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Contact:

Department of Mathematics,
Massachusetts Institute of Technology,
Cambridge, Massachusetts

eranasaf@mit.edu

Eran Assaf

At the moment, I am a Research Scientist at MIT.

Research:

My latest interest is computing modular forms, in various settings, with applications to equations of modular curves (classical modular forms), the paramodular conjecture and understanding endoscopy in the Langlands program (orthogonal forms). I am interested in algebraic number theory, representation theory and the interplay between them. In particular, the $p$-adic Langlands programme and Shimura Varieties.

Papers and preprints:

• The Fibonacci Zeta Function and Continuation, with Chan Ieong Kuan, David Lowry-Duda and Alexander Walker. arXiv preprint, 2024.


• A note on the trace formula. arXiv preprint, 2024.


• A database of paramodular forms from quinary orthogonal modular forms, with Watson Ladd, Gustavo Rama, Gonzalo Tornaría and John Voight. Contemporary Mathematics, Volume 796, 2024, Pages 285-312.


• A database of basic numerical invariants of Hilbert modular surfaces, with Angelica Babei, Ben Breen, Edgar Costa, Juanita Duque-Rosero, Aleksander Horawa, Jean Kieffer, Avinash Kulkarni, Grant Molnar, Sam Schiavone and John Voight. Contemporary Mathematics, Volume 796, 2024, Pages 243-259.


• On smooth plane models for modular curves of Shimura type, with Samuele Anni and Elisa Lorenzo García. Research in Number Theory 9:21, Springer, 2023.


• Definite orthogonal modular forms: Computations, Excursions and Discoveries, with Dan Fretwell, Colin Ingalls, Adam Logan, Spencer Secord and John Voight. Research in Number Theory 8:70, Springer, 2022.


• An asymptotic formula for the variance of the number of zeroes of a stationary Gaussian process, with Jeremiah Buckley and Naomi Feldheim. Probability Theory and Related Fields, Springer, 2023.


• Computing classical modular forms for arbitrary congruence subgroups. Arithmetic Geometry, Number Theory and Computation, Simons Symposia, Springer, 2022, pp. 43-104.


• Existence of invariant norms in $p$-adic representations of $\mathrm{GL}_2(F)$ of large weights. Journal of Number Theory, Volume 224, 2021, Pages 95-141.


• Kirillov models and the Breuil-Schneider conjecture for $\mathrm{GL}_2(F)$, with David Kazhdan and Ehud de Shalit. arXiv preprint, 2013.


• Pay for a sliding Bloom filter and get counting, distinct elements, and entropy for Free, with Ran Ben Basat, Gil Eizinger and Roy Friedman. IEEE INFOCOM 2018 - IEEE Conference on Computer Communications.


• Counting distinct elements over sliding windows, with Ran Ben Basat, Gil Eizinger, Roy Friedman and Yaron Kassner. SYSTOR 2017 Proceedings of the 10th ACM International Systems and Storage Conference.


• Characterization of regular Diophantine quadruples, with Shay Gueron. Elemente der Mathematik 2001.

Teaching:

• Summer 2024 - Math 35, Dartmouth College.


• Winter 2023 - Math 81/111, Dartmouth College.


• Fall 2021 - Math 9, Dartmouth College.


• Winter 2021 - Math 3, Dartmouth College.

Talks and Notes:

• AGNES scavenger hunt. Algebraic Geometry Northeastern Series, November 2024.


• Hilbert modular forms from orthogonal modular forms on quaternary lattices. Wesleyan Algebra and Number Theory Seminar, October 2024.


• Hilbert modular forms from orthogonal modular forms on quaternary lattices. MIT Number Theory Seminar, October 2024.


• Hilbert modular forms from orthogonal modular forms on quaternary lattices. Langenhop lecture and SIU Math conference, May 2024.


• Canonical rings of stacky surfaces. Dartmouth-UVM math day, Feb 2024.


• Algebraic modular forms - from computations to insights. Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation Annual Meeting, Jan 2024.


• Canonical rings of stacky surfaces. JMM - AMS Special Session on Explicit Computation with Stacks, Jan 2024.


• Lattice methods in modularity. University of Haifa, Math Colloquium, Dec 2023.


• Hilbert modular surfaces, revisited. Hebrew University of Jerusalem, Number Theory & Algebraic Geometry seminar, Dec 2023.


• Smooth plane models of modular curves. Brown Algebra, Number Theory and Algebraic Geometry Seminar, November 2023.


• Equations for modular curves. Dartmouth Algebra and Number Theory Seminar, October 2020.


• The canonical model of a Shimura variety. Dartmouth Reading Seminar, July 2020.


• General Shimura varieties. Dartmouth Reading Seminar, June 2020.


• Motives. Dartmouth Reading Seminar, June 2020.


• The Siegel modular variety. Dartmouth Reading Seminar, April 2020.


• Computing modular forms. Quebec Vermont Number Theory Seminar, February 2020.


• Diagrams mod $p$ and integral structures in representations of reductive groups of semisimple rank one. $p$-adic Langlands correspondence: a constructive and algorithmic approach, Rennes, September 2019.


• The Newton stratification on flag varieties. London Number Theory Study Group, June 2019.


• The complex realization of the derived Hecke algebra. London Number Theory Study Group, March 2019.


• Serre weight conjecture. January 2019.


• Poster - $p$-adic Langlands correspondence. September 2016.


• The $p$-adic local Langlands programme and invariant norms. May 2015.


• A criterion for integral structures in $p$-adic representations of $U_3(F)$. March 2015.

Code Packages:

• Modular symbols to arbitrary congruence subgroups.


• Algebraic modular forms.

PhD Thesis

Existence of Invariant Norms in p-adic Representations of Reductive Groups

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