Applications Of Modular Forms In Congruences For Partition Functions

The topic of congruences for partition functions is an interesting and far-reaching subject in the theory of partitions and the number theory. It has received much attention as it is strongly connected to diverse areas of mathematics such as the representation theory of Lie algebras, modular forms and combinatorics. G.E. Andrews pointed out that "Partitions:at the interface of q-series and modular forms."In this thesis, we consider the congruences for certain partition functions, including ped-r （n） which enumerate the number of r-tuple multipartitions of n with even parts distinct and broken k-diamond partition functions△k（n）. The methods we mainly utilized are the properties of modular forms and related op-erators, such as the U-operators, the V-operators, the Hecke operators and the Hecke eigenforms together with the analysis of q-series.This thesis is organized as follows. Chapter1is assigned to give a brief introduction to the congruences for the partition functions. Starting with the background and some breakthroughs in Ramanujan’s original congruences for ordinary partition function p（n）, then we give some definitions and related works for ped-r（n） and△k（n） in two subsections. Some results worked by author will be listed in the end of each subsection.In Chapter2, we give an overview of some definitions and properties of mod-ular forms. First we list some terminology and notation concerning with modular forms on the subgroups （?）o（N） and （?）1（N） of SL2（Z）. Then some operators act-ing on modular forms will be introduced. These operators enable us to construct suitable modular forms related to the generating functions of partition functions. In the end, we discuss an important class of modular forms—the η-quotient.The objective of Chapter3is presenting some congruences for ped_r（n） mod-ulo powers of2. First we give the arithmetic properties for ped_r（n） modulo2and the generating functions of ped₄（2n+1） and ped₄（4n+3） by using the dissection formula of q-series. Then by utilizing the recurrence relation for the coefficients of Hecke eigenforms, we find various families of congruences satisfied by ped₂（n） and ped-4（n） modulo powers of2and a family of congruences for ped₂（n） modulo24. For example, we show that for α≥0,n≥0,In Chapter4, we derive two infinite families of congruences for△2（n） modulo3as well as some general congruence results. Based on a formula of Radu and Sellers, we present the proof for the congruences of△2（n） modulo3by using the3-dissection formula of the generating function of triangular number due to Berndt, and the properties of the U-operator, the V-operator, the Hecke operator and the Hecke eigenform. Specifically, we obtain that, for l>1, n≥0, Then we deduce a general congruence for△2（n） modulo powers of primes m≡1（mod8） by using a classical result owing to Deligne and Serre. In the end, we find a modular form Gm,k（z） on ro（10） that is congruent to the generating function Fm,k（z） of△2（（mkn+1）/4） modulo powers of primes m≠2,5. Which is where...