Anna MEDVEDOVSKY

Mod-2 Hecke algebras of level 3 and 5. With Shaunak Deo. Submitted.
We define a local deformation condition to prove that the big Hecke algebra acting on mod-2 modular forms of prime level 3 or 5 is, up to taking maximal reduced quotients, isomorphic to the corresponding universal deformation ring. We determine the structure of this big mod-2 Hecke algebra completely, and prove an “R = T” theorem for the related partially full Hecke algebra. We also describe compatible gradings on spaces of mod-p modular forms and their Hecke algebras more generally.
- ArXiv version

Deep congruences + the Brauer-Nesbitt theorem. With Samuele Anni and Alexandru Ghitza. In preparation.
Let M be a finite free Z_p-module with an action of a linear operator T. How much must one know about the trace of Tⁿ on M to pin down the semisimplification of M modulo p? We give a precise mod-power-of-p criterion, refining the Brauer-Nesbitt theorem for a single operator over Z_p — or more generally over a torsion free Z_(p)-algebra with a divided-power ideal.
- Draft (updated more frequently than ArXiv)
- ArXiv version (theorem etc. numbering here differs radically from the version above)
The motivating application — counting Hecke eigensystems by exhibiting up-to-semisimplification Hecke-module isomorphisms between spaces of mod-p modular forms — will appear separately. Related material.

Data slides
Samuele Anni’s slides from a talk at the Novenas Jornadas de Teoría de Números 2022

Big images of two-dimensional Galois pseudorepresentations. With Andrea Conti and Jaclyn Lang. Math. Annalen (2022).
We use Bellaïche-Pink-Lie theory to prove that, under mild conditions, the image of a continuous two-dimensional pseudorepresentation ρ of a p-finite profinite group on a local pro-p domain A is as big as it can be: it contains a nontrivial congruence subgroup of SL₂(B) for a subring B of A that we relate both to the conjugate self-twists of ρ and to the adjoint traces of ρ. This purely algebraic result recovers and extends known p-adic big-image theorems for Galois representations arising from elliptic, Hilbert, and Bianchi modular forms and p-adic Hida and Coleman families of elliptic and Hilbert modular forms.

Journal version

ArXiv version

Jaclyn Lang’s slides from CNTA 2018

Jaclyn Lang’s report in “Algebraische Zahlentheorie”, Oberwohlfach reports 15:2 (2018)

Newforms mod p in squarefree level, with applications to Monsky’s Hecke-stable filtration.
With Shaunak Deo, and an appendix by Alexandru Ghitza. Transactions of the AMS, Series B 6 (2019).
We propose an algebraic definition of the space of ℓ-new mod-p modular forms for Γ₀(Nℓ) for ℓ, N, p pairwise coprime. Along the way we renormalize the Atkin-Lehner involution to obtain an algebra automorphism of the space of modular forms, even mod p.

Journal version

ArXiv version

Mod-2 dihedral Galois representations of prime conductor.
With Kiran Kedlaya. Proceedings of the Thirteenth Algorithmic Number Theory Symposium. Open Book Series 2 (2019).
For odd primes N up to 500000, we compute the action of the Hecke operator T₂ on weight-2 cuspforms of level N, and determine whether 0 and 1 appear as mod-2 eigenvalues. We then partially explain our observations in terms of class field theory and modular mod-2 Galois representations. Our methods recover and extend prior results of Setzer, Hadano, and Kida on the nonexistence of elliptic curves and modular forms with certain mod-2 reductions.

Journal version

ArXiv version

Kiran Kedlaya’s slides from a talk at ANTS-XIII

An explicit universal Galois representation on the mod-3 Hecke algebra. In preparation.
Following published and unpublished work of Serre, Nicolas, and Bellaïche on mod-2 modular forms of level one, we analyze the space of mod-3 modular forms of level one and the structure of its Hecke algebra, including the compatible (Z/3Z)^×-grading on both. We construct a universal Galois pseudorepresentation on this Hecke algebra and study its localizations modulo prime ideals, identifying the ideals of reducibility and dihedrality. We finish with an explicit matrix realization of this representation.

Paper draft

Slides from a speed talk at Building Bridges 2016

Data: A convenient basis for the space of mod-3 modular forms of level one, adapted, in the spirit of Nicolas and Serre, to T₇–2 and T₂.

Nilpotence order growth of recursion operators in characteristic p. Algebra and Number Theory 12 (2018) no. 3.
We prove that the killing rate of certain degree-lowering “recursion operators” on a polynomial algebra over a finite field grows slower than linearly in the degree of the polynomial attacked. We explain the motivating application: obtaining lower bounds on the Krull dimension of local components of big mod-p Hecke algebras of level N if X₀(Np) has genus zero. We sketch this application for p = 2, 3 in level one. For p = 2 we thus recover results of Nicolas and Serre obtained by different methods; for p = 3 we thereby complete the proof of Theorem 24 of Bellaïche–Khare.

Journal version
ArXiv version
Poster slides on this project. An overview. Some pretty pictures.

Lower bounds on dimensions of mod-p Hecke algebras: The nilpotence method. Ph.D. dissertation, Brandeis, 2015.
Develops the nilpotence method for obtaining lower bounds on Krull dimensions of local components of mod-p Hecke algebras of level one for p = 2, 3, 5, 7, 13. Determines the structure of each Hecke component explicitly in each of these cases.

The key technical theorem (Theorem 5.1) has been published, with corrected proof (!), in the ANT paper above.

Other parts — in particular, the theory of recursion operators (Chapter 4), managing multiple local components (Chapter 3), and finding explicit generators for the Hecke algebra at a reducible component (Chapter 7) have yet to be published.

Data: Hecke recursion polynomials in level one mod small primes.

Research

Teaching