bundles for every positive integer e. Conversely, if a smooth projective variety X satisfies this property and the Kodaira dimension of X is non-negative, t…
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The Shafarevich conjecture for hypersurfaces in abelian varieties Open
Faltings proved that there are finitely many abelian varieties of genus $g$ over a number field $K$, with good reduction outside a finite set of primes $S$. Fixing one of these abelian varieties $A$, we prove that there are finitely many s…
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Almost ordinary abelian varieties over finite fields Open
We provide a characterization of almost ordinary abelian varieties over\nfinite fields, and use this characterization to provide lower bounds for the\nsizes of some almost ordinary isogeny classes.\n
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On K3 Surface Quotients of K3 or Abelian Surfaces Open
The aim of this paper is to prove that a K3 surface is the minimal model of the quotient of an Abelian surface by a group G (respectively of a K3 surface by an Abelian group G ) if and only if a certain lattice is primitively embedded in i…
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Faltings heights of abelian varieties with complex multiplication Open
Let M be the Shimura variety associated with the group of spinor similitudes of a rational quadratic space over of signature (n,2). We prove a conjecture of Bruinier-Kudla-Yang, relating the arithmetic intersection multiplicities of specia…
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On the growth of Mordell–Weil ranks in $p$-adic Lie extensions Open
Let $p$ be an odd prime and $F_{\\infty}$ a $p$-adic Lie extension of a number\nfield $F$. Let $A$ be an abelian variety over $F$ which has ordinary reduction\nat every primes above $p$. Under various assumptions, we establish asymptotic\n…
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An arithmetic method of counting the subgroups of a finite abelian group Open
The main goal of this paper is to apply the arithmetic method developed in our previous paper \cite{13} to determine the number of some types of subgroups of finite abelian groups.
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Co-prime order graphs of finite Abelian groups and dihedral groups Open
The \\textbf{Co-Prime Order Graph} $\\Theta (G)$ of a given finite group is a\nsimple undirected graph whose vertex set is the group $G$ itself, and any two\nvertexes x,y in $\\Theta (G)$ are adjacent if and only if $gcd(o(x),o(y))=1$ or\n…
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A locally quasi-convex abelian group without a Mackey group topology Open
We give the first example of a locally quasi-convex (even countable reflexive and $k_\omega$) abelian group $G$ which does not admit the strongest compatible locally quasi-convex group topology. Our group $G$ is the Graev free abelian grou…
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The equivariant complex cobordism ring of a finite abelian group Open
We compute the equivariant (stable) complex cobordism ring (MUG)∗ for finite abelian groups G.
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The number of additive triples in subsets of abelian groups Open
A set of elements of a finite abelian group is called sum-free if it contains no Schur triple, i.e., no triple of elements x , y , z with x + y = z . The study of how large the largest sum-free subset of a given abelian group is had starte…
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Level structures on abelian varieties and Vojta’s conjecture Open
Assuming Vojta’s conjecture, and building on recent work of the authors, we prove that, for a fixed number field $K$ and a positive integer $g$ , there is an integer $m_{0}$ such that for any $m>m_{0}$ there is no principally polarized abe…
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On the distribution of rational points on ramified covers of abelian varieties Open
We prove new results on the distribution of rational points on ramified covers of abelian varieties over finitely generated fields $k$ of characteristic zero. For example, given a ramified cover $\pi : X \to A$ , where $A$ is an abelian va…
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The Betti map associated to a section of an abelian scheme (with an appendix by Z. Gao) Open
Given a point $ξ$ on a complex abelian variety $A$, its abelian logarithm can be expressed as a linear combination of the periods of $A$ with real coefficients, the Betti coordinates of $ξ$. When $(A, ξ)$ varies in an algebraic family, the…