Group Γ0 Research Articles

Automorphisms of a Chevalley Group of Type G2 Over a Commutative Ring R with 1/3 Generated by the Invertible Elements and 2R

Automorphisms of a Chevalley Group of Type G2 Over a Commutative Ring R with 1/3 Generated by the Invertible Elements and 2R

Affine Grassmannians for G2

We study affine Grassmannians for the exceptional group of type G2. This group can be given as automorphisms of octonion algebras (or para-octonion algebras). By using this automorphism group, we consider all maximal parahoric subgroups in G2, and give a description of affine Grassmannians for G2 as functors classifying suitable orders in a fixed space.

Location and double interlacing of zeros of certain combination of the Eisenstein series for [formula omitted]

We show that the zeros of aE2k+(z)−(Ek+(z))2 and bE2k+(z)+(Ek+(z))2 lie on the arc of the fundamental domain for the Fricke group Γ0+(2) of level 2, where Ek+(z) is the Eisenstein series for Γ0+(2), and we investigate the doubly interlacing property between non-elliptic zeros of aE2k+(z)−(Ek+(z))2 and bE2k+(z)+(Ek+(z))2.

On a conjecture of Chen and Yui: Fricke groups

In this work, we establish formulas for the prime decompositions of resultant and discriminant of the ring class polynomial associated with Thompson series for Fricke group Γ0(p)+ for p prime and imaginary quadratic field, and as a consequence, validate a conjecture of Chen and Yui on the upper bound of prime factors of such resultants and discriminants. All associated numerical conjectures of Chen and Yui are verified using the Magma code for our formulas written by Chao Qin.

Infinitely many zeros of additively twisted L-functions on the critical line

For f a cuspidal modular form for the group Γ0(N) of integral or half-integral weight, N a multiple of 4 in case the weight is half-integral, we study the zeros of the L-function attached to f twisted by an additive character e2πinpq with pq∈Q. We prove that for certain f and pq∈Q, the additively twisted L-function has infinitely many zeros on the critical line. We develop a variant of the Hardy-Littlewood method which uses automorphic distributions to prove the result.

Jutila's circle method and [formula omitted] shifted convolution sums

Let λf(n) be the normalized Fourier coefficients of a Hecke-Maass or Hecke holomorphic cusp form f for congruence group Γ0(N) with level N and nebentypus χN. Let Q(x) be a positive definite integral quadratic form, and r(n,Q) denote the number of representations of n by the quadratic form Q. In this paper, we apply Jutila's circle method to derive a sharp bound for the shifted convolution sum of Fourier coefficients λf(n) and r(n,Q). We generalize and improve previous results without the Ramanujan conjecture.

Derivations of the Positive Part of the Two-parameter Quantum Group of Type G2

In this paper, we compute the derivations of the positive part of the two-parameter quantum group of type G2 by embedding it into a quantum torus. We also show that the first Hochschild cohomology group of this algebra is a two-dimensional vector space over the complex field.

On the zeros of period functions associated to the Eisenstein series for [formula omitted

We consider the period functions associated to the Eisenstein series for the Fricke group Γ0+(N), the odd parts of the period functions and certain polynomials obtained from the period functions for Γ0+(N), and we prove that all zeros of each of them lie on the circle |z|=1N by applying the properties of a self-inversive polynomial. In particular, our result proves Berndt and Straub's suggested problem.

The Hasse invariant of the Tate normal form E5 and the class number of [formula omitted

It is shown that the number of irreducible quartic factors of the form g(x)=x4+ax3+(11a+2)x2−ax+1 which divide the Hasse invariant of the Tate normal form E5 in characteristic l is a simple linear function of the class number h(−5l) of the field Q(−5l), when l≡2,3 modulo 5. A similar result holds for irreducible quadratic factors of g(x), when l≡1,4 modulo 5. This implies a formula for the number of linear factors over Fp of the supersingular polynomial ssp(5⁎)(x) corresponding to the Fricke group Γ0⁎(5).

Pagoda[5]arene with Large and Rigid Cavity for the Formation of 1∶2 Host–Guest Complexes and Acid/Base-Responsive Crystalline Vapochromic Properties

Pagoda[5]arene (P5), which is composed of five 2,6-dimethoxylanthracene (2,6-DMA) subunits bridged by methylene groups at 1,5 positions, was conveniently synthesized in 43% by trifluoroacetic acid ...

