Search

For an ordinary prime p ⩾ 3 , we consider the Hida family associated to modular forms of a fixed tame level, and their Selmer groups defined over certain Galois extensions of Q ( μ p ) whose Galois group is G ≅ Z p ⋊ Z p . For Selmer groups defined over the cyclotomic Z p -extension of Q ( μ p ) , we show that if the μ-invariant of one member of the Hida family is zero, then so are the μ-invariants of the other members, while the λ-invariants remain the same only in a branch of the Hida family. We use these results to study the behavior of some invariants from non-commutative Iwasawa theory in the Hida family.

We study the linear relations among the Fourier coefficients of modular forms on the group Γ0(N) of genus zero. Applying these linear relations, congruence properties of Hecke eigenforms, replicable properties of Hauptmoduln and congruences of representation numbers of the sums of n squares can be obtained. The eta-quotient expression of the unique normalized modular form ΔN(z) of weight 12 on Γ0(N) with a zero of maximum order at ∞ is listed.

Let rQ(n) be the representation number of a nonnegative integer n by the quaternary quadratic form Q=x12+2x22+x32+x42+x1x3+x1x4+x2x4. We first prove the identity rQ(p2n)=rQ(p2)rQ(n)/rQ(1) for any prime p different from 13 and any positive integer n prime to p, which was conjectured in Eum et al. (2011) [2]. And, we explicitly determine a concise formula for the number rQ(n2) as well for any integer n.

Given modular forms f and g of weights k and l, respectively, their Rankin–Cohen bracket [ f , g ] n ( k , l ) corresponding to a nonnegative integer n is a modular form of weight k + l + 2 n , and it is given as a linear combination of the products of the form f ( r ) g ( n − r ) for 0 ⩽ r ⩽ n . We use a correspondence between quasimodular forms and sequences of modular forms to express the Dirichlet series of a product of derivatives of modular forms as a linear combination of the Dirichlet series of Rankin–Cohen brackets.

We find some modularity criterion for a product of Klein forms of the congruence subgroup Γ 1 ( N ) (Theorem 2.6) and, as its application, construct a basis of the space of modular forms for Γ 1 ( 13 ) of weight 2 (Example 3.4). In the process we face with an interesting property about the coefficients of certain theta function from a quadratic form and prove it conditionally by applying Hecke operators (Proposition 4.3).

This paper is concerned with modular lattices over cyclotomic fields. In particular, the notion of Arakelov modular ideal lattice is introduced. All the cyclotomic fields over which there exists an Arakelov modular lattice of given level are characterised.

Given a prime p and cusp forms f1 and f2 on some Γ1(N) that are eigenforms outside Np and have coefficients in the ring of integers of some number field K, we consider the problem of deciding whether f1 and f2 have the same eigenvalues mod pm (where p is a fixed prime of K over p) for Hecke operators Tℓ at all primes ℓ∤Np. When the weights of the forms are equal the problem is easily solved via an easy generalization of a theorem of Sturm. Thus, the main challenge in the analysis is the case where the forms have different weights. Here, we prove a number of necessary and sufficient conditions for the existence of congruences mod pm in the above sense. The prime motivation for this study is the connection to modular mod pm Galois representations, and we also explain this connection.

A formula for the dimension of the space of cuspidal modular forms on Γ0(N) of weight k (k⩾2 even) has been known for several decades. More recent but still well-known is the Atkin–Lehner decomposition of this space of cusp forms into subspaces corresponding to newforms on Γ0(d) of weight k, as d runs over the divisors of N. A recursive algorithm for computing the dimensions of these spaces of newforms follows from the combination of these two results, but it would be desirable to have a formula in closed form for these dimensions. In this paper we establish such a closed-form formula, not only for these dimensions, but also for the corresponding dimensions of spaces of newforms on Γ1(N) of weight k (k⩾2). This formula is much more amenable to analysis and to computation. For example, we derive asymptotically sharp upper and lower bounds for these dimensions, and we compute their average orders; even for the dimensions of spaces of cusp forms, these results are new. We also establish sharp inequalities for the special case of weight-2 newforms on Γ0(N), and we report on extensive computations of these dimensions: we find the complete list of all N such that the dimension of the space of weight-2 newforms on Γ0(N) is less than or equal to 100 (previous such results had only gone up to 3).

A review is given of several aspects of the work of Basil Gordon. These include: Rogers–Ramanujan identities, plane partitions, the method of weighted words, modular forms and partition congruences, and the asymptotics of partitions and related q-series.