We establish a one-to-one correspondence between conjugacy classes of any Hecke group and irreducible systems of poles of rational period functions for automorphic integrals on the same group. We use this correspondence to construct irreducible systems of poles and to count poles. We characterize Hecke-conjugation and Hecke-symmetry for poles of rational period functions in terms of the transpose of matrices in conjugacy classes. We construct new rational period functions and families of rational period functions.
In this paper we continue work in the direction of a characterization of rational period functions on the Hecke groups. We examine the role that Hecke-symmetry of poles plays in this setting, and pay particular attention to non-symmetric irreducible systems of poles for a rational period function. This gives us a new expression for a class of rational period functions of any positive even integer weight on the Hecke groups. We illustrate these properties with examples of specific rational period functions. We also correct the wording of a theorem from an earlier paper.
This survey article, is written as an extended note and supplement of my lectures in the current developments in mathematics conference in 2015. We discuss some recent developments on the conjugacy classes of affine Weyl groups and $p$-adic groups, and some applications to Shimura varieties and to representations of affine Hecke algebras.
We describe a novel method for bounding the dimension $d$ of the largest simple Hecke submodule of $S_2(Γ_0(N);\mathbb{Q})$ from below. Such bounds are of interest because of their relevance to the structure of $J_0(N)$, for instance. In contrast with previous results of this kind, our bound does not rely on the equidistribution of Hecke eigenvalues. Instead, it is obtained via a Hecke-compatible congruence between the target space and a space of modular forms whose Hecke eigenvalues are easily controlled. For prime levels $N\equiv 7\mod 8$ our method yields an unconditional bound of $d\ge\log…
We give a concrete characterization of the rational conjugacy classes of maximal tori in groups of type $D_n$, with specific emphasis on the case of number fields and p-adic fields. This includes the forms associated to quadratic spaces, all of their inner and outer forms as well as the Spin groups, their simply connected covers. In particular, in this work we handle all (simply connected) outer forms of $D_4$.
It is well-known for an elliptic curve with complex multiplication that the existence of a $\mathbb{Q}$-rational model is equivalent to its field of moduli being equal to $\mathbb{Q}$, or its endomorphism ring being the ring of integers of 9 possible fields ($\ast$). Murabayashi and Umegaki proved analogous results for abelian surfaces. For three dimensional CM abelian varieties with rational fields of moduli, Chun narrowed down to a list of 37 possible CM fields. In this paper, we show that his list is exact. By constructing certain Hecke characters that satisfy a theorem of Shimura, we prove…
We study polynomials interpolating the (rational) constant terms of certain meromorphic modular forms for Hecke groups. We make observations about the divisibility properties of the constant terms and connect them to several sequences, for example, to O.E.I.S. sequence A005148 \cite{OEISNewmanShanks}, which was studied by Newman, Shanks and Zagier \cite{newman2004sequence}, \cite{newman2004sequenceAppendix} in an article on its use in series approximations to $π$.
Hecke operators relate characters of rational conformal field theories (RCFTs) with different central charges, and extend the previously studied Galois symmetry of modular representations and fusion algebras. We show that the conductor $N$ of a RCFT and the quadratic residues mod $N$ play an important role in the computation and classification of Galois permutations. We establish a field correspondence in different theories through the picture of effective central charge, which combines Galois inner automorphisms and the structure of simple currents. We then make a first attempt to extend Heck…
We define Hecke operators on vector-valued modular forms of the type that appear as characters of rational conformal field theories (RCFTs). These operators extend the previously studied Galois symmetry of the modular representation and fusion algebra of RCFTs to a relation between RCFT characters. We apply our results to derive a number of relations between characters of known RCFTs with different central charges and also explore the relation between Hecke operators and RCFT characters as solutions to modular linear differential equations. We show that Hecke operators can be used to construct…
We study in this paper Hida's p-adic Hecke algebra for GL_n over a CM field F. Hida has made a conjecture about the dimension of these Hecke algebras, which he calls the non-abelian Leopoldt conjecture, and shown that his conjecture in the case of F being the rationals implies the classical Leopoldt conjecture for a number field K of degree n over the rationals, if one assumes further the existence of automorphic induction of characters for the extension K over the rationals. We study Hida's conjecture using the automorphy lifting techniques adapted to the GL_n setting by Calegari--Geraghty. W…
Grothendieck's dessins d'enfants arise with ever-increasing frequency in many areas of 21st century mathematical physics. In this paper, we review the connections between dessins and the theory of Hecke groups. Focussing on the restricted class of highly symmetric dessins corresponding to the so-called Archimedean solids, we apply this theory in order to provide a means of computing representatives of the associated conjugacy classes of Hecke subgroups in each case. The aim of this paper is to demonstrate that dessins arising in mathematical physics can point to new and hitherto unexpected dir…
We study the rational Bianchi newforms (weight 2, trivial character, with rational Hecke eigenvalues) in the LMFDB that are not associated to elliptic curves, but instead to abelian surfaces with quaternionic multiplication. Two of these examples exhibit a rather special kind of behaviour: we show they arise from twisted base change of a classical newform with nebentypus character of order 4 and eight inner twists.
