Related papers: Nahm's Conjecture: Asymptotic Computations and Cou…

Our paper originated from a generalization of the Volume Conjecture to multisums of $q$-hypergeometric terms. This generalization was sketched by Kontsevich in a problem list in Aarhus University in 2006; \cite{Ko}. We introduce the notion…

This article surveys some recent work of the author on Hilbert modular fourfolds X. After some preliminaries on the cohomology and special, codimension 2 cycles Z on X of Hirzebruch-Zagier type, a proof of the Tate conjecture for X over…

Let $(R,\mathfrak{m},\mathbb{k})$ be an equicharacteristic one-dimensional complete local domain over an algebraically closed field $\mathbb{k}$ of characteristic 0. R. Berger conjectured that R is regular if and only if the universally…

The Quantum Modularity Conjecture of Zagier predicts the existence of a formal power series with arithmetically interesting coefficients that appears in the asymptotics of the Kashaev invariant at each root of unity. Our goal is to…

For quantum groups at a root of unity, there is a web of theorems (due to Bezrukavnikov and Ostrik, and relying on work of Lusztig) connecting the following topics: (i) tilting modules; (ii) vector bundles on nilpotent orbits; and (iii)…

In this article we prove a version of Kolyvagin's conjecture for modular forms at non-ordinary primes. In particular, we generalize the work of Wang on a converse to a higher weight Gross-Zagier-Kolyvagin theorem in order to prove the…

In this note, we report on the results of a computer search performed to find possible modular companions to certain $q$-series identities and conjectures. For the search, we use conditions arising from the asymptotics of Nahm sums. We…

Recently, Mizuno studied generalized Nahm sums associated with symmetrizable matrices. He provided 14 sets of candidates of modular Nahm sums in rank two and justified four of them. We prove the modularity for eight other sets of candidates…

Let $f$ be a newform of weight $k\geq 2$, level $N$ with coefficients in a number field $K$, and $A$ the adjoint motive of the motive $M$ associated to $f$. We carefully discuss the construction of the realisations of $M$ and $A$, as well…

It is conjectured that for fixed $A$, $r \ge 1$, and $d \ge 1$, there is a uniform bound on the size of the torsion submodule of a Drinfeld $A$-module of rank $r$ over a degree $d$ extension $L$ of the fraction field $K$ of $A$. We verify…

In [7], G. Navarro proposed a refinement of the McKay conjecture involving a special class of Galois automorphisms. In [6] this new conjecture was verified by the author for the alternating groups A(n) when p=2. In this note the Navarro…

For the BGG category of $\mathfrak{q}(n)$-modules of half-integer weights, a Kazhdan-Lusztig conjecture \`a la Brundan is formulated in terms of categorical canonical basis of the $n$th tensor power of the natural representation of the…

We prove the following conjecture of Furstenberg (1969): if $A,B\subset [0,1]$ are closed and invariant under $\times p \mod 1$ and $\times q \mod 1$, respectively, and if $\log p/\log q\notin \mathbb{Q}$, then for all real numbers $u$ and…

Let R be a commutative Noetherian domain, and let M and N be finitely generated R-modules. We give new criteria for determining when M tensor N has torsion. We also give constructive formulas for producing a module in the isomorphism class…

Let $M$ be a finitely generated module over a free twisted commutative algebra $A$ that is finitely generated in degree one. We show that the projective dimension of $M({\bf C}^n)$ as an $A({\bf C}^n)$-module is eventually linear as a…

Let N be a square-free positive integer and let f be a newform of weight 2 on \Gamma_0(N). Let A denote the abelian subvariety of J_0(N) associated to f and let m be a maximal ideal of the Hecke algebra T that contains Ann_T(f) and has…

Algebraic Nahm equations, considered in the paper, are polynomial equations, governing the $q\rightarrow 1$ limit of the $q$-hypergeometric Nahm sums. They make an appearance in various fields: hyperbolic geometry, knot theory, quiver…

Let $G$ be a simply-connected semisimple algebraic group over an algebraically closed field of characteristic $p$, assumed to be larger than the Coxeter number. The "support variety" of a $G$-module $M$ is a certain closed subvariety of the…

Let $k$ be an algebraically closed field and $A$ the polynomial algebra in $r$ variables with coefficients in $k$. In case the characteristic of $k$ is $2$, Carlsson conjectured that for any $DG$-$A$-module $M$ of dimension $N$ as a free…

The main result of this paper is an instance of the conjecture made by Gouvea and Mazur (Math. Res. Lett., 1995) which asserts that for certain values of r the space of r-overconvergent p-adic modular forms of tame level N and weight k…