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A. Silverberg

Publications and source records attributed to A. Silverberg.

At least 19 recordsLinked to original sources
arXiv2022

Realizing orders as group rings

An order is a commutative ring that as an abelian group is finitely generated and free. A commutative ring is reduced if it has no non-zero nilpotent elements. In this paper we use a new tool, namely, the fact that every reduced order has a universal grading, to answer questions about realizing orders as group rings. In particular, we address the Isomorphism Problem for group rings in the case where the ring is a reduced order. We prove that any non-zero reduced order $R$ can be written as a group ring in a unique ``maximal'' way, up to isomorphism. More precisely, there exist a ring $A$ and a finite abelian group $G$, both uniquely determined up to isomorphism, such that $R\cong A[G]$ as rings, and such that if $B$ is a ring and $H$ is a group, then $R\cong B[H]$ as rings if and only if there is a finite abelian group $J$ such that $B\cong A[J]$ as rings and $J\times H\cong G$ as groups. Computing $A$ and $G$ for given $R$ can be done by means of an algorithm that is not quite polynomial-time. We also give a description of the automorphism group of $R$ in terms of $A$ and $G$.

math.AC↗
arXiv2018

Universal gradings of orders

For commutative rings, we introduce the notion of a {\em universal grading}, which can be viewed as the "largest possible grading". While not every commutative ring (or order) has a universal grading, we prove that every {\em reduced order} has a universal grading, and this grading is by a {\em finite} group. Examples of graded orders are provided by group rings of finite abelian groups over rings of integers in number fields. We generalize known properties of nilpotents, idempotents, and roots of unity in such group rings to the case of graded orders; this has applications to cryptography. Lattices play an important role in this paper; a novel aspect is that our proofs use that the additive group of any reduced order can in a natural way be equipped with a lattice structure.

math.AC↗
arXiv2016

Algorithms for commutative algebras over the rational numbers

The algebras considered in this paper are commutative rings of which the additive group is a finite-dimensional vector space over the field of rational numbers. We present deterministic polynomial-time algorithms that, given such an algebra, determine its nilradical, all of its prime ideals, as well as the corresponding localizations and residue class fields, its largest separable subalgebra, and its primitive idempotents. We also solve the discrete logarithm problem in the multiplicative group of the algebra. While deterministic polynomial-time algorithms were known earlier, our approach is different from previous ones. One of our tools is a primitive element algorithm; it decides whether the algebra has a primitive element and, if so, finds one, all in polynomial time. A methodological novelty is the use of derivations to replace a Hensel-Newton iteration. It leads to an explicit formula for lifting idempotents against nilpotents that is valid in any commutative ring.

math.AC↗
arXiv2016

Lattices with Symmetry

For large ranks, there is no good algorithm that decides whether a given lattice has an orthonormal basis. But when the lattice is given with enough symmetry, we can construct a provably deterministic polynomial-time algorithm to accomplish this, based on the work of Gentry and Szydlo. The techniques involve algorithmic algebraic number theory, analytic number theory, commutative algebra, and lattice basis reduction.

math.NT↗
arXiv2016

Roots of unity in orders

We give deterministic polynomial-time algorithms that, given an order, compute the primitive idempotents and determine a set of generators for the group of roots of unity in the order. Also, we show that the discrete logarithm problem in the group of roots of unity can be solved in polynomial time. As an auxiliary result, we solve the discrete logarithm problem for certain unit groups in finite rings. Our techniques, which are taken from commutative algebra, may have further potential in the context of cryptology and computer algebra.

math.AC↗
arXiv2015

Isogenies of abelian varieties over finite fields

In this paper we give conditions under which two abelian varieties that are defined over a finite field $F$, and are isogenous over some larger field, are $F$-isogenous. Further, we give conditions under which a given isogeny is defined over $F$.

math.NT↗
arXiv2015

Determining cyclicity of finite modules

We present a deterministic polynomial-time algorithm that determines whether a finite module over a finite commutative ring is cyclic, and if it is, outputs a generator.

math.AC↗
arXiv2012

On elliptic curves with an isogeny of degree 7

We show that if $E$ is an elliptic curve over $\mathbf{Q}$ with a $\mathbf{Q}$-rational isogeny of degree 7, then the image of the 7-adic Galois representation attached to $E$ is as large as allowed by the isogeny, except for the curves with complex multiplication by $\mathbf{Q}(\sqrt{-7})$. The analogous result with 7 replaced by a prime $p > 7$ was proved by the first author in [7]. The present case $p = 7$ has additional interesting complications. We show that any exceptions correspond to the rational points on a certain curve of genus 12. We then use the method of Chabauty to show that the exceptions are exactly the curves with complex multiplication. As a by-product of one of the key steps in our proof, we determine exactly when there exist elliptic curves over an arbitrary field $k$ of characteristic not 7 with a $k$-rational isogeny of degree 7 and a specified Galois action on the kernel of the isogeny, and we give a parametric description of such curves.

