Narrow Results

In this paper, we describe the set of prime numbers splitting completely in the non-abelian splitting field of certain monic irreducible polynomials of degree $3.$ As an application, we establish some divisibility properties of the associated ternary recurrence sequence by primes $p$, thus greatly extending recent work of Evink and Helminck [Tribonacci numbers and primes of the form $p=x^2+11y^2$, Math. Slovaca69(3) (2019) 521–532] and of Faisant [On the Padovan sequence (2019), https://arxiv.org/abs/1905.07702]. We also prove some new results on the number of solutions of the characteristic equation of the recurrence sequence modulo $p$.

Global conformal invariance (GCI) of quantum field theory (QFT) in two and higher space-time dimensions implies the Huygens' principle, and hence, rationality of correlation functions of observable fields [29]. The conformal Hamiltonian H has discrete spectrum assumed here to be finitely degenerate. We then prove that thermal expectation values of field products on compactified Minkowski space can be represented as finite linear combinations of basic (doubly periodic) elliptic functions in the conformal time variables (of periods 1 and τ) whose coefficients are, in general, formal power series in q^½ = e^iπτ involving spherical functions of the "space-like" fields' arguments. As a corollary, if the resulting expansions converge to meromorphic functions, then the finite temperature correlation functions are elliptic. Thermal 2-point functions of free fields are computed and shown to display these features. We also study modular transformation properties of Gibbs energy mean values with respect to the (complex) inverse temperature . The results are used to obtain the thermodynamic limit of thermal energy densities and correlation functions.

In the studies on the modularity conjecture for rigid Calabi–Yau threefolds several examples with the unique level 8 cusp form were constructed. According to the Tate conjecture, correspondences inducing isomorphisms on the middle cohomologies should exist between these varieties. In the paper, we construct several examples of such correspondences. In the constructions elliptic fibrations play a crucial role. In fact we show that all but three examples are in some sense built upon two modular curves from the Beauville list.

Invariance under finite conformal transformations in Minkowski space and the Wightman axioms imply strong locality (Huygens principle) and rationality of correlation functions, thus providing an extension of the concept of vertex algebra to higher dimensions. Gibbs (finite temperature) expectation values appear as elliptic functions in the conformal time. We survey and further pursue our program of constructing a globally conformal invariant model of a Hermitian scalar field ℒ of scale dimension four in Minkowski space–time which can be interpreted as the Lagrangian density of a gauge field theory.

Mock theta functions were introduced by Ramanujan in 1920 but a proper understanding of mock modularity has emerged only recently with the work of Zwegers in 2002. In these lectures, we describe three manifestations of this apparently exotic mathematics in three important physical contexts of holography, topology and duality where mock modularity has come to play an important role.

Our results in this paper are threefold: First, we establish the modular properties of the graded dimensions of principal subspaces of level one standard modules for , and of principal subspaces of certain higher level standard modules for . Second, we establish the modular properties of families of q-series that appear in identities due to Warnaar and Zudilin, which generalize Macdonald's identities and the Rogers–Ramanujan identities. Third, we formulate a number of conjectures regarding the modularity of series of this type related to A_N-1 root systems.

We present a formula for vector-valued modular forms, expressing the value of the Hilbert-polynomial of the module of holomorphic forms evaluated at specific arguments in terms of traces of representation matrices, restricting the weight distribution of the free generators.

False theta functions are functions that are closely related to classical theta functions and mock theta functions. In this paper, we study their modular properties at all ranks by forming modular completions analogous to modular completions of indefinite theta functions of any signature and thereby develop a structure parallel to the recently developed theory of higher depth mock modular forms. We then demonstrate this theoretical base on a number of examples up to depth three coming from characters of modules for the vertex algebra $W^0{(p)}_{A_n}$, $1\le n\le 3$, and from $Ẑ$-invariants of three-manifolds associated with gauge group SU(3).

For a positive integer m, let R be either the ring ℤ_2m of integers modulo 2m or the quaternionic ring Σ_2m = ℤ_2m + αℤ_2m + βℤ_2m + γℤ_2m with α = 1 + î, β = 1 + ĵ and , where are elements of the ring ℍ of real quaternions satisfying , , and . In this paper, we obtain Jacobi forms (or Siegel modular forms) of genus r from byte weight enumerators (or symmetrized byte weight enumerators) in genus r of Type I and Type II codes over R. Furthermore, we derive a functional equation for partial Epstein zeta functions, which are summands of classical Epstein zeta functions associated with quadratic forms.

In this paper, we study holomorphic $\mathbb{Z}$-graded vertex superalgebras. We prove that all such vertex superalgebras of central charge $8$ and $16$ are purely even. For the case of central charge $24$ we prove that the weight-one Lie superalgebra is either zero, of superdimension $24$, or else is one of an explicit list of 1332 semisimple Lie superalgebras.

By a combined use of analytical, algebraic and computational tools, we derive a description of the algebra of modular forms with respect to a certain congruence subgroup of SL₂(ℤ) × SL₂(ℤ) of level 3.

Let r₃(n) be the number of representations of a positive integer n as a sum of three squares of integers. We give two alternative proofs of a conjecture of Wagon concerning the asymptotic value of the mean square of r₃(n).

It is a classical observation of Serre that the Hecke algebra acts locally nilpotently on the graded ring of modular forms modulo 2 for the full modular group. Here we consider the problem of classifying spaces of modular forms for which this phenomenon continues to hold. We give a number of consequences of this investigation as they relate to quadratic forms, partition functions, and central values of twisted modular L-functions.

We give a short proof of Milne's formulas for sums of 4n² and 4n² + 4n integer squares using the theory of modular forms. Other identities of Milne are also discussed.

Kohnen showed that the zeros of the Eisenstein series E_k in the standard fundamental domain other than i and ρ are transcendental. In this paper, we obtain similar results for a more general class of modular forms, using the earlier works of Kanou, Kohnen and the recent work of Getz.

Let f be a normalized Hecke eigenform of weight k ≥ 4 on Γ₀(N). Let λ_f(n) denote the eigenvalue of the nth Hecke operator acting on f. We show that the number of n such that λ_f(n) takes a given value coprime to 2, is finite. We also treat the case of levels 2^aN₀ with a arbitrary and N₀ = 1, 3, 5, 15 and 17. We discuss the relationship of these results to the classical conjecture of Lang and Trotter.

An elliptic curve defined over the field of rational numbers can be considered as a complex torus. We can describe its complex periods in terms of integration of the weight-2 cusp form corresponding to the elliptic curve. In this paper, we will study an analogous description of the p-adic periods of the elliptic curve, considering the elliptic curve as a p-adic torus. An essential tool for the proof of such a description is the level-lowering theorem of Ribet, which is one of the main ingredients used in the proof of Fermat's Last Theorem.

Let b_ℓ(n) denote the number of ℓ-regular partitions of n, where ℓ is prime and 3 ≤ ℓ ≤ 23. In this paper we prove results on the distribution of b_ℓ(n) modulo m for any odd integer m > 1 with 3 ∤ m if ℓ ≠ 3.

The rank of an overpartition pair is a generalization of Dyson's rank of a partition. The purpose of this paper is to investigate the role that this statistic plays in the congruence properties of , the number of overpartition pairs of n. Some generating functions and identities involving this rank are also presented.

In 2002, Milne [5, 6] obtained ten infinite families of formulas for the sums of integer squares. Recently, Long and Yang [4] reproved four of these identities using modular forms on various subgroups. In this paper, we prove the remaining six, and show that all of the identities can be proved by interpreting them in terms of modular forms for Γ₀(4).