$λ$-invariant stability in families of modular Galois representations

Constructing Galois representations with large Iwasawa $$\lambda $$-invariant

“…Furthermore, not only is one able to construct infinitely many modular Galois representations giving rise to a prescribed λ-invariant, one also obtains an explicit and satisfactory quantitative description of their levels. We contrast Theorem 1.1 to the results of recent work by Hatley and Kundu [HK22], where it is shown that there are infinitely many modular forms f that are p-congruent to a fixed modular form g, for which λ p (f ) = λ p (g). This λ-stability result requires an additional assumption on g: that the λ-invariant of g is minimal in the family of all λ-invariants for modular forms that are p-congruent to g. For further details, we refer to p.15 of loc.…”

“…This λ-stability result follows from Theorem 1.1 in the special case when n = 0, without the additional hypothesis. The method used in proving our results does draw some inspiration from [Ray23,HK22].…”

Constructing Galois representations with prescribed Iwasawa λ$\lambda$‐invariant

“…Furthermore, not only is one able to construct infinitely many modular Galois representations giving rise to a prescribed 𝜆-invariant, one also obtains an explicit and satisfactory quantitative description of their levels. We contrast Theorem 1.1 to the results of recent work by Hatley and Kundu [8],…”

“…This 𝜆-stability result follows from Theorem 1.1 in the special case when 𝑛 = 0, without the additional hypothesis. The method used in proving our results does draw some inspiration from [8,16].…”