math.NT↗
arXiv2009

Point counting on reductions of CM elliptic curves

We give explicit formulas for the number of points on reductions of elliptic curves with complex multiplication by any imaginary quadratic field. We also find models for CM $\mathbf{Q}$-curves in certain cases. This generalizes earlier results of Gross, Stark, and others.

math.NT↗
arXiv2007

Twisting commutative algebraic groups

If $V$ is a commutative algebraic group over a field $k$, $O$ is a commutative ring that acts on $V$, and $I$ is a finitely generated free $O$-module with a right action of the absolute Galois group of $k$, then there is a commutative algebraic group $I \otimes_O V$ over $k$, which is a twist of a power of $V$. These group varieties have applications to cryptography (in the cases of abelian varieties and algebraic tori over finite fields) and to the arithmetic of abelian varieties over number fields. For purposes of such applications we devote this article to making explicit this tensor product construction and its basic properties.

math.AG↗
arXiv2000

Symplectic representations of inertia groups

Suppose $\ell$ is a prime number, $\ell >3$, $K$ is a field that is an unramified finite extension of the field $\Q_\ell$ of $\ell$-adic numbers, and $G$ is a finite group that is a semi-direct product of a normal $\ell'$-subgroup $H$ and a cyclic $\ell$-group $L$. Suppose that the group algebra $K[H]$ is decomposable. If there exists an embedding of $G$ in the symplectic group $\Sp_{2d}(K)$ for some positive integer $d$, then there exists an embedding of $G$ in $\Sp_{2d}({\mathcal O}_K)$, where ${\mathcal O}_K$ is the ring of integers of $K$.

math.NT↗
arXiv2000

Polarizations on abelian varieties

The results in this paper imply that for every number field F and positive integer r, there exists an F-isogeny class of abelian varieties such that r divides the degree of every F-polarization on every abelian variety in the isogeny class.

math.AG↗
arXiv1999

Étale cohomology and reduction of abelian varieties

In this paper we study the étale cohomology groups associated to abelian varieties. We obtain necessary and sufficient conditions for an abelian variety to have semistable reduction (or purely additive reduction which becomes semistable over a quadratic extension) in terms of the action of the absolute inertia group on the étale cohomology groups with finite coefficients.

math.AG↗
arXiv1998

Modular representations arising from self-dual $\ell$-adic representations of finite groups

Suppose $\ell$ is a prime number, ${\mathbf Q}_\ell$ is the field of $\ell$-adic numbers, ${\mathbf F}_\ell$ is the finite field of $\ell$ elements, and $d$ is a positive integer. Suppose $G$ is a finite subgroup of a symplectic group $Sp_{2d}({\mathbf Q}_\ell)$. We prove that $G$ can be embedded in $Sp_{2d}({\mathbf F}_\ell)$ in such a way that the characteristic polynomials are preserved (mod $\ell$), as long as $\ell>3$.

math.GR↗
arXiv1998

Reduction of abelian varieties

We study semistable reduction and torsion points of abelian varieties. In particular, we give necessary and sufficient conditions for an abelian variety to have semistable reduction. We also study Néron models of abelian varieties with potentially good reduction and torsion points of small order. We study some invariants that measure the extent to which an abelian variety with potentially good reduction fails to have good reduction.

alg-geom↗
arXiv1998

Subgroups of inertia groups arising from abelian varieties

Given an abelian variety over a field with a discrete valuation, Grothendieck defined a certain open normal subgroup of the absolute inertia group. This subgroup encodes information on the extensions over which the abelian variety acquires semistable reduction. We study this subgroup, and use it to obtain information on the extensions over which the abelian variety acquires semistable reduction.

alg-geom↗
arXiv1996

Connectedness extensions for abelian varieties

Suppose $A$ is an abelian variety over a field $F$, and $\ell$ is a prime not equal to the characteristic of $F$. Let $F_{Φ,\ell}(A)$ denote the smallest extension of $F$ such that the Zariski closure of the image of the $\ell$-adic representation associated to $A$ is connected. Serre introduced this field, and proved that when $F$ is a finitely generated extension of ${\mathbf Q}$, $F_{Φ,\ell}(A)$ does not depend on the choice of $\ell$. In this paper we study extensions $F_{Φ,\ell}(B)/F$ for twists $B$ of a given abelian variety, especially when the abelian varieties are of Weil type.

alg-geom↗