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A Local Analogue of the Ghost Conjecture of Bergdall–Pollack
We formulate a local analogue of the ghost conjecture of Bergdall and Pollack, which essentially relies purely on the representation theory of
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An explicit computation of the Hecke operator and the ghost conjecture
In this paper, we investigate the Hecke operator at p = 5 and show that the upper minors of the matrix have non zero corank and, interestingly,...
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Diagonal restrictions of p-adic Eisenstein families
We compute the diagonal restriction of the first derivative with respect to the weight of a p -adic family of Hilbert modular Eisenstein series...
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Two Recent p-adic Approaches Towards the (Effective) Mordell Conjecture
We give an introductory account of two recent approaches towards an effective proof of the Mordell conjecture, due to Lawrence–Venkatesh and Kim. The...
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Stark points on elliptic curves via Perrin-Riou’s philosophy
In the early 90’s, Perrin-Riou (Ann Inst Fourier 43(4):945–995, 1993) introduced an important refinement of the Mazur–Swinnerton-Dyer p -adic L -functio...
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Integral models of Hilbert modular varieties in the ramified case, deformations of modular Galois representations, and weight one forms
We prove the strong Artin conjecture for continuous, totally odd, two-dimensional representations of the absolute Galois group of a totally real...
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On the Hilbert eigenvariety at exotic and CM classical weight 1 points
Let F be a totally real number field and let f be a classical cuspidal p -regular Hilbert modular eigenform over F of parallel weight 1. Let x be the...
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Interpolation of Beilinson–Kato elements and p-adic L-functions
Our objective in this series of two articles, of which the present article is the first, is to give a Perrin-Riou-style construction of p -adic L -funct...
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The Adjoint p-Adic L-Function and the Ramification Locus of the Eigencurve
In this chapter, we introduce the theory of the L-ideal of a finite extension of rings, and use it to construct (locally) a p-adic analytic function...
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Differential equations and Lie group representations
We discuss the role of differential equations in Lie group representation theory. We use Kashiwara’s pentagon as a reference frame for the real...
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Canonical subgroups of generalized Drinfeld modules
We study basic properties of canonical subgroups of generalized Drinfeld modules over a complete valuation ring of positive characteristic. In...
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A geometric perspective on p-adic properties of mock modular forms
Bringmann et al. (Trans Am Math Soc 364(5):2393–2410,
2012 ) showed how to ‘regularize’ mock modular forms by a certain linear combination of the... -
Explicit computations of Hida families via overconvergent modular symbols
In Pollack and Stevens (Ann Sci Éc Norm Supér 44(1):1–42,
2011 ), efficient algorithms are given to compute with overconvergent modular symbols. These... -
A Note on Critical p-adic L-functions
We study the adjunction property of the Jacquet-Emerton functor in certain neighborhoods of critical points in the eigencurve. As an application, we...