On the space of theta functions whose levels are square-free

Hecke conjectured that an explicit set of theta series obtained from a quaternion algebra defined over ℚ ramified at a prime N is a basis of a space of holomorphicmodular forms of weight 2 for the Hecke congruence group Γ0(N). However, Eichler noticed that Hecke’s conjecture is not true in general. Hence it is natural to ask the dimension of the subspace of M2(Γ0(N)) spanned by the theta series, and this question is called Hecke’s basis problem, which we have shown an answer in [K. Sugiyama, On the space of theta functions for a prime level, Comment. Math. Univ. St. Pauli, 67(1):66–81, 2019]. In this paper, we generalize the results for a square-free positive integer N.

The number of zeros of certain combinations of the Eisenstein series for [formula omitted

We give a lower bound of the number of zeros of the modular forms Ek+Eℓ+−Ek+ℓ+ for large enough k and ℓ on each of the arc boundary and on the vertical boundary of the fundamental domain for the Fricke group Γ0+(2), where Ek+ is the Eisenstein series of weight k for Γ0+(2).

On the Hasse invariants of the Tate normal forms E5 and E7

A formula is proved for the number of linear factors over Fl of the Hasse invariant of the Tate normal form E5(b) for a point of order 5, as a polynomial in the parameter b, in terms of the class number of the imaginary quadratic field K=Q(−l), proving a conjecture of the author from 2005. A similar theorem is proved for quadratic factors with constant term −1, and a theorem is stated for the number of quartic factors of a specific form in terms of the class number of Q(−5l). These results are shown to imply a recent conjecture of Nakaya on the number of linear factors over Fl of the supersingular polynomial ssl(5⁎)(X) corresponding to the Fricke group Γ0⁎(5). The degrees and forms of the irreducible factors of the Hasse invariant of the Tate normal form E7 for a point of order 7 are determined, which is used to show that the polynomial ssl(N⁎)(X) for the group Γ0⁎(N) has roots in Fl2, for any prime l≠N, when N∈{2,3,5,7}.

Drinfeld cusp forms: oldforms and newforms

Let p=(P) be any prime of Fq[t], let m be any ideal of Fq[t] not divisible by p and consider the space of Drinfeld cusp forms of levelmp, i.e. for the modular group Γ0(mp). Using degeneracy maps, traces and Fricke involutions we offer definitions for p-oldforms and p-newforms which turn out to be subspaces stable with respect to the action of the Atkin operator UP. We provide eigenvalues and/or slopes for p-oldforms and p-newforms and a condition to get the whole space of cusp forms as the direct sum between them.

Word Maps of Chevalley Groups Over Infinite Fields

Let G be a simply connected Chevalley group over an infinite field K, and let $$ \tilde{w} $$ : Gn → G be a word map that corresponds to a nontrivial word w. In 2015, it has been proved that if w = w1w2w3w4 is the product of four words in independent variables, then every noncentral element of G is contained in the image of $$ \tilde{w} $$. A similar result for a word w = w1w2w3, which is the product of three independent words, was obtained in 2019 under the condition that the group G is not of type B2 or G2. In the present paper, it is proved that for a group of type B2 or G2, all elements of the large Bruhat cell B nw0B are contained in the image of the word map $$ \tilde{w} $$, where w = w1w2w3 is the product of three independent words. For a group G of type Ar, Cr, or G2 (respectively, for a group of type Ar) or a group over a perfect field K (respectively, over a perfect field K the characteristic of which is not a bad prime for G) with dim K ≤ 1 (here, dim K is the cohomological dimension of K), it is proved that all split regular semisimple elements (respectively, all regular unipotent elements) of G are contained in the image of $$ \tilde{w} $$, where w = w1w2 is the product of two independent words. Also, for any isotropic (but not necessary split) simple algebraic group G over a field K of characteristic zero, it is shown that for a word map $$ \tilde{w} $$ : G(K)n → G(K), where w = w1w2 is a product of two independent words, all unipotent elements are contained in Im $$ \tilde{w} $$.

