Modular supercuspidal lifts, potentially diagonalizable lifts and Langlands base change of Hida families

Iván Blanco Chacón, University of Alcalá

The present talk aims at presenting several results developed by the speaker since 2021 until 2025, jointly with Dieulefait. These results deal with the problem of producing modular lifts of a given modular Galois representation with different local conditions.

In [1] we proved that given a cuspidal newform of level N and weight k and a prime p>max{k+1,N} (p not dividing N), there exist infinitely many potentially diagonalizable modular lifts of the residual representation. In [2] we proved that under the same hypotheses, there exist a modular p-supercuspidal lift of weight 2 (but we can indeed show this for any weight). Finally, in [3] (jointly with Dieulefait and my PhD student Haavikko) we introduced the Langlands base change of Hida families to arbitrary totally real fields. In order to obtain this, our starting point is Dieulefait's work [4] which proves the existence of Langlands base change of modular representations of Gal(Qalg/Q) to Gal(Falg/F) with F totally real. Then, we generalize earlier results by Saito to explicitly present the Hecke eigenvalues of the lifted Hilbert modular form from those of the classical modular form, and let them vary in a Hida family, being able to lift it to a Hida family to F. As a corollary, we generalize [1] being able to F-base change the aforementioned modular lifts (in the ordinary case via lifted Hida families, in the non-ordinary case via a different approach). We also discuss the motivations behind these results, which are, in a nutshell, the study of automorphicity of tensor products of automorphic representations, still under research. We warn that a very recent preprint by Emerton and Gee [5] superseeds our works [1] and [2] but, time permiting, we will also discuss a more recent work of us, not superseeded by [5] which produces (not necessarily modular) crystalline ordinary lifts of arbitrary weights of a given ordinary crystalline Galois representation (not necessarily modular).



[1] I. Blanco-Chacón, L. Dieulefait. Potentially diagonalizable modular lifts of arbitrary weight. Journal of Number Theory, 2021.

[2] I. Blanco-Chacón, L. Dieulefait. Modular supercuspidal lifts of weight 2. Journal of Number Theory, 2025.

[3]. I. Blanco-Chacón, L. Dieulefait, A. Haavikko. On the computation of base change lifts and Hida families https://arxiv.org/pdf/2604.05618 2026

[4] L. Dieulefait. Langlands base change for GL(2). Annals of Mathematics, 2012.

[5] M. Emerton, T. Gee, L. Pan, X. Zhu. Prescribed lifts of 2-dimensional representations. https://arxiv.org/pdf/2606.25485 2026.