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)

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.

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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.

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To appear in International Mathematics Research Notices (IMRN).

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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".

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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.

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We give a sufficient condition, namely "Buzzard irregularity", for there to exist a cuspidal eigenform which does not have integral p-adic slope.

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Compositio Math., 153(1):132-174, 2017.

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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.

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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.

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J. Number Theory, 134(1):226–239, 2014.

[journal]

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.

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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.

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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 Γ_{0}(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 Z_{p}.

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Submitted.

We study p-adic families of eigenforms for which the p-th Hecke eigenvalue a_{p} 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 a_{p} 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.

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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.

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.

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Joint with Robert Pollack.

Note: The authors are not professional coders. Use at your own risk!

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 U_{p}-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.

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Joint with Przemyslaw Chojecki.

Note: There is no plan to publish this article at the moment.

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.

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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*.

Note: Some material, especially Chapter 4, is generalized and subsumed by the paper

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.

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

Fall 2013-2016

Fall 2015

Fall 2014

Spring 2014

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

Michigan State University

619 Red Cedar Road

C212 Wells Hall

East Lansing, MI 48824