Related papers: Combinatorial Congruences and $\psi$-Operators

Let $q$ be a prime power and $F=\mathbb{F}_q(T)$ be the rational function field over $\mathbb{F}_q$, the field with $q$ elements. Let $\phi$ be a Drinfeld module over $F$ and $\mathfrak{p}$ be a non-zero prime ideal of $A:=\mathbb{F}_q[T]$.…

Let $\Phi_n^{(k)}(x)$ be the $k$-th derivative of $n$-th cyclotomic polynomial. Extending a work of D.~H.~Lehmer, we show some curious congruences: $2\Phi^{(3)}_n(1)$ is divisible by $\phi(n)-2$ and $\Phi^{(2k+1)}_n(1)$ is divisible by…

We consider the problem of defining an action of Hecke operators on the coherent cohomology of certain integral models of Shimura varieties. We formulate a general conjecture describing which Hecke operators should act integrally and solve…

For a real quadratic field $K=\mathbb{Q}(\sqrt{D})$, let $K_{\infty}$ denote the cyclotomic $\mathbb{Z}_{p}$-extension of $K$. Greenberg conjectured that the corresponding Iwasawa module $X_{\infty}$ is finite. Building on the work of…

We prove an Equivariant Main Conjecture in Iwasawa Theory along any rank one, sign-normalized Drinfeld modular, split at infinity Iwasawa tower of a general function field of characteristic p, for the Iwasawa modules recently considered by…

In this paper, we study the complex symmetry of weighted composition-differentiation operator $D_{n, \psi, \phi}$ on weighted Bergman spaces $\mathcal{A}^2_{\alpha}$ with respect to the conjugation $C_{\mu, \eta}$ for $\mu, \eta \in \{z\in…

In this paper, we study $(\varphi,\Gamma)$-modules over rings which are "combinations of discrete algebras and affinoid $\mathbb{Q}_p$-algebras", and prove basic results such as the existence of a fully faithful functor from the category of…

A modular relation of the form $F(\alpha, w)=F(\beta, iw)$, where $i=\sqrt{-1}$ and $\alpha\beta=1$, is obtained. It involves the generalized digamma function $\psi_w(a)$ which was recently studied by the authors in their work on developing…

We prove the exactness of the reduction map from \'etale $(\phi,\Gamma)$-modules over completed localized group rings of compact open subgroups of unipotent $p$-adic algebraic groups to usual \'etale $(\phi,\Gamma)$-modules over Fontaine's…

Generalized digamma functions $\psi_k(x)$, studied by Ramanujan, Deninger, Dilcher, Kanemitsu, Ishibashi etc., appear as the Laurent series coefficients of the zeta function associated to an indefinite quadratic form. In this paper, a…

We prove a "twist-compatibility" result for p-adic families of cohomology classes associated to symmetric spaces. This shows that a single family of classes (lying in a finitely-generated Iwasawa module) interpolates classical cohomology…

We consider finite iterated generalized harmonic sums weighted by the binomial $\binom{2k}{k}$ in numerators and denominators. A large class of these functions emerges in the calculation of massive Feynman diagrams with local operator…

Based on an explicit formula of the generating series for the $n$-point psi-class intersection numbers (cf. Bertola et. al. [4]), we give a novel proof of a conjecture of Delecroix et. al. [9] regarding the large genus uniform leading…

In this paper, we obtain the asymptotic expansions of super intersection numbers and prove that the associated coefficients are polynomials. Moreover, we give an algorithm which can explicitly compute these coefficients. As an application,…

Rowland and Zeilberger devised an approach to algorithmically determine the modulo $p^r$ reductions of values of combinatorial sequences representable as constant terms (building on work of Rowland and Yassawi). The resulting $p$-schemes…

Let $K$ be a finite extension of $\mathbf{Q}_p$ and let $G_K = \mathrm{Gal}(\bar{\mathbf{Q}}_p/K)$. There is a very useful classification of $p$-adic representations of $G_K$ in terms of cyclotomic $(\varphi,\Gamma)$-modules (cyclotomic…

In this article, we study the complex symmetry of compositions operators $C_{\phi}f=f\circ \phi$ induced on weighted Bergman spaces $A^2_{\beta}(\mathbb{D}),\ \beta\geq -1,$ by analytic self-maps of the unit disk. One of ours main results…

In this article, we completely characterize the complex symmetry, cyclicity and hypercyclicity of composition operators $C_\phi f=f\circ\phi$ induced by affine self-maps $\phi$ of the right half-plane $\mathbb{C}_+$ on the Hardy-Hilbert…

For a given prime $p$, we study the properties of the $p$-dissection identities of Ramanujan's theta functions $\psi(q)$ and $f(-q)$, respectively. Then as applications, we find many infinite family of congruences modulo 2 for some…

The essential parts of the operad algebra are concisely presented, which should be useful when confronting with the operadic physics. It is also clarified how the Gerstenhaber algebras can be associated with the linear pre-operads (comp…