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Let be the number of -regular partitions of . Recently, Hou et al. established several infinite families of congruences for modulo , where and . In this paper, using the vanishing property given by Hou et al., we prove an infinite family of congruence for modulo . Moreover, for and , we obtain three infinite families of congruences for modulo and respectively using the theory of Hecke eigenforms. [ABSTRACT FROM AUTHOR]

The generating functions of divisor functions are quasimodular forms of weight 2 and the product of them is a quasimodular form of higher weight. In this work, we evaluate the convolution sums for the positive integers , and with lcm. We reprove the known formulas for the number of representations of a positive integer by each of the quadratic forms as an application of the new identities proved in this paper. [ABSTRACT FROM AUTHOR]

The modular properties of generalized theta-functions with characteristics are used to obtain the formulas for the number of representations of positive integers by the diagonal quadratic forms in nine variables with coefficients that are ones and twos. [ABSTRACT FROM AUTHOR]

In this paper, we give a criterion for the dual Selmer group of an elliptic curve which has either good ordinary reduction or multiplicative reduction at every prime above p to satisfy the 픐H(G)-conjecture. As a by-product of our calculations, we are able to define the Akashi series of the dual Selmer groups assuming the conjectures of Mazur and Schneider. Previously, the Akashi series are defined under the stronger assumption that the dual Selmer group satisfies the 픐H(G)-conjecture. We then establish a criterion for the vanishing of the dual Selmer groups using the Akashi series. We will apply this criterion to prove some results on the characteristic elements of the dual Selmer groups. Our methods in this paper are inspired by the work of Coates-Schneider-Sujatha and can be extended to the Greenberg Selmer groups attached to other ordinary representations, for instance, those coming from a p-ordinary modular form. [ABSTRACT FROM AUTHOR]

We study modular Galois representations mod pm. We show that there are three progressively weaker notions of modularity for a Galois representation mod pm: We have named these "strongly", "weakly", and "dc-weakly" modular. Here, "dc" stands for "divided congruence" in the sense of Katz and Hida. These notions of modularity are relative to a fixed level M. Using results of Hida we display a level-lowering result ("stripping-of-powers of p away from the level"): A mod pm strongly modular representation of some level Npr is always dc-weakly modular of level N (here, N is a natural number not divisible by p). We also study eigenforms mod pm corresponding to the above three notions. Assuming residual irreducibility, we utilize a theorem of Carayol to show that one can attach a Galois representation mod pm to any "dc-weak" eigenform, and hence to any eigenform mod pm in any of the three senses. We show that the three notions of modularity coincide when m = 1 (as well as in other particular cases), but not in general. [ABSTRACT FROM AUTHOR]