This Research-in-Groups project concerns the Galois counterpart of small parabolic eigenvarieties inspired by thework of Bellaïche—Chenevier. This proposal aims to achieve the following:
• Construct families of Galois representations over the small parabolic eigenvarieties.
• Use the Galois information to study the local geometry of small parabolic eigenvarieties.
• Construct p-adic L-functions over the small parabolic eigenvarieties and study their relationships withBloch—Kato Selmer groups.
This research will be particularly interesting for mathematicians working in algebraic number theory andarithmetic geometry. Successful results will lead to important applications in understanding newbehaviours of p-adic families of automorphic forms and proving new cases of the Bloch—Kato conjecture.
The proposed research grew when the three researchers all attended the ICMS workshop ‘p-adic Families of Automorphic Forms: Theories and Applications’. During the conference,we were inspired by the talk of David Loeffler and had chances to discuss with experts, including DanielBarrera Salazar and Chris Williams.
