Related papers: Notes On Hilbert's 12th Problem

Let $p$ be an irregular prime and $K=\Q(\zeta)$ the $p$-cyclotomic field. Let $\sigma$ be a $\Q$-isomorphism of $K$ generating $Gal(K/\Q)$. Let $S/K$ be a cyclic unramified extension of degree $p$, defined by $S= K(A^{1/p})$ where $A\in…

Given a modular form f of even weight larger than two and an imaginary quadratic field K satisfying a relaxed Heegner hypothesis, we construct a collection of CM cycles on a Kuga-Sato variety over a suitable Shimura curve which gives rise…

Using mirror symmetry, we resolve an old puzzle in the linear sigma model description of the spacetime Higgs mechanism in a heterotic string compactification with (2,2) worldsheet supersymmetry. The resolution has a nice spacetime…

The MinRank problem is a simple linear algebra problem: given matrices with coefficients in a field, find a non trivial linear combination of the matrices that has a small rank. There are several algebraic modeling of the problem. The main…

We axiomatically define (pre-)Hilbert categories. The axioms resemble those for monoidal Abelian categories with the addition of an involutive functor. We then prove embedding theorems: any locally small pre-Hilbert category whose monoidal…

The aim of this short note is to prove the formula of the Hilbert series of the preprojective algebras in arbitrary characteristic by making effective use of the formulas of the Hilbert series of differential graded (dg) algebras with Adams…

We begin by defining general hypergeometric functions over finite fields and obtaining a finite field analogue of a classical symmetry in their complex counterparts. We give a geometric proof for the symmetry by constructing isomorphisms…

The introduction of strongly-interacting mirror fermions with masses between the weak scale and 1 TeV could offer a viable alternative to the Higgs mechanism. The framework provided solves the hierarchy problem naturally and predicts a rich…

We use Macaulay's inverse system to study the Hilbert series for almost complete intersections generated by uniform powers of general linear forms. This allows us to give a classification of the Weak Lefschetz property for these algebras,…

One can consider the Hilbert scheme as a natural compactification of the space of smooth projective curves with fixed Hilbert polynomial. Here we consider a different modular compactification, namely the functor CM parameterizing curves…

The inverse spectral transform for the Zakharov-Shabat equation on the semi-line is reconsidered as a Hilbert problem. The boundary data induce an essential singularity at large k to one of the basic solutions. Then solving the inverse…

We argue that presymmetry, a hidden predynamical electroweak quark-lepton symmetry that explains the fractional charges and triplication of families, must be extended beyond the Standard Model as to have a residual presymmetry that embraces…

We use the method of Bruinier--Raum to show that symmetric formal Fourier--Jacobi series, in the cases of norm-Euclidean imaginary quadratic fields, are Hermitian modular forms. Consequently, combining a theorem of Yifeng Liu, we deduce…

Arnold, Falk, and Winther recently showed [Bull. Amer. Math. Soc. 47 (2010), 281-354] that linear, mixed variational problems, and their numerical approximation by mixed finite element methods, can be studied using the powerful, abstract…

In this work we explore the construction of abelian extensions of number fields with exactly one complex place using multivariate analytic functions in the spirit of Hilbert's 12th problem. To this end we study the special values of the…

The work of the first author on the moment map for representations of quivers included a classification of the possible dimension vectors of simple modules for deformed preprojective algebras. That classification was later used to solve an…

We construct a minimal viable extension of the standard model (SM) with classical scale symmetry. Its scalar sector contains a complex singlet in addition to the SM Higgs doublet. The scale-invariant and CP-symmetric Higgs potential…

We introduce an explicit class of tempered Laurent polynomials in the sense of Villegas and Doran--Kerr in $n \leqslant 4$ variables including all Landau--Ginzburg models for smooth Fano threefolds with very ample anticanonical class. We…

In this paper we deal with the connection of frames with the class of Hilbert Schmidt operators. First we give an easy criteria for operators being in this class using frames. It is the equivalent to the criteria using orthonormal bases.…

The first part is expository: it explains how finite fields may be used to prove theorems on infinite fields by a reduction mod p process. The second part gives a variant of P.Smith's fixed point theorem which applies in any characteristic.