Let $K$ be a non-discrete non-Archimedean local field with residue characteristic $p$. Let $G$ be the group of $K$ rational points of a algebraic connected reductive group defined over $K$. In this article we compute the extensions between characters of affine pro-$p$ Iwahori--Hecke algebra $\mathcal{H}^{\text{aff}}$ over an algebraically closed field $R$ of characteristic $p$. In rank one case we deduce the relation between the blocks and $L$-packets.
Let $K$ be an algebraically closed field of characteristic $0$. In this paper we classify the $\text{PGL}_3(K)$-conjugacy classes of semi-stable dominant degree $2$ rational maps $f:{\mathbb P}^2_K\dashrightarrow{\mathbb P}^2_K$ whose automorphism group $$\text{Aut}(f):=\{ϕ\in\text{PGL}_3(K): ϕ^{-1}\circ f\circϕ=f\}$$ is finite and of order at least $3$. In particular, we prove that $\#\text{Aut}(f)\le24$ in general, that $\#\text{Aut}(f)\le21$ for morphisms, and that $\#\text{Aut}(f)\le6$ for all but finitely many conjugacy classes of $f$.
Rationality problem of three-dimensional monomial group actions
设$K$是特征不为二的域，$K(x,y,z)$是$K$上的有理函数域，其中$x,y,z$为三元。设$G$是一个有限群，它通过单项$K$ -自同构作用在$K(x,y,z)$上.我们考虑固定域$K(x,y,z)^G$在$G$作用下的合理性问题，即$K(x,y,z)^G$在$K$上是否是合理的（即纯超越的）.我们可以假设$G$是$\mathrm{GL}(3,\mathbb{Z}) and the problem is determined up to conjugacy in $ \mathrm{GL}（3，\mathbb{Z}）$. There are 73 conjugacy classes of $ G $ in $ \mathrm{GL}（3，\mathbb{Z}）$. By results of Endo-Miyata, Voskresenski\uı, Lenstra, Saltman, Hajja, Kang and Yamasaki, 8 conjugacy classes of 2-groups in $ \mathrm{GL}（3，\mathbb{Z}）$ have negative answers to the problem under certain monomial actions over some base field $ K $, and the nec…
In this article, we construct the Hitchin fibration for groups following the scheme outlined by Frenkel-Ngo in the case of SL_{2}. This construction uses as a decisive tool the Vinberg's semigroup. The total space of Hitchin is obtained by taking the fiber product of the Hecke stack with the diagonal of the stack of G-bundles $Bun_{G}$; we prove a transversality statement between the intersection complex of the Hecke stack and the diagonal of $Bun_{G}$, over a sufficiently big open subset, in order to get local applications, such that the fundamental lemma for the spherical Hecke algebra. Alon…
We give a survey of Denef's rationality theorem on $p$-adic integrals, its uniform in $p$ versions, the relevant model theory, and a number of applications to counting subgroups of finitely generated nilpotent groups and conjugacy classes in congruence quotients of Chevalley groups over rings of integers of local fields. We then state results on analytic properties of Euler products of such $p$-adic integrals over all $p$, and an application to counting conjugacy classes in congruence quotients of certain algebraic groups over the rationals. We then briefly discuss zeta functions arising from …
We show that there is a dual description of conformal blocks of $d=2$ rational CFT in terms of Hecke eigenfields and eigensheaves. In particular, partition functions, conformal characters and lattice theta functions may be reconstructed from the action of Hecke operators. This method can be applied to: 1) rings of integers of Galois number fields equipped with the trace (or anti-trace) form; 2) root lattices of affine Kac-Moody algebras and WZW-models; 3) minimal models of Belavin-Polyakov-Zamolodchikov and related $d=2$ spin-chain/lattice models; 4) vertex algebras of Leech and Niemeier latti…
