scholarly journals Hecke’s modular forms by functional equations

2018 ◽  
Vol 14 (05) ◽  
pp. 1247-1256
Author(s):  
Bernhard Heim
Keyword(s):  
Modular Forms ◽  
Functional Equations ◽  
Hecke Operators ◽  
General Context ◽  
Periodic Functions

We investigate the interplay between multiplicative Hecke operators, including bad primes, and the characterization of modular forms studied by Hecke. The operators are applied on periodic functions, which lead to functional equations characterizing certain eta-quotients. This can be considered as a prototype for functional equations in the more general context of Borcherds products.

scholarly journals CONSTRUCTING SIMULTANEOUS HECKE EIGENFORMS

2010 ◽  
Vol 06 (05) ◽  
pp. 1117-1137 ◽  
Author(s):  
T. SHEMANSKE ◽  
S. TRENEER ◽  
L. WALLING
Keyword(s):  
Hecke Operators ◽  
Cusp Forms ◽  
Hecke Eigenforms ◽  
Linearly Independent ◽  
Integral Weight ◽  
Trivial Character ◽  
Free Level ◽  
Fixed Space ◽  
Space Of Cusp Forms

It is well known that newforms of integral weight are simultaneous eigenforms for all the Hecke operators, and that the converse is not true. In this paper, we give a characterization of all simultaneous Hecke eigenforms associated to a given newform, and provide several applications. These include determining the number of linearly independent simultaneous eigenforms in a fixed space which correspond to a given newform, and characterizing several situations in which the full space of cusp forms is spanned by a basis consisting of such eigenforms. Part of our results can be seen as a generalization of results of Choie–Kohnen who considered diagonalization of "bad" Hecke operators on spaces with square-free level and trivial character. Of independent interest, but used herein, is a lower bound for the dimension of the space of newforms with arbitrary character.

scholarly journals A Characterization of Polynomial Density on Curves via Matrix Algebra

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1231
Author(s):  
Carmen Escribano ◽  
Raquel Gonzalo ◽  
Emilio Torrano
Keyword(s):  
Matrix Algebra ◽  
Optimization Problems ◽  
Positive Semidefinite ◽  
Point Of View ◽  
Alternative Proof ◽  
General Context ◽  
Positive Semidefinite Matrices ◽  
Jordan Curves ◽  
Point Evaluations

In this work, our aim is to obtain conditions to assure polynomial approximation in Hilbert spaces L 2 ( μ ) , with μ a compactly supported measure in the complex plane, in terms of properties of the associated moment matrix with the measure μ . To do it, in the more general context of Hermitian positive semidefinite matrices, we introduce two indexes, γ ( M ) and λ ( M ) , associated with different optimization problems concerning theses matrices. Our main result is a characterization of density of polynomials in the case of measures supported on Jordan curves with non-empty interior using the index γ and other specific index related to it. Moreover, we provide a new point of view of bounded point evaluations associated with a measure in terms of the index γ that will allow us to give an alternative proof of Thomson’s theorem, by using these matrix indexes. We point out that our techniques are based in matrix algebra tools in the framework of Hermitian positive definite matrices and in the computation of certain indexes related to some optimization problems for infinite matrices.

scholarly journals Siegel modular forms and theta series attached to quaternion algebras

1991 ◽  
Vol 121 ◽  
pp. 35-96 ◽  
Author(s):  
Siegfried Böcherer ◽  
Rainer Schulze-Pillot
Keyword(s):  
Modular Forms ◽  
Orthogonal Group ◽  
Automorphic Forms ◽  
Theta Series ◽  
Siegel Modular Forms ◽  
Theta Correspondence ◽  
Quaternion Algebras ◽  
Metaplectic Group

The two main problems in the theory of the theta correspondence or lifting (between automorphic forms on some adelic orthogonal group and on some adelic symplectic or metaplectic group) are the characterization of kernel and image of this correspondence. Both problems tend to be particularly difficult if the two groups are approximately the same size.

scholarly journals Hecke operators on differential modular forms mod p

2012 ◽  
Vol 132 (5) ◽  
pp. 966-997 ◽  
Author(s):  
Alexandru Buium ◽  
Arnab Saha
Keyword(s):  
Modular Forms ◽  
Hecke Operators ◽  
Differential Modular Forms ◽  
Mod P

scholarly journals Nutritional Characterization of Two Rare Landraces of Turnip (Brassica rapa. var. rapa) Tops and Their On-Farm Conservation in Foggia Province

2020 ◽  
Vol 12 (9) ◽  
pp. 3842
Author(s):  
Giulia Conversa ◽  
Corrado Lazzizera ◽  
Anna Bonasia ◽  
Paolo La Rotonda ◽  
Antonio Elia
Keyword(s):  
Brassica Rapa ◽  
Urban Areas ◽  
General Context ◽  
Antioxidant Compounds ◽  
Economic Activities ◽  
Marginal Lands ◽  
On Farm Conservation ◽  
Landscape Units ◽  
On Farm

The study of nutritional properties in landrace products and the general context of its cultivation site are crucial to designing a sustainable on-farm strategy for landrace conservation. The present study describes the main nutritional aspects of two Brassica rapa subspecies rapa landraces collected in Puglia, Italy along with agroecological and socioeconomical traits where they are cultivated. The two B. rapa landraces (‘Cima di rapa dalla testa’ and ‘Cima di rapa antica’) are only found in sites at 700–800 m asl and in two landscape units (the Southern Daunian Mountains (SDM) and the Umbra Forest (UF), respectively) of the Foggia province. These rare landraces were selected by farmers to produce turnip greens/tops from ancient root turnip crops. They are named and consumed by local people in the same way as turnip tops of Brassica rapa subspecies sylvestris (‘Cima di rapa’), which are widely cultivated in Puglia. Compared to the most common ‘Cima di rapa’, the two highlighted landraces have a better nutritional profile linked to an improved content in antioxidant compounds—i.e., vitamin C (458 mg kg−1 FW), total phenols (347 mg ac. gallic equivalent kg−1 FW)—in glucosinolate (741 µmol kg FW−1, in ‘Cima di rapa antica’) and in minerals, such as K. Both landraces are deemed as having a high risk of erosion. Few exemplars are cultivated in marginal lands and urban/peri-urban areas (SDM), or in isolated sites within the UF, which is a special protection zone within Gargano National Park. However, natural, cultural, and recreational tourism are the main economic activities in both landscape units.

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