John F Bergdall


Welcome to my webpage. I am a postdoctoral researcher at Michigan State University. My position presents me the opportunity to combine teaching, personal research and active participation in seminars. Prior to arriving at MSU, I was a postdoctoral fellow at Boston University, partially supported by the National Science Foundation.

You may want to view my CV. (Last update: November 3, 2017)

Research

My research is in number theory. I am ultimately interested in all aspects of the Langlands program, but my focus at the moment is on the theme of p-adic variation. Below you will find my writings along with a short abstract for each article.

For articles which have appeared in peer-reviewed journals, the arXiv versions may differ slightly from the published versions (especially in the numbering of statements). I have tried to provide links to the official journal versions if possible.

Publications
Smoothness on definite unitary eigenvarieties at critical points
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To appear in J. reine angew. Math. (Crelle's journal).
We compute an upper bound for the dimension of the tangent spaces of certain definite eigenvarieties at classical points, especially including the so-called critically refined cases. Our bound is given in terms of "critical types" and when our bound is minimized it matches the dimension of the eigenvariety. In those cases, which we explicitly determine, the eigenvariety is necessarily smooth and our proof also shows that the completed local ring on the eigenvariety is naturally a certain universal Galois deformation ring.
Slopes of modular forms and the ghost conjecture
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To appear in International Mathematics Research Notices (IMRN).
Joint with Robert Pollack.
We formulate a conjecture on slopes of overconvergent p-adic cuspforms of any p-adic weight in the Gamma_0(N)-regular case. This conjecture unifies a conjecture of Buzzard on classical slopes and more recent conjectures on slopes "at the boundary of weight space".
An adjunction formula for the Emerton-Jacquet functor
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To appear in Israel J. Math.
Joint with Przemyslaw Chojecki.
The Emerton-Jacquet functor is a tool for studying locally analytic representations of p-adic Lie groups. It provides a way to access the theory of p-adic automorphic forms. Here we give an adjunction formula for the Emerton-Jacquet functor, relating it directly to locally analytic inductions, under a strict hypothesis that we call non-critical. We also further study the relationship to socles of principal series in the non-critical setting.
A remark on non-integral p-adic slopes for modular forms
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C. R. Acad. Sci. Math. Sci. Paris, 355(3):260-262, 2017.
Joint with Robert Pollack.
We give a sufficient condition, namely "Buzzard irregularity", for there to exist a cuspidal eigenform which does not have integral p-adic slope.
Paraboline variation of p-adic families of (φ,Γ)-modules
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Compositio Math., 153(1):132-174, 2017.
We study the p-adic variation of triangulations over p-adic families of (φ,Γ)-modules. In particular, we study certain canonical sub-filtrations of the pointwise triangulations and show that they extend to affinoid neighborhoods of crystalline points. This generalizes results of Kedlaya, Pottharst and Xiao and (independently) Liu in the case where one expects the entire triangulation to extend. As an application, we study the ramification of weight parameters over natural p-adic families.
Arithmetic properties of Fredholm series for p-adic modular forms
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Proc. Lon. Math. Soc., 113(3):419-444, 2016.
Joint with Robert Pollack.
We study the relationship between recent conjectures on slopes of overconvergent p-adic modular forms "near the boundary" of p-adic weight space. We also prove in tame level 1 that the coefficients of the Fredholm series of the U_p operator never vanish modulo p, a phenomenon that fails at higher level. In higher level, we do check that infinitely many coefficients are non-zero modulo p using a modular interpretation of the mod p reduction of the Fredholm series recently discovered by Andreatta, Iovita and Pilloni.
Ordinary modular forms and companion points on the eigencurve
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J. Number Theory, 134(1):226–239, 2014.
We give a new proof of a result due to Breuil and Emerton which relates the splitting behavior at p of the p-adic Galois representation attached to a p-ordinary eigenform to the existence of an overconvergent p-adic companion form for f.
Preprints
On p-adic L-functions for Hilbert modular forms
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Submitted.
Joint with David Hansen.
