scholarly journals THE SHIMURA CURVE OF DISCRIMINANT 15 AND TOPOLOGICAL AUTOMORPHIC FORMS

2015 ◽  
Vol 3 ◽  
Author(s):  
TYLER LAWSON
Keyword(s):  
Automorphic Forms ◽  
Homotopy Groups ◽  
Graded Ring ◽  
Shimura Curve

We find defining equations for the Shimura curve of discriminant 15 over $\mathbb{Z}[1/15]$. We then determine the graded ring of automorphic forms over the 2-adic integers, as well as the higher cohomology. We apply this to calculate the homotopy groups of a spectrum of ‘topological automorphic forms’ associated to this curve, as well as one associated to a quotient by an Atkin–Lehner involution.

scholarly journals Schwarzian differential equations and Hecke eigenforms on Shimura curves

2012 ◽  
Vol 149 (1) ◽  
pp. 1-31 ◽  
Author(s):  
Yifan Yang
Keyword(s):  
Differential Equations ◽  
Automorphic Forms ◽  
Hecke Operators ◽  
Shimura Curves ◽  
Genus Zero ◽  
Shimura Curve ◽  
Hecke Eigenforms ◽  
Image Position

AbstractLet X be a Shimura curve of genus zero. In this paper, we first characterize the spaces of automorphic forms on X in terms of Schwarzian differential equations. We then devise a method to compute Hecke operators on these spaces. An interesting by-product of our analysis is the evaluation and other similar identities.

Algebraic Invariants of Simple 4-Knots

1997 ◽  
Vol 06 (03) ◽  
pp. 307-318
Author(s):  
J. A. Hillman ◽  
C. Kearton
Keyword(s):  
Necessary Conditions ◽  
Universal Covering ◽  
Homotopy Groups ◽  
Graded Ring ◽  
Whitehead Product ◽  
Algebraic Invariant ◽  
Universal Covering Space ◽  
Algebraic Invariants ◽  
Hopf Map ◽  
Module Structures

We propose as an algebraic invariant for a simple 4-knot K with exterior X the triple (L, η, [λ]), where L = Z ⊕ π2(X)⊕π3(X) is a commutative graded ring with unit whose multiplication in positive degrees is determined by Whitehead product, η is composition with the Hopf map and [λ] is the orbit of the homotopy class of the longitude in π4(X) under the group of self homotopy equivalences of the universal covering space X′ which induce the identity on L. If K is fibred these invariants determine the fibre, and the natural Z[t,t-1]-module structures on the homotopy groups capture part of the monodromy. Every such triple with L finitely generated as an abelian group (and satisfying the order obviously necessary conditions) may be realized by some fibred simple 4-knot. In certain cases we can show that the triple determines the knot up to a finite ambiguity.

scholarly journals Computing fundamental domains for the Bruhat–Tits tree for , -adic automorphic forms, and the canonical embedding of Shimura curves

2014 ◽  
Vol 17 (1) ◽  
pp. 1-23 ◽  
Author(s):  
Cameron Franc ◽  
Marc Masdeu
Keyword(s):  
High Precision ◽  
Modular Forms ◽  
Automorphic Forms ◽  
Shimura Curves ◽  
Discrete Groups ◽  
Canonical Embedding ◽  
Quaternion Algebras ◽  
Shimura Curve ◽  
Fundamental Domains

AbstractWe describe algorithms that allow the computation of fundamental domains in the Bruhat–Tits tree for the action of discrete groups arising from quaternion algebras. These algorithms are used to compute spaces of rigid modular forms of arbitrary even weight, and we explain how to evaluate such forms to high precision using overconvergent methods. Finally, these algorithms are applied to the calculation of conjectural equations for the canonical embedding of p-adically uniformizable rational Shimura curves. We conclude with an example in the case of a genus 4 Shimura curve.

scholarly journals Whitehead products in the complex Stiefel manifolds

1983 ◽  
Vol 26 (2) ◽  
pp. 241-251 ◽  
Author(s):  
Yasukuni Furukawa
Keyword(s):  
Stiefel Manifold ◽  
Isotropy Group ◽  
Homotopy Groups ◽  
Stiefel Manifolds

The complex Stiefel manifoldWn,k, wheren≦k≦1, is a space whose points arek-frames inCn. By using the formula of McCarty [4], we will make the calculations of the Whitehead products in the groups π*(Wn,k). The case of real and quaternionic will be treated by Nomura and Furukawa [7]. The product [[η],j1l] appears as generator of the isotropy group of the identity map of Stiefel manifolds. In this note we use freely the results of the 2-components of the homotopy groups of real and complex Stiefel manifolds such as Paechter [8], Hoo-Mahowald [1], Nomura [5], Sigrist [9] and Nomura-Furukawa [6].

scholarly journals On the Bloch–Kato conjecture for Hilbert modular forms

2021 ◽  
Author(s):  
Matteo Tamiozzo
Keyword(s):  
Modular Forms ◽  
Automorphic Forms ◽  
Euler System ◽  
Central Point ◽  
Shimura Curves ◽  
Hilbert Modular Forms ◽  
Reciprocity Laws ◽  
Hilbert Modular ◽  
Special Points

AbstractThe aim of this paper is to prove inequalities towards instances of the Bloch–Kato conjecture for Hilbert modular forms of parallel weight two, when the order of vanishing of the L-function at the central point is zero or one. We achieve this implementing an inductive Euler system argument which relies on explicit reciprocity laws for cohomology classes constructed using congruences of automorphic forms and special points on several Shimura curves.

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