Algebra & Number Theory最新文献

We construct two groups of size $2^{365}�\cdot 3^{105}�\cdot 7^{104}$: a solvable group $G$ and a nonsolvable group $H$ such that for every integer $n$ the groups have the same number of elements of order $n$. This answers a question posed in 1987 by John G. Thompson.

On an orthogonal Shimura variety, one has a collection of arithmetic special cycles in the Gillet–Soulé arithmetic Chow group. We describe how these cycles behave under pullback to an embedded orthogonal Shimura variety of lower dimension. The bulk of the paper is devoted to cases in which the special cycles intersect the embedded Shimura variety improperly, which requires that we analyze logarithmic expansions of Green currents on the deformation to the normal bundle of the embedding.

Let $L$ be an affine Kac–Moody algebra, with central element $c$, and let $\lambda �\in �ℂ$. We study two-sided ideals in the central quotient $U_{\lambda }(L)�:=�U(L)∕(c�-�\lambda )$ of the universal enveloping algebra of $L$ and prove:

1. If $\lambda \ne 0$ then $U_{\lambda }(L)$ is simple.

2. The algebra $U_0(L)$ has just-infinite growth, in the sense that any proper quotient has polynomial growth.

As an immediate corollary, we show that the annihilator of any nontrivial integrable highest-weight representation of $L$ is centrally generated, extending a result of Chari for Verma modules.

We also show that universal enveloping algebras of loop algebras and current algebras of finite-dimensional simple Lie algebras have just-infinite growth, and prove similar results to the two results above for quotients of symmetric algebras of these Lie algebras by Poisson ideals.

We develop new methods for counting integral orbits having bounded invariants that lie inside the cusps of fundamental domains for coregular representations. We illustrate these methods for a representation of cardinal interest in number theory, namely that of the split orthogonal group acting on the space of quadratic forms.

We construct an explicit sequence $V_{k_n,a_n}$ of crystalline representations of exceptional weights converging to a given irreducible two-dimensional semistable representation $V_{k,\mathcal{ℒ}}$ of $\mathrm{Gal}<!--FUNCTION\; APPLICATION-->({\overline{\mathbb{Q}}}_p$/${\mathbb{Q}}_p)$. The convergence takes place in the blow-up space of two-dimensional trianguline representations studied by Colmez and Chenevier. The process of blow-up is described in detail in the rigid-analytic setting and may be of independent interest. Also, we recover a formula of Stevens expressing the $\mathcal{ℒ}$-invariant as a logarithmic derivative.

Our result can be used to compute the reduction of $V_{k,\mathcal{ℒ}}$ in terms of the reductions of the $V_{k_n,a_n}$. For instance, using the zig-zag conjecture we recover (resp. extend) the work of Breuil and Mézard and Guerberoff and Park computing the reductions of the $V_{k,\mathcal{ℒ}}$ for weights $k$ at most $p�-1$ (resp. $p�+1),$ at least on the inertia subgroup. In the cases where zig-zag is known, we are further able to obtain some new information about the reductions for small odd weights.

In the cases where zig-zag is known, we are further able to obtain some new information about the reduct

We give a formula for the Chow rings of weighted blow-ups. Along the way, we also compute the Chow rings of weighted projective stack bundles, a formula for the Gysin homomorphism of a weighted blow-up, and a generalization of the splitting principle. In addition, in the Appendix we compute the Chern class of a weighted blow-up.

We provide an explicit and algorithmic version of a theorem of Momose classifying isogenies of prime degree of elliptic curves over number fields, which we implement in Sage and PARI/GP. Combining this algorithm with recent work of Box, Gajović and Goodman we obtain the first classifications of the possible prime degree isogenies of elliptic curves over cubic number fields, as well as for several quadratic fields not previously known. While the correctness of the general algorithm relies on the generalised Riemann hypothesis, the algorithm is unconditional for the restricted class of semistable elliptic curves.

Motivated by the work of Lubotzky, we use Galois cohomology to study the difference between the number of generators and the minimal number of relations in a presentation of $G_S(k)$, the Galois group of the maximal extension of a global field $k$ that is unramified outside a finite set $S$ of places, as $k$ varies among a certain family of extensions of a fixed global field $Q$. We define a group $B_S(k,A)$, for each finite simple $G_S(k)$-module $A$, to generalize the work of Koch and Shafarevich on the pro-$\ell$ completion of $G_S(k)$. We prove that $G_S(k)$ always admits a balanced presentation when it is finitely generated. In the setting of the nonabelian Cohen–Lenstra heuristics, we prove that the unramified Galois groups studied by the Liu–Wood–Zureick-Brown conjecture always admit a balanced presentation in the form of the random group in the conjecture.

This work concerns asymptotical stabilisation phenomena occurring in the moduli space of sections of certain algebraic families over a smooth projective curve, whenever the generic fibre of the family is a smooth projective Fano variety, or not far from being Fano.

We describe the expected behaviour of the class, in a ring of motivic integration, of the moduli space of sections of given numerical class. Up to an adequate normalisation, it should converge, when the class of the sections goes arbitrarily far from the boundary of the dual of the effective cone, to an effective element given by a motivic Euler product. Such a principle can be seen as an analogue for rational curves of the Batyrev–Manin–Peyre principle for rational points.

The central tool of this article is the property of equidistribution of curves. We show that this notion does not depend on the choice of a model of the generic fibre, and that equidistribution of curves holds for smooth projective split toric varieties. As an application, we study the Batyrev–Manin–Peyre principle for curves on a certain kind of twisted products.

A smooth cuboid can be identified with a $3\times 3$ matrix of linear forms in three variables, with coefficients in a field $K$, whose determinant describes a smooth cubic in the projective plane. To each such matrix one can associate a group scheme over $K$. We produce isomorphism invariants of these groups in terms of their adjoint algebras, which also give information on the number of their maximal abelian subgroups. Moreover, when $K$ is finite, we give a characterization of the isomorphism types of the groups in terms of isomorphisms of elliptic curves and also describe their automorphism groups. We conclude by applying our results to the determination of the automorphism groups and isomorphism testing of finite $p$-groups of class $2$ and exponent $p$ arising in this way.