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Register a new userQuantum modular forms and singular combinatorial series with repeated roots of unity
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In 2007, G.E. Andrews introduced the $(n+1)$-variable combinatorial generating function $R_n(x_1,x_2,cdots,x_n;q)$ for ranks of $n$-marked Durfee symbols, an $(n+1)$-dimensional multisum, as a vast generalization to the ordinary two-variable partition rank generating function. Since then, it has been a problem of interest to understand the automorphic properties of this function; in special cases and under suitable specializations of parameters, $R_n$ has been shown to possess modular, quasimodular, and mock modular properties when viewed as a function on the upper half complex plane $mathbb H$, in work of Bringmann, Folsom, Garvan, Kimport, Mahlburg, and Ono. Quantum modular forms, defined by Zagier in 2010, are similar to modular or mock modular forms but are defined on the rationals $mathbb Q$ as opposed to $mathbb H$, and exhibit modular transformations there up to suitably analytic error functions in $mathbb R$; in general, they have been related to diverse areas including number theory, topology, and representation theory. Here, we establish quantum modular properties of $R_n$.
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Understanding the relationship between mock modular forms and quantum modular forms is a problem of current interest. Both mock and quantum modular forms exhibit modular-like transformation properties under suitable subgroups of $rm{SL}_2(mathbb Z)$, up to nontrivial error terms; however, their domains (the upper half-plane $mathbb H$, and the rationals $mathbb Q$, respectively) are notably different. Quantum modular forms, originally defined by Zagier in 2010, have also been shown to be related to the diverse areas of colored Jones polynomials, meromorphic Jacobi forms, partial theta functions, vertex algebras, and more. In this paper we study the $(n+1)$-variable combinatorial rank generating function $R_n(x_1,x_2,dots,x_n;q)$ for $n$-marked Durfee symbols. These are $n+1$ dimensional multisums for $n>1$, and specialize to the ordinary two-variable partition rank generating function when $n=1$. The mock modular properties of $R_n$ when viewed as a function of $tauinmathbb H$, with $q=e^{2pi i tau}$, for various $n$ and fixed parameters $x_1, x_2, cdots, x_n$, have been studied in a series of papers. Namely, by Bringmann and Ono when $n=1$ and $x_1$ a root of unity; by Bringmann when $n=2$ and $x_1=x_2=1$; by Bringmann, Garvan, and Mahlburg for $ngeq 2$ and $x_1=x_2=dots=x_n=1$; and by the first and third authors for $ngeq 2$ and the $x_j$ suitable roots of unity ($1leq j leq n$). The quantum modular properties of $R_1$ readily follow from existing results. Here, we focus our attention on the case $ngeq 2$, and prove for any $ngeq 2$ that the combinatorial generating function $R_n$ is a quantum modular form when viewed as a function of $x in mathbb Q$, where $q=e^{2pi i x}$, and the $x_j$ are suitable distinct roots of unity.
We identify a class of semi-modular forms invariant on special subgroups of $GL_2(mathbb Z)$, which includes classical modular forms together with complementary classes of functions that are also nice in a specific sense. We define an Eisenstein-like series summed over integer partitions, and use it to construct families of semi-modular forms.
Generalizing a result of cite{Z1991, CPZ} about elliptic modular forms, we give a closed formula for the sum of all Hilbert Hecke eigenforms over a totally real number field with strict class number $1$, multiplied by their period polynomials, as a single product of the Kronecker series.
In recent work, M. Just and the second author defined a class of semi-modular forms on $mathbb C$, in analogy with classical modular forms, that are half modular in a particular sense; and constructed families of such functions as Eisenstein-like series using symmetries related to integer partitions. Looking for further natural examples of semi-modular behavior, here we construct a family of Eisenstein-like series to produce semi-modular forms, using symmetries related to Fibonacci numbers instead of partitions. We then consider other Lucas sequences that yield semi-modular forms.
Motivated by questions in number theory, Myerson asked how small the sum of 5 complex nth roots of unity can be. We obtain a uniform bound of O(n^{-4/3}) by perturbing the vertices of a regular pentagon, improving to O(n^{-7/3}) infinitely often. The corresponding configurations were suggested by examining exact minimum values computed for n <= 221000. These minima can be explained at least in part by selection of the best example from multiple families of competing configurations related to close rational approximations.
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