scholarly journals PROOF OF SOME CONJECTURAL CONGRUENCES OF DA SILVA AND SELLERS

2021 ◽  
pp. 1-5
Author(s):  
AJIT SINGH ◽  
RUPAM BARMAN
Keyword(s):  
Modular Forms ◽  
Arithmetic Properties ◽  
Regular Partitions

Abstract Let $p_{\{3, 3\}}(n)$ denote the number of $3$ -regular partitions in three colours. Da Silva and Sellers [‘Arithmetic properties of 3-regular partitions in three colours’, Bull. Aust. Math. Soc.104(3) (2021), 415–423] conjectured four Ramanujan-like congruences modulo $5$ satisfied by $p_{\{3, 3\}}(n)$ . We confirm these conjectural congruences using the theory of modular forms.

scholarly journals EFFICIENTLY GENERATED SPACES OF CLASSICAL SIEGEL MODULAR FORMS AND THE BÖCHERER CONJECTURE

2010 ◽  
Vol 89 (3) ◽  
pp. 393-405 ◽  
Author(s):  
MARTIN RAUM
Keyword(s):  
Modular Forms ◽  
Class Number ◽  
Siegel Modular Forms ◽  
Generating Set ◽  
Arithmetic Properties ◽  
Fourier Expansions

AbstractWe state and verify up to weight 172 a conjecture on the existence of a certain generating set for spaces of classical Siegel modular forms. This conjecture is particularly useful for calculations involving Fourier expansions. Using this generating set, we verify the Böcherer conjecture for nonrational eigenforms and discriminants with class number greater than one. As a further application we verify another conjecture for weights up to 150 and investigate an analog of the Victor–Miller basis. Additionally, we describe some arithmetic properties of the basis we found.

scholarly journals Modular embeddings of Teichmüller curves

2016 ◽  
Vol 152 (11) ◽  
pp. 2269-2349 ◽  
Author(s):  
Martin Möller ◽  
Don Zagier
Keyword(s):  
Modular Forms ◽  
Fuchsian Groups ◽  
Triangle Groups ◽  
Flat Surfaces ◽  
Arithmetic Properties ◽  
Hilbert Modular Surfaces ◽  
Hilbert Modular ◽  
Genus Two ◽  
Teichmuller Curves ◽  
Teichmüller Curves

Fuchsian groups with a modular embedding have the richest arithmetic properties among non-arithmetic Fuchsian groups. But they are very rare, all known examples being related either to triangle groups or to Teichmüller curves. In Part I of this paper we study the arithmetic properties of the modular embedding and develop from scratch a theory of twisted modular forms for Fuchsian groups with a modular embedding, proving dimension formulas, coefficient growth estimates and differential equations. In Part II we provide a modular proof for an Apéry-like integrality statement for solutions of Picard–Fuchs equations. We illustrate the theory on a worked example, giving explicit Fourier expansions of twisted modular forms and the equation of a Teichmüller curve in a Hilbert modular surface. In Part III we show that genus two Teichmüller curves are cut out in Hilbert modular surfaces by a product of theta derivatives. We rederive most of the known properties of those Teichmüller curves from this viewpoint, without using the theory of flat surfaces. As a consequence we give the modular embeddings for all genus two Teichmüller curves and prove that the Fourier developments of their twisted modular forms are algebraic up to one transcendental scaling constant. Moreover, we prove that Bainbridge’s compactification of Hilbert modular surfaces is toroidal. The strategy to compactify can be expressed using continued fractions and resembles Hirzebruch’s in form, but every detail is different.

scholarly journals Sequences, modular forms and cellular integrals

2018 ◽  
Vol 168 (2) ◽  
pp. 379-404 ◽  
Author(s):  
DERMOT McCARTHY ◽  
ROBERT OSBURN ◽  
ARMIN STRAUB
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Arithmetic Properties ◽  
General Conjecture

AbstractIt is well-known that the Apéry sequences which arise in the irrationality proofs for ζ(2) and ζ(3) satisfy many intriguing arithmetic properties and are related to the pth Fourier coefficients of modular forms. In this paper, we prove that the connection to modular forms persists for sequences associated to Brown's cellular integrals and state a general conjecture concerning supercongruences.

GENERALISATION OF KEITH’S CONJECTURE ON 9-REGULAR PARTITIONS AND 3-CORES

2014 ◽  
Vol 90 (2) ◽  
pp. 204-212 ◽  
Author(s):  
BERNARD L. S. LIN ◽  
ANDREW Y. Z. WANG
Keyword(s):  
Modular Forms ◽  
Infinite Family ◽  
Regular Partitions

AbstractRecently, Keith used the theory of modular forms to study 9-regular partitions modulo 2 and 3. He obtained one infinite family of congruences modulo 3, and meanwhile proposed an analogous conjecture. In this note, we show that 9-regular partitions and 3-cores satisfy the same congruences modulo 3. Thus, we first derive several results on 3-cores, and then generalise Keith’s conjecture and get a stronger result, which implies that all of Keith’s results on congruences modulo 3 are consequences of our result.

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