We construct p-adic L-functions associated with p-refined cohomological Hilbert modular forms over any totally real field under a mild hypothesis. Our construction is canonical, varies naturally in p-adic families, and does not require any small slope or non-criticality assumptions on the p-refinement. The main new ingredients are an adelic definition of a canonical map from overconvergent cohomology to a space of locally analytic distributions on the relevant Galois group and a smoothness theorem for certain eigenvarieties at critically refined points.
Slopes of modular forms and the ghost conjecture, II
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Submitted.
Joint with Robert Pollack.
In a previous work, we constructed an entire power series over p-adic weight space (the 'ghost series') and conjectured, in the Γ0(N)-regular case, that this series encodes the slopes of overconvergent modular forms of any p-adic weight. In this paper, we construct 'abstract ghost series' which can be associated to various natural subspaces of overconvergent modular forms. This abstraction allows us to generalize our conjecture to, for example, the case of slopes of overconvergent modular forms with a fixed residual representation locally reducible at p. Ample numerical evidence is given for this new conjecture. Further, we prove that the slopes computed by any abstract ghost series satisfy a distributional result at classical weights (consistent with the conjectures of Gouvêa) while they form unions of arithmetic progressions at all weights not in Zp.
Upper bounds for constant slope p-adic families of modular forms
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Submitted.
We study p-adic families of eigenforms for which the p-th Hecke eigenvalue ap has constant p-adic valuation ("constant slope families"). We prove two separate upper bounds for the size of such families. The first is in terms of the logarithmic derivative of ap while the second depends only on the slope of the family. We also investigate the numerical relationship between our results and the former Gouvêa-Mazur conjecture.
Other
Slopes of modular forms and the ghost conjecture (unabridged version)
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Joint with Robert Pollack.
Note: This manuscript is a combined version of the two papers we have written with the title "Slopes of modular forms and the ghost conjecture." It is here for historical purposes only.
We formulate a conjecture on slopes of overconvergent p-adic cuspforms of any p-adic weight in the Gamma_0(N)-regular case. This conjecture unifies a conjecture of Buzzard on classical slopes and more recent conjectures on slopes "at the boundary of weight space".
Website: Slopes of modular forms
[ | url]
Joint with Robert Pollack.
Note: The authors are not professional coders. Use at your own risk!
We have collected together all the code and examples we used in our series of joint articles. This includes an implementation of Koike's formula to compute the characteristic series of the Up-operator, and code to compute slopes of p-adic modular forms wiith a fixed mod p Galois representation. Also included is software implementing the ghost series construction on a computer.
Ordinary representations and companion points for U(3) in the indecomposable case
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Joint with Przemyslaw Chojecki.
Note: There is no plan to publish this article at the moment.
We prove that certain p-adic Banach representations, associated to local or- dinary Galois representations, constructed by Breuil and Herzig appears in the completed cohomology of a definite unitary group in three variables. This confirms part of their con- jecture. Our main technique is making use of p-adic automorphic forms for definite unitary groups and the eigenvarieties which parameterize them.
On the variation of (φ,Γ)-modules over p-adic families of automorphic forms
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Ph.D. thesis, Brandeis University, 2013.
Note: Some material, especially Chapter 4, is generalized and subsumed by the paper Paraboline variation of p-adic families of (φ,Γ)-modules posted above. The rest is contained in Smoothness on definite eigenvarieties at critical points.
We present new results on the variation of Galois representations in p-adic families of automorphic forms. As an application of our main result, we obtain new smoothness results in such families.

Teaching

Fall 2017
Fall 2016
Fall 2015
MA123: Calculus I (login required)
Summer 2015
PROMYS: Dirichlet's theorem on primes in arithmetic progressions (no website, but exercises sheets are available here.)
Spring 2014
Fall 2013

Other activities

Contact information
Email:
bergdall@math.msu.edu
Office:
Wells Hall, Room C237
Hours:
M 1:30-3:00, W 4:00-5:30
Mail:
Dept. of Mathematics
Michigan State University
619 Red Cedar Road
C212 Wells Hall
East Lansing, MI 48824