Related papers: On Drinfeld modular forms of higher rank VI: The s…

\begin{document} \begin This paper continues work of the earlier articles with the same title. For two classes of modular forms $f$: \begin{itemize} \item para-Eisenstein series $\alpha_{k}$ and \item coefficient forms ${}_a \ell_{k}$,…

We study expansions of Drinfeld modular forms of rank \(r \geq 2\) along the boundary of moduli varieties. Product formulas for the discriminant forms \(\Delta_{\mathfrak{n}}\) are developed, which are analogous with Jacobi's formula for…

This is the third part of a series of articles providing a foundation for the theory of Drinfeld modular forms of arbitrary rank. In the present article we construct and study some examples of Drinfeld modular forms. In particular we define…

Continuing the work of \cite{7} and \cite{8}, we derive an analogue of the classical "$k/12$-formula" for Drinfeld modular forms of rank $r \geq 2$. Here the vanishing order $\nu_{\omega}(f)$ of one modular form at some point $\omega$ of…

We show that the absolute value $|f|$ of an invertible holomorphic function $f$ on the Drinfeld symmetric space $\OM^r$ $(r \geq 2)$ is constant on fibers of the building map to the Bruhat-Tits building $\MB\MT$. Its logarithm $\log|f|$ is…

We give closed formulas for the first few expansion coefficients of the basic modular forms for \(\mathrm{GL}(r, \mathbb{F}_{q}[T])\). Here the rank \(r\) is larger or equal to \(3\), and the forms in question include the coefficient forms…

We give an explicit formula for the correspondence between simple Yetter-Drinfeld modules for certain finite-dimensional pointed Hopf algebras $H$ and those for cocycle twists $H^{\sigma}$ of $H$. This implies an equivalence between modules…

For a simplicial complex X on {1,2, ..., n} we define enriched homology and cohomology modules. They are graded modules over k[x_1, ..., x_n] whose ranks are equal to the dimensions of the reduced homology and cohomology groups. We…

Let $\mathbb{F}_q$ be the field of $q$ elements and let $A=\mathbb{F}_q[t]$ be the polynomial ring over $\mathbb{F}_q$. Let $\mathfrak{n}\in A\setminus \mathbb{F}_q$ be a monic polynomial with a prime factor of degree prime to $q-1$. Let…

We prove the existence of a Quillen Flat Model Structure in the category of unbounded complexes of h-unitary modules over a nonunital ring (or a $k$-algebra, with $k$ a field). This model structure provides a natural framework where a…

We construct and study a natural compactification $\overline{M}^r(N)$ of the moduli scheme $M^r(N)$ for rank-$r$ Drinfeld $\F_q[T]$-modules with a structure of level $N \in \F_q[T]$. Namely, $\overline{M}^r(N) = {\rm Proj}\,{\bf Eis}(N)$,…

The aim of this article is to study the derivative of "incoherent" Siegel-Eisenstein series on symplectic groups over function fields. By the Siegel-Weil formula for "coherent" Siegel-Eisenstein series, we can relate the non-singular…

For each positive integer $r$, we construct a nowhere-vanishing, single-cuspidal Drinfeld modular form for $\GL_r(\FF_q[\theta])$, necessarily of least possible weight, via determinants using rigid analytic trivializations of the universal…

We work with detail the Drinfeld module over the ring $$A=F_2[x,y]/(y^2+y=x^3+x+1).$$ The example in question is one of the four examples that come from quadratic imaginary fields with class number $h = 1$ and rank one. We develop specific…

We show that complex semisimple quantum groups, that is, Drinfeld doubles of $ q $-deformations of compact semisimple Lie groups, satisfy a categorical version of the Baum-Connes conjecture with trivial coefficients. This approach, based on…

Automorphic forms on a bounded symmetric domain D=G/K can be viewed as holomorphic sections of $L^{\otimes k}$, where L is a quantizing line bundle on a compact quotient of D and k is a positive integer. Let $\Gamma$ be a cocompact discrete…

We compute the Euler-Poincar\'e characteristic of quotients of the Bruhat-Tits building of PGL(n) under the action of arithmetic groups arising from central division algebras over rational function fields of positive characteristic. We use…

Assume that the link of a complex normal surface singularity is a rational homology sphere. Then its Seiberg-Witten invariant can be computed as the `periodic constant' of the topological multivariable Poincar\'e series (zeta function).…

Let $G$ be a higher-rank semisimple Lie group over a nonarchimedean local field, for example $G={\rm PGL}(n,Q_P)$. To any lattice $L$ in $G$ there is an associated simplicial complex $B_L$, given by the quotient by $L$ of the Bruhat-Tits…

We consider free and proper cotangent-lifted symmetries of Hamiltonian systems. For the special case of G = SO(3), we construct symplectic slice coordinates around an arbitrary point. We thus obtain a parametrisation of the phase space…