What question did this study set out to answer?

The aim is to create p-adic L-functions that generate the congruence ideal related to Hida families.

February 2, 2026Open Access

A canonical generator for congruence ideals of Hida families

AMAlexandre MaksoudMax Planck Institute for Mathematics

Key Points

The aim is to create p-adic L-functions that generate the congruence ideal related to Hida families.
Constructed adjoint p-adic L-functions for Hida families.
Interpolate Petersson norms of classical ordinary newforms.
Adjusted by Euler factors for accuracy.
Established connections to Hida’s universal ordinary Hecke algebra.
Generated functions lead to congruence ideals of Hida families.
Showed interpolation by regular elements of the Hecke algebra.
Linked p-adic L-functions to primitive adjoint Selmer groups.

Abstract

Abstract We construct adjoint p -adic L -functions generating the congruence ideal attached to Hida families. These functions interpolate the Petersson norm of any classical ordinary newform, normalized by a product of Shimura’s canonical periods. We show that, after adjusting by suitable Euler factors, they are interpolated by a regular element of Hida’s universal ordinary Hecke algebra. We also establish a link between these p -adic L -functions and the characteristic series of primitive adjoint Selmer groups.

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