A note on trilinear forms for reducible representations and Beilinson's conjectures

Michael Harris; Anthony Scholl

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Classical and overconvergent modular forms of higher level

Robert F. Coleman (1997)

Journal de théorie des nombres de Bordeaux

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We define the notion overconvergent modular forms on $Γ_{1} (N p^{n})$ where $p$ is a prime, $N$ and $n$ are positive integers and $N$ is prime to $p$ . We show that an overconvergent eigenform on $Γ_{1} (N p^{n})$ of weight $k$ whose $U_{p}$ -eigenvalue has valuation strictly less than $k - 1$ is classical.

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Representations of a module over a *-algebra $_{}$ are considered and some related seminorms are constructed and studied, with the aim of finding bounded *-representations of $_{}$ .

Overconvergent modular forms

Vincent Pilloni (2013)

Annales de l’institut Fourier

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We give a geometric definition of overconvergent modular forms of any $p$ -adic weight. As an application, we reprove Coleman’s theory of $p$ -adic families of modular forms and reconstruct the eigencurve of Coleman and Mazur without using the Eisenstein family.

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Jordi Quer (2010)

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Robert Harron, Liang Xiao (2014)

Annales de l’institut Fourier

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We interpolate the Gauss–Manin connection in $p$ -adic families of nearly overconvergent modular forms. This gives a family of Maass–Shimura type differential operators from the space of nearly overconvergent modular forms of type $r$ to the space of nearly overconvergent modular forms of type $r + 1$ with $p$ -adic weight shifted by $2$ . Our construction is purely geometric, using Andreatta–Iovita–Stevens and Pilloni’s geometric construction of eigencurves, and should thus generalize to higher rank...

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We show that it is possible in rather general situations to obtain a finite-dimensional modular representation $ρ$ of the Galois group of a number field $F$ as a constituent of one of the modular Galois representations attached to automorphic representations of a general linear group over $F$ , provided one works “potentially.” The proof is based on a close study of the monodromy of the Dwork family of Calabi–Yau hypersurfaces; this in turn makes use of properties of rigid local systems and...

On the number of representations of a positive integer by certain quadratic forms

Ernest X. W. Xia (2014)

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For natural numbers a,b and positive integer n, let R(a,b;n) denote the number of representations of n in the form $\sum_{i = 1}^{a} (x ²_{i} + x_{i} y_{i} + y ²_{i}) + 2 \sum_{j = 1}^{b} (u ²_{j} + u_{j} v_{j} + v ²_{j})$ . Lomadze discovered a formula for R(6,0;n). Explicit formulas for R(1,5;n), R(2,4;n), R(3,3;n), R(4,2;n) and R(5,1;n) are determined in this paper by using the (p;k)-parametrization of theta functions due to Alaca, Alaca and Williams.

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This note is based on a short talk presented at the “42nd Winter School Geometry and Physics” held in Srni, Czech Republic, January 15th–22nd 2022. We review the notion of Lie superalgebra cohomology and extend it to different form complexes, typical of the superalgebraic setting. In particular, we introduce pseudoforms as infinite-dimensional modules related to sub-superalgebras. We then show how to extend the Koszul-Hochschild-Serre spectral sequence for pseudoforms as a computational...

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In this note, we study formal deformations of derived representations of the principal series representations of $SL (2, ℝ)$ . In particular, we recover all the representations of the derived principal series by deforming one of them. Similar results are also obtained for $SL (2, ℂ)$ .

Local Indecomposability of Hilbert Modular Galois Representations

Bin Zhao (2014)

Annales de l’institut Fourier

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We prove the indecomposability of the Galois representation restricted to the $p$ -decomposition group attached to a non CM nearly $p$ -ordinary weight two Hilbert modular form over a totally real field $F$ under the assumption that either the degree of $F$ over $ℚ$ is odd or the automorphic representation attached to the Hilbert modular form is square integrable at some finite place of $F$ .

Stacks of group representations

Paul Balmer (2015)

Journal of the European Mathematical Society

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We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extension-of-scalars. We deduce that, given a group $G$ , the derived and the stable categories of representations of a subgroup $H$ can be constructed out of the corresponding category for $G$ by a purely triangulated-categorical construction, analogous to étale extension in algebraic geometry. In the case of finite groups, we then use descent methods...

Quadratic modular symbols on Shimura curves

Pilar Bayer, Iván Blanco-Chacón (2013)

Journal de Théorie des Nombres de Bordeaux

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We introduce the concept of modular symbol and study how these symbols are related to $p$ -adic $L$ -functions. These objects were introduced in [] in the case of modular curves. In this paper, we discuss a method to attach quadratic modular symbols and quadratic $p$ -adic $L$ -functions to more general Shimura curves.

Algebras of the cohomology operations in some cohomology theories

A. Jankowski

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Contents0. Introduction............................................................................................................................................. 51. Preliminaries.......................................................................................................................................... 62. Generalized cohomology theories with a coefficient group $Z_{p}$ .............................................. 83. Cohomology theory BP* ( , $Z_{p}$ )........................................................................................................

The expansion of $\prod_{k = 1}^{\infty} (1 - q^{a k}) (1 - q^{b k})$

Zhi-Hong Sun (2008)

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