Related papers: On class groups of random number fields

One can often encounter claims that classical (Kolmogorovian) probability theory cannot handle, or even is contradicted by, certain empirical findings or substantive theories. This note joins several previous attempts to explain that these…

Let H(G) be the Hecke algebra of a reductive p-adic group G. We formulate a conjecture for the ideals in the Bernstein decomposition of H(G). The conjecture says that each ideal is geometrically equivalent to an algebraic variety. Our…

These notes describe some links between the group $\mathrm{SL}_2(\mathbb{R})$, the Heisenberg group and hypercomplex numbers---complex, dual and double numbers. Relations between quantum and classical mechanics are clarified in this…

As an alternative to the well-known methods of "chaining" and "bracketing" that have been developed in the study of random fields, a new method, which is based on a {\em stochastic maximal inequality} derived by using the formula for…

The purpose of this article is to initiate Arakelov theory in a noncommutative setting. More precisely, we are concerned with Arakelov theory of noncommutative arithmetic curves. Our first main result is an arithmetic Riemann-Roch formula…

Group robustness has become a major concern in machine learning (ML) as conventional training paradigms were found to produce high error on minority groups. Without explicit group annotations, proposed solutions rely on heuristics that aim…

The investigation of the ideal class group $Cl_K$ of an algebraic number field $K$ is one of the key subjects of inquiry in algebraic number theory since it encodes a lot of arithmetic information about K. There is a considerable amount of…

The Multivariate Hensel Lemma for local rings is usually proved as a consequence of the Grothendieck version of Zariski's Main Theorem. This version deals with a more general situation that is a priori much more difficult. In this paper, we…

Using compactifications in the logarithmic cotangent bundle, we obtain a formula for the Chern classes of the pushforward of Lagrangian cycles under an open embedding with normal crossing complement. This generalizes earlier results of…

In [R2] and [RO] the Arnold conjecture for closed symplectic manifolds with trivial second homotopy group was proved. This proof used surgery and cobordism theory. Here we give a purely cohomological proof of this result.

Based on the discovery of the duality between Jackiw-Teitelboim quantum gravity and a double-scaled matrix ensemble by Saad, Shenker and Stanford in 2019, we show how consistency between the two theories in the universal Random Matrix…

Let us consider the following matrix $B_n$. The columns of $B_n$ are indexed with $[n]=\{1,2,\dots,n\}$ and the rows are indexed with $[n]^3$. The row corresponding to $(x_1,x_2,x_3)\in [n]^3$ is given by $\sum_{i=1}^3 e_{x_i}$, where…

We present an algorithm for the computation of logarithmic l-class groups of number fields. Our principal motivation is the effective determination of the l-rank of the wild kernel in the K-theory of number fields.

In [7] we proposed a non-generational conjectural derivation of all first class constraints (involving, only, variables compatible with canonical Poisson brackets) for realistic gauge (singular) field theories; and we verified the…

The purpose of this short note is to prove a formula for the Chern-Mather classes of a toric variety in terms of its orbits and the local Euler obstructions at general points of each orbit (Theorem 2). We use the general definition of the…

In this article, we review the proofs of the first Zassenhaus Conjecture on conjugacy of torsion units in integral group rings for the alternating groups of degree 5 and 6, by Luthar-Passi and Hertweck. We describe how the study of these…

The purpose of this paper is to combine classical methods from transcendental number theory with the technique of restriction to real scalars. We develop a conceptual approach relating transcendence properties of algebraic groups to results…

We introduce the concepts of an amazing hypercube decomposition and a double shortcut for it, and use these new ideas to formulate a conjecture implying the Combinatorial Invariance Conjecture of the Kazhdan--Lusztig polynomials for the…

We give a proof of the Universality Conjecture for orthogonal and symplectic ensembles of random matrices in the scaling limit for a class of weights w(x)=exp(-V(x)) where V is a polynomial, V(x)=kappa_{2m}x^{2m}+..., kappa_{2m}>0. For such…

Suppose $E$ is an elliptic curve over $\mathbb{Q}$ and $\chi$ is a Dirichlet character. We use statistical properties of modular symbols to estimate heuristically the probability that $L(E,\chi,1) = 0$. Via the Birch and Swinnerton-Dyer…