Exceptional zeros and $\mathcal {L}$-invariants of Bianchi modular forms

Salazar, Daniel Barrera; Williams, Chris

doi:10.1090/tran/7436

Cited by 10 publications

(46 citation statements)

References 26 publications

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“…Proof. This is a reformulation of 5 of [BSW19a], which we briey sketch. First we show surjectivity; let ψ ∈…”

Section: Denote the Space Of Such Forms Bymentioning

confidence: 96%

“…. By [BSW19a,Thm. 8.6], the right-hand side is one-dimensional; hence we deduce that H 1 (Γ, ∆ 0 ⊗ V k,k ) (f ) is a one-dimensional L-vector space.…”

Section: Denote the Space Of Such Forms Bymentioning

confidence: 99%

“…Using the fact that f is p-new, the results of [BSW19a] allow us to extend the overconvergent Bianchi modular symbols of [Wil17] over the BruhatTits tree T p for GL 2 /F p , giving a space which we denote MS Γ (L). Our integration theory yields pairings…”

Section: Let R Be the Eichlermentioning

confidence: 99%

“…Note that the Galois structure we conjecture on the analytic monodromy module, via the trivial zero conjecture, is a natural one to consider. In particular, the main result of [BSW19a] related the analytic L-invariant we consider in this paper to the arithmetic of the standard Lfunction of the Bianchi form f , that is, the L-function of its standard Galois…”

Section: Let R Be the Eichlermentioning

confidence: 99%

“…In this section we recall some results on the BruhatTits tree and p-adic integration from [BSW19a]. One of the key technical results op.…”

Section: P-adic Integration and An L-invariantmentioning

confidence: 99%

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Stark–Heegner cycles attached to Bianchi modular forms

Venkat

Williams

2021

J. London Math. Soc.

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“…Proof. This is a reformulation of 5 of [BSW19a], which we briey sketch. First we show surjectivity; let ψ ∈…”

Section: Denote the Space Of Such Forms Bymentioning

confidence: 96%

“…. By [BSW19a,Thm. 8.6], the right-hand side is one-dimensional; hence we deduce that H 1 (Γ, ∆ 0 ⊗ V k,k ) (f ) is a one-dimensional L-vector space.…”

Section: Denote the Space Of Such Forms Bymentioning

confidence: 99%

Section: Let R Be the Eichlermentioning

confidence: 99%

Section: Let R Be the Eichlermentioning

confidence: 99%

“…In this section we recall some results on the BruhatTits tree and p-adic integration from [BSW19a]. One of the key technical results op.…”

Section: P-adic Integration and An L-invariantmentioning

confidence: 99%

See 3 more Smart Citations

Stark–Heegner cycles attached to Bianchi modular forms

Venkat

Williams

2021

J. London Math. Soc.

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Families of Bianchi modular symbols: critical base-change p-adic L-functions and p-adic Artin formalism

Salazar

Williams

2021

Sel. Math. New Ser.

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Let K be an imaginary quadratic field. In this article, we study the eigenvariety for $$\mathrm {GL}_2/K$$ GL 2 / K , proving an étaleness result for the weight map at non-critical classical points and a smoothness result at base-change classical points. We give three main applications of this; let f be a p-stabilised newform of weight $$k \ge 2$$ k ≥ 2 without CM by K. Suppose f has finite slope at p and its base-change $$f_{/K}$$ f / K to K is p-regular. Then: (1) We construct a two-variable p-adic L-function attached to $$f_{/K}$$ f / K under assumptions on f that conjecturally always hold, in particular with no non-critical assumption on f/K. (2) We construct three-variable p-adic L-functions over the eigenvariety interpolating the p-adic L-functions of classical base-change Bianchi cusp forms. (3) We prove that these base-change p-adic L-functions satisfy a p-adic Artin formalism result, that is, they factorise in the same way as the classical L-function under Artin formalism.

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Parabolic eigenvarieties via overconvergent cohomology

Salazar¹,

Williams²

2021

Math. Z.

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Let $$\mathcal {G}$$ G be a connected reductive group over $$\mathbf {Q}$$ Q such that $$G = \mathcal {G}/\mathbf {Q}_p$$ G = G / Q p is quasi-split, and let $$Q \subset G$$ Q ⊂ G be a parabolic subgroup. We introduce parahoric overconvergent cohomology groups with respect to Q, and prove a classicality theorem showing that the small slope parts of these groups coincide with those of classical cohomology. This allows the use of overconvergent cohomology at parahoric, rather than Iwahoric, level, and provides flexible lifting theorems that appear to be particularly well-adapted to arithmetic applications. When Q is a Borel, we recover the usual theory of overconvergent cohomology, and our classicality theorem gives a stronger slope bound than in the existing literature. We use our theory to construct Q-parabolic eigenvarieties, which parametrise p-adic families of systems of Hecke eigenvalues that are finite slope at Q, but that allow infinite slope away from Q.

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Exceptional zeros and $\mathcal {L}$-invariants of Bianchi modular forms

Cited by 10 publications

References 26 publications

Stark–Heegner cycles attached to Bianchi modular forms

Stark–Heegner cycles attached to Bianchi modular forms

Families of Bianchi modular symbols: critical base-change p-adic L-functions and p-adic Artin formalism

Parabolic eigenvarieties via overconvergent cohomology

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