Related papers: A Whipple $_7F_6$ formula revisited

We investigate the system of holomorphic differential identities implied by special K\"ahlerian geometry of four-dimensional N=2 supergravity. For superstring compactifications on \cy threefolds these identities are equivalent to the…

Greither and Pareigis have established a connection between Hopf Galois structures on a Galois extension $L/K$ with Galois group $G$, and the regular subgroups of the group of permutations on $G$, which are normalized by $G$. Byott has…

Let $p$ be a prime number, $n$ an integer $\geq 2$ and $\rho$ an $n$-dimensional automorphic $p$-adic Galois representation (for a compact unitary group) such that $r:=\rho\vert_{\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)}$ is…

Let $p\geq 5$ be a prime number, $\mathbb{F}$ a finite field of characteristic $p$ and let $\bar{\chi}$ be the mod-$p$ cyclotomic character. Let $\bar{\rho}:\operatorname{G}_{\mathbb{Q}}\rightarrow \operatorname{GL}_2(\mathbb{F})$ be a…

Building on the developments of many people including Evans, Greene, Katz, McCarthy, Ono, Roberts, and Rodriguez-Villegas, we consider period functions for hypergeometric type algebraic varieties over finite fields and consequently study…

Comodule algebras of a Hopf algebroid H with a bijective antipode, i.e. algebra extensions B\subseteq A by H, are studied. Assuming that a lifted canonical map is a split epimorphism of modules of the non-commutative base algebra of H,…

Let $H$ be a finite dimensional Hopf algebra, and let $A$ be a left $H$-module algebra. Motivated by the study of the isolated singularities of $A^H$ and the endomorphism ring $\mathrm{End}_{A^H}(A)$, we introduce the concept of Hopf dense…

In this note, we firstly establish an extended Gauss's summation identity. Using this identity, we compute values of a family of $_4F_3$ hypergeometric functions, which generalize the results obtained by Ferretti et al..

We construct plane models of the modular curve $X_H(\ell)$, and use their explicit equations to compute Galois representations associated to modular forms for values of $\ell$ that are significantly higher than in prior works.

We show that the Galois cohomology groups of $p$-adic representations of a direct power of $\operatorname{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ can be computed via the generalization of Herr's complex to multivariable…

Recently, there emerges different versions of beta function and hypergeometric functions containing extra parameters. Gaining enlightenment from these ideas, we will first introduce a new extension of generalized hypergeometric function and…

A connection between the Galois-theoretic approach to semi-abelian homology and the homological closure operators is established. In particular, a generalised Hopf formula for homology is obtained, allowing the choice of a new kind of…

As a natural generalization of the notion of `higher rank Euler system', we develop a theory of `higher special elements' in the exterior power biduals of the Galois cohomology of $p$-adic representations. We show, in particular, that such…

We extend the lifting methods of our previous paper to lift reducible odd representations $\bar{\rho}:\mathrm{Gal}(\overline{F}/F) \to G(k)$ of Galois groups of global fields $F$ valued in Chevalley groups $G(k)$. Lifting results, when…

In his work on modularity theorems, Wiles proved a numerical criterion for a map of rings $R\to T$ to be an isomorphism of complete intersections. He used this to show that certain deformation rings and Hecke algebras associated to a mod…

We call a real algebraic hypersurface in $(\mathbb{C}^*)^n$ simplicial if it is given by a real Laurent polynomial in $n$-variables that has exactly $n+1$ monomials with non-zero coefficients and such that the convex hull in $\mathbb{R}^n$…

Let p be a prime and F a totally real field in which p is unramified. We consider mod p Hilbert modular forms for F, defined as sections of automorphic line bundles on Hilbert modular varieties of level prime to p in characteristic p. For a…

The global deformation theory of residually reducible Galois representations with fixed auxiliary conditions is studied. We show that $\bar{\rho}:\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow…

The arithmetic of Hilbert modular forms has been extensively studied under the assumption that the forms concerned are "paritious" -- all the components of the weight are congruent modulo 2. In contrast, non-paritious Hilbert modular forms…

We prove hypergeometric type identities for a function defined in terms of quotients of the $p$-adic gamma function. We use these identities to prove a supercongruence conjecture of Rodriguez-Villegas between a truncated $_4F_3$…