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The notion of broken k-diamond partitions was introduced by Andrews and Paule. Let $$\Delta _k(n)$$ denote the number of broken k-diamond partitions of n for a fixed positive integer k. Recently, a number of parity results satisfied by $$\Delta _k(n)$$ for small values of k have been proved by Radu and Sellers and others. However, congruences modulo 4 for $$\Delta _k(n)$$ are unknown. In this paper, we will prove five congruences modulo 4 for $$\Delta _5(n)$$ , four infinite families of congruences modulo 4 for $$\Delta _7(n)$$ and one congruence modulo 4 for $$\Delta _{11}(n)$$ by employing theta function identities. Furthermore, we will prove a new parity result for $$\Delta _2(n)$$ . [ABSTRACT FROM AUTHOR]

We present an algorithmic approach to the verification of identities on multiple theta functions in the form of products of theta functions [(1) a1 1 a 2 2 a r r qs; qt]1, where i are integers, = 0 or 1, s 2 Q, t 2 Q+, and the exponent vectors (1; 2; : : : ; r) are linearly independent over Q. For an identity on such multiple theta functions, we provide an algorithm to compute a system of contiguous relations satisfied by all the involved multiple theta functions. Using Stanley's lemma on the fundamental parallelepiped, we show that a multiple theta function can be determined by a nite number of its coeficients. Thus such an identity can be reduced to a finite number of simpler relations. Many classical multiple theta function identities fall into this framework, including Riemann's addition formula and the extended Riemann identity. [ABSTRACT FROM AUTHOR]

Text We develop geometric methods to study the generating weights of free modules of vector-valued modular forms of half-integral weight, taking values in a complex representation of the metaplectic group. We then compute the generating weights for modular forms taking values in the Weil representation attached to cyclic quadratic modules of order 2 p r , where p ≥ 5 is a prime. We also show that the generating weights approach a simple limiting distribution as p grows, or as r grows and p remains fixed. Video For a video summary of this paper, please visit https://youtu.be/QNbPSXXKot4 . [ABSTRACT FROM AUTHOR]

Let $$\overline{p}(n)$$ denote the number of overpartitions of n. In this paper, we shall show that for $$n\ge 0$$ , where $$r=8,52,68$$ , and 72. In addition, we present a short alternative proof of the congruence which is conjectured by Hirschhorn and Sellers. [ABSTRACT FROM AUTHOR]

Let k be an odd positive integer, L a lattice on a regular positive definite k-dimensional quadratic space over $$\mathbb {Q}$$ , $$N_L$$ the level of L, and $$\mathscr {M}(L)$$ be the linear space of $$\theta $$ -series attached to the distinct classes in the genus of L. We prove that, for an odd prime $$p|N_L$$ , if $$L_p=L_{p,1}\,\bot \, L_{p,2}$$ , where $$L_{p,1}$$ is unimodular, $$L_{p,2}$$ is ( p)-modular, and $$\mathbb {Q}_pL_{p,2}$$ is anisotropic, then $$\mathscr {M}(L;p):=$$ $$\mathscr {M}(L)$$ $$+T_{p^2}.$$ $$\mathscr {M}(L)$$ is stable under the Hecke operator $$T_{p^2}$$ . If $$L_2$$ is isometric to $$\left( \begin{array}{ll}0&{}\frac{1}{2}\\ \frac{1}{2}&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle \varepsilon \rangle $$ or $$\left( \begin{array}{ll}0&{}\frac{1}{2}\\ \frac{1}{2}&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle 2\varepsilon \rangle $$ or $$\left( \begin{array}{ll}0&{}1\\ 1&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle \varepsilon \rangle $$ with $$\varepsilon \in \mathbb {Z}_2^{\times }$$ and $$\kappa :=\frac{k-1}{2}$$ , then $$\mathscr {M}(L;2):=T_{2^2}.\mathscr {M}(L)+T_{2^2}^2.\,\mathscr {M}(L)$$ is stable under the Hecke operator $$T_{2^2}$$ . Furthermore, we determine some invariant subspaces of the cusp forms for the Hecke operators. [ABSTRACT FROM AUTHOR]

In this paper, we study congruence properties of overcubic partition pairs. Let b(n) denote the number of overcubic partition pairs of n. We will establish some new Ramanujan type congruences and several infinite families of congruences modulo powers of 2 satisfied by b(n). [ABSTRACT FROM AUTHOR]

In 2007, Andrews and Paule introduced the notion of broken k-diamond partitions. Let $$\Delta _k(n)$$ denote the number of broken k-diamond partitions of n for a fixed positive integer k. Recently, Paule and Radu presented some conjectures on congruences modulo 7 for $$\Delta _3(n)$$ which were proved by Jameson and Xiong based on the theory of modular forms. Very recently, Xia proved several infinite families of congruences modulo 7 for $$\Delta _3(n)$$ using theta function identities. In this paper, many new infinite families of congruences modulo 7 for $$\Delta _3(n)$$ are derived based on an identity of Newman and the ( p; k)-parametrization of theta functions due to Alaca, Alaca and Williams. In particular, some non-standard congruences modulo 7 for $$\Delta _3(n)$$ are deduced. For example, we prove that for $$\alpha \ge 0$$ , $$\Delta _3\left( \frac{14\times 757^{\alpha }+1}{3}\right) \equiv 6 -\alpha \ (\mathrm{mod}\ 7)$$ . [ABSTRACT FROM AUTHOR]

We show that for any lattice L ⊆ ℝn and vectors x, y ∈ ℝn, ρ(L + x)² ρ(L + y)² ≤ ρ(L)² ρ(L + x + y) ρ(L + x - y), where ρ is the Gaussian mass function ρ(A) := Σw ∈ A exp(-π||w||²). We show a number of applications, including bounds on the moments of the discrete Gaussian distribution, various monotonicity properties of the heat kernel on flat tori, and a positive correlation inequality for Gaussian measures on lattices. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the arithmetic properties of ℓ-regular overpartition pairs. Let Bℓ(n) denote the number of ℓ-regular overpartition pairs of n. We will prove the number of Ramanujan-like congruences and infinite families of congruences modulo 3, 8, 16, 36, 48, 96 for B3(n) and modulo 3, 16, 64, 96 for B4(n). For example, we find that for all nonnegative integers α and n, B3(3α(3n + 2)) ≡ 0 (mod 3), B3(3α(6n + 4)) ≡ 0 (mod 3), and B4(8n + 7) ≡ 0 (mod 64). [ABSTRACT FROM AUTHOR]

We present some applications of the Kudla–Millson and the Millson theta lift. The two lifts map weakly holomorphic modular functions to vector valued harmonic Maass forms of weight 3/2 and 1/2, respectively. We give finite algebraic formulas for the coefficients of Ramanujan's mock theta functions f ( q ) and ω ( q ) in terms of traces of CM-values of a weakly holomorphic modular function. Further, we construct vector valued harmonic Maass forms whose shadows are unary theta functions, and whose holomorphic parts have rational coefficients. This yields a rationality result for the coefficients of mock theta functions, i.e., harmonic Maass forms whose shadows lie in the space of unary theta functions. Moreover, the harmonic Maass forms we construct can be used to evaluate the Petersson inner products of unary theta functions with harmonic Maass forms, giving formulas and rationality results for the Weyl vectors of Borcherds products. [ABSTRACT FROM AUTHOR]