John Bergdall

Assistant Professor
University of Arkansas

Research

My research focuses on p-adic automorphic forms and Galois representations. Below I list the written products of my research. If you are interested in viewing recorded talks I have given, you can find some at the bottom of this page.

Preprints

Preprints are listed chronologically by their first appearance, newest at top, regardless of revisions. All links provided point to the arXiv.

i.
New phenomena for L-invariants of modular forms
Joint with Robert Pollack
 
 
Publications

All linked titles refer to arXiv versions. Journal links, by digital object identifier (doi), are given in the far right column when available. It happens that some of the journal versions of differ from the final arXiv versions.

15.
A p-adic adjoint L-function and the ramification locus of the Hilbert eigenvariety
Joint with Baskar Balasubramanyam and Matteo Longo
 
To appear in Tunisian Journal of Mathematics.
14.
Huber rings and valuation spectra
 
 
To appear in Münster Lectures in Mathematics.
13.
On p-adic L-functions for Hilbert modular forms
Joint with David Hansen.
Journal
To appear in Memoirs of the Amer. Math. Soc.
12.
Reductions of 2-dimensional semi-stable representations with large L-invariant
Joint with Brandon Levin and Tong Liu.
Journal
J. Inst. Math. Jussieu, 22, no. 6, p. 2619–2644, 2023.
11.
Slopes of modular forms and reducible Galois representations: an oversight in the ghost conjecture
Joint with Robert Pollack.
Journal
Proc. Amer. Math. Soc. Ser. B, 9:432-444, 2022.
10.
Reductions of some two-dimensional crystalline representations via Kisin modules
Joint with Brandon Levin.
Journal
Int. Math. Res. Not., 2022(4):3170–3197, 2022.
9.
Smoothness of definite unitary eigenvarieties at critical points
 
Journal
J. reine angew. Math., 759:29-60, 2020.
8.
Upper bounds for constant slope p-adic families of modular forms
 
Journal
Selecta Mathematica, 25(4):Art. 59, pp. 24, 2019.
7.
Slopes of modular forms and the ghost conjecture, II
Joint with Robert Pollack.
Journal
Trans. Amer. Math. Soc., 372(1):357–388, 2019.
6.
Slopes of modular forms and the ghost conjecture
Joint with Robert Pollack.
Journal
Int. Math. Res. Not., 2019(4):1125-1144, 2019.
5.
An adjunction formula for the Emerton-Jacquet functor
Joint with Przemyslaw Chojecki.
Journal
Israel J. Math., 223(1):1-52, 2018.
4.
A remark on non-integral p-adic slopes for modular forms
Joint with Robert Pollack.
Journal
C. R. Acad. Sci. Math. Sci. Paris, 355(3):260-262, 2017.
3.
Paraboline variation of p-adic families of (φ,Γ)-modules
 
Journal
Compositio Math., 153(1):132-174, 2017.
2.
Arithmetic properties of Fredholm series for p-adic modular forms
Joint with Robert Pollack.
Journal
Proc. Lon. Math. Soc., 113(3):419-444, 2016.
1.
Ordinary modular forms and companion points on the eigencurve
 
Journal
J. Number Theory, 134(1):226–239, 2014.
Other

Sometimes I have produced things that are not destined to be published. Those appear here, with a brief explanation.

 
Github: L-invariants of modular forms
Joint with Robert Pollack
Github repository of L-invariant code from project with Pollack.
 
Github: slopes of modular forms
Joint with Robert Pollack
Github repository of slopes code from projects with Pollack.
 
Slopes of modular forms and the ghost conjecture (unabridged version)
Joint with Robert Pollack
This manuscript split into two publications titled "Slopes of modular forms and the ghost conjecture" above. It is here for historical purposes.
 
Website: slopes of modular forms
Joint with Robert Pollack
Code and data from the ghost conjecture project with Pollack.
 
Ordinary representations and companion points for U(3) in the indecomposable case
Joint with Przemyslaw Chojecki.
Preprint never to be published. Representation theory appeared in "Adjunction" paper. Applications known now, cf. the paper of Breuil and Herzig regarding the "finite slope" part.
 
On the variation of (φ,Γ)-modules over p-adic families of automorphic forms
Ph.D. thesis. Main arguments streamlined into "Paraboline..." above.
Video lectures
2019
Lecture on the paper "Upper bounds for constant slope families of p-adic modular forms''
At: Princeton/IAS number theory seminar
Courtesy of the Institute for Advanced Study
Direct: link to YouTube.
2016
Lecture on the paper "Slopes of modular forms and the ghost conjecture''
At: The p-adic Langlands Program and Related Topics
Courtesy of Matthias Strauch
Direct: link to YouTube.