Summary
The Bianchi eigenvariety is studied around non-cuspidal points, specifically ordinary non-cuspidal base change points. Multiplicity-one results are proven for Bianchi Eisenstein eigensystems, and the geometry of the Bianchi eigenvariety is analyzed.
Highlights
- Bianchi eigenvariety is studied around non-cuspidal points.
- Multiplicity-one results are proven for Bianchi Eisenstein eigensystems.
- Geometry of the Bianchi eigenvariety is analyzed.
- Ordinary non-cuspidal base change points are considered.
- p-stabilizations of Bianchi Eisenstein eigensystems are introduced.
- p-adic L-functions of Bianchi Eisenstein eigensystems are constructed.
- Coleman-Mazur eigencurve is related to the Bianchi eigenvariety.
Key Insights
- The Bianchi eigenvariety is a rigid analytic space that parametrizes p-adic automorphic representations. Studying its geometry around non-cuspidal points is crucial for understanding arithmetic applications.
- The multiplicity-one results for Bianchi Eisenstein eigensystems imply that the cohomology groups are one-dimensional, which is essential for constructing p-adic L-functions.
- The ordinary non-cuspidal base change points are related to the base change of cuspidal automorphic representations of GL(2,Q) to GL(2,K), where K is an imaginary quadratic field.
- The p-stabilizations of Bianchi Eisenstein eigensystems are used to construct p-adic L-functions, which are expected to satisfy an interpolation formula related to special values of Hecke L-functions.
- The geometry of the Bianchi eigenvariety around non-cuspidal points is related to the Coleman-Mazur eigencurve, which is a curve that parametrizes p-adic modular forms.
- The study of the Bianchi eigenvariety has implications for the arithmetic of elliptic curves and the distribution of prime numbers.
- The construction of p-adic L-functions of Bianchi Eisenstein eigensystems is a crucial step towards understanding the arithmetic of automorphic forms.
Mindmap
Citation
Salazar, D. B., & Palacios, L. S. (2024). Geometry of the Bianchi eigenvariety around non-cuspidal points and strong multiplicity-one results (Version 1). arXiv. https://doi.org/10.48550/ARXIV.2412.18045