Indirect Selection of Tolerant Barley (Hordeum vulgare L.) Genotypes under Semi Arid Conditions Based on the Numerical Images Analysis Indices

Image analysis systems have been increasingly utilized for the assessment of plant growth and health for decades. We used in this study the software Mesurim Pro to evaluate the variation of the leaf reflectance at Red, Green and Blue and the variation of the senescence parameters. The analysis of variance revealed that the reflectance at different wavelengths (Red, Blue and Green) was highly significant genotypes effects (P 0.001); for this parameter the good genotypes are those we have the lowest values such as G19. In addition, the preferable genotypes were those which have low values for the mean senescence and senescence velocity; based on this raison the best genotype was the introduce genotype G12. The genotypes effect was significant for the grain yield and thousand-kernel weight, for the chlorophyll content and the analysis of variance showed a significant effect of genotypes, the highest values registered by the introduced genotype G5 this one was in the same homogenize group of G2, G4, G8 and G18. The ranking of genotypes based on all parameters suggested that the genotypes G11, G12, G5, G15 and G18, respectively (introduce genotypes) were the ideal genotypes under these conditions.

Place of cardiovascular comorbidities in hospitalized patients for exacerbation of chronic obstructive pulmonary disease

The chronic obstructive pulmonary disease (COPD) is a chronic systemic disease. Recently, emphasis has been placed on the importance of the associated comorbidities, especially cardiovascular, which have a considerable impact on the quality of life and the prognosis of patients. In this study, we attempted to assess the association between COPD and different comorbidities and to evaluate a possible dependence with the profile of a frequent exacerbator. It is a randomized, double-blind, balanced, prospective study including 43 out-patient, aged from 50 to 92 years, affected with COPD followed-up in the department of pneumology. Each patient was subjected for 12-lead ECG, 24-Hour Holter for arrhythmia detection and two-dimensional echocardiography in the department of cardiology. We compared a group of patients with frequent exacerbations of COPD G1 ( n = 23) to a group of patients G2 with infrequent exacerbations of COPD ( n = 20). The mean of age was 71 years (ranging from 47 years to 92 years) in G1 versus 65 years in G2 ( P = 0.04). Among comorbidities, cardiovascular pathologies were the most frequently associated with COPD observed in 15 patients (35%) followed by diabetes in 7 patients (16%) and by anemia in 5 patients (11%). We noted that different cardiovascular comorbidities were significantly more common in the group of patients with frequent exacerbations than the patients with infrequent exacerbations of COPD ( P = 0.04). In our study, there is a close relationship between exacerbation of COPD and association with different cardiovascular comorbidities. These comorbidities influence both the symptoms and the life expectancy, hence the importance of a global management of comorbidities by a multidisciplinary team.

The transcendence of zeros of canonical basis elements of the space of weakly holomorphic modular forms for Γ0(2)

We consider the canonical basis elements fk,mε for the space of weakly holomorphic modular forms of weight k for the Hecke congruence group Γ0(2) and we prove that for all m≥c(k) for some constant c(k), if z0 in a fundamental domain for Γ0(2) is a zero of fk,mε, then either z0 is in {i2,−12+i2,12+i2,−1+i74,1+i74}, or z0 is transcendental.

The Hauptmodul at elliptic points of certain arithmetic groups

Let N be a square-free integer such that the arithmetic group Γ0(N)+ or Γ0(N)⁎ has genus zero; there are 52 such groups. Let jN+,⁎ denote the associated Hauptmodul normalized to have residue equal to one and constant term equal to zero in its q-expansion. In this article we prove that the Hauptmodul at any elliptic point of the surface associated to Γ0(N)+ or Γ0(N)⁎ is an algebraic integer. Moreover, for each such N and elliptic point e, we show how to explicitly evaluate jN+,⁎(e), and we provide the list of generating polynomials (with small coefficients) of the class fields or their subfields corresponding to the associated orders over the imaginary quadratic extension of rationals.

The nullcone of the Lie algebra of [formula omitted

This paper investigates the nilpotent conjugacy classes of the Lie algebra of the simple algebraic group of type G2. These classes are determined by first finding the stratification, and then finding the classes within the strata. Except for characteristic 3, the classes coincide with the strata. In characteristic 3, one stratum splits into two orbits. If the characteristic differs from 2 and 3, the classes are determined by the singularities of the nilpotent variety. In characteristic 3, the matter is undecided yet. In characteristic 2, different classes have the same singularities.