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October 2, 2025Open Access

The Asai--Flach Euler system in p-adic families

Key Points

The Asai Euler system is effectively interpolated p-adically as Hilbert modular forms change over a Hida family, offering a new perspective.
A critical aspect of this work is the assessment of how the Euler system corresponds with the Hilbert modular form and its implications for number theory.
Evolving from previous research (2018), this work aids in addressing the Bloch-Kato conjecture's analytic rank zero context for the Asai representation.
The findings consolidate the theory behind Euler systems and their behavior across families, indicating potential for broader applications in mathematics.

Abstract

We show that the Euler system for the Asai representation corresponding to a Hilbert modular eigenform over a real quadratic field, constructed by Lei, Loeffler and Zerbes (2018), can be interpolated p-adically as the Hilbert modular form varies in a Hida family. This work is used as an important input in recent work of Grossi, Loeffler and Zerbes (2025) on the proof of the Bloch--Kato conjecture in analytic rank zero for the Asai representation.

The Asai--Flach Euler system in p-adic families

Key Points

Abstract

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