Anna Medvedovsky

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• Bellaïche, J., & Medvedovsky, A. (2025). Formes modulaires spéciales modulo 2. Tunisian Journal of Mathematics, 7(Issue 3). doi:10.2140/tunis.2025.7.453
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  In this posthumously published work, Bellaïche continues the study of modular forms modulo 2 of level 1 begun by Nicolas, Serre, and himself in 2012. After a careful analysis of the Galois group in question— the Galois group of the maximal pro-2 extension of Q unramified outside 2—Bellaïche, in collaboration with Serre, gives explicit matrices realizing the representation of this Galois group on the big Hecke algebra whose trace is the universal in the sense of Chenevier and Mazur. Bellaïche then analyzes the ideals of the Hecke algebra corresponding to certain special forms mod 2—those whose prime Fourier coefficients depend on Frobenius conjugacy classes in finite abelian or dihedral extensions of Q—and gives a basis of these forms.
• Charlton, S., Mauth, L., & Medvedovsky, A. (2025). On the parity of coefficients of eta powers. Research in Mathematical Sciences, 12(Issue 3). doi:10.1007/s40687-025-00507-9
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  We study a new notion of mod-ptwisted density for a modular form f supported on an arithmetic progression: the proportion of primes ℓ for which the mod-p order of infinity of Uℓf is minimal. We show that for classical modular forms, twisted density is always defined and rational, connecting it with an earlier notion of density studied by Bellaïche. Finally, we specialize to p=2 and f a positive power of the Dedekind eta function, studying densities via Galois-theoretic techniques developed by Bellaïche in level 1 and extending them to level 9. In particular, we explicitly calculate twisted densities for certain eta powers corresponding to CM/dihedral mod-2 modular forms in the sense of Nicolas and Serre. En passant we take the opportunity to communicate proofs of two of Bellaïche’s unpublished results on densities of mod-2 modular forms.
• Anni, S., Ghitza, A., & Medvedovsky, A. (2024). ELEMENTARY SYMMETRIC FUNCTIONS AND DEEP POWER-SUM CONGRUENCES. Integers, 24(Issue). doi:10.5281/zenodo.11221647
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  We prove that congruences modulo p between polynomials in Zp [X] are equivalent to deeper p-power congruences between power-sum functions of their roots. This result generalizes to torsion-free Z(p)-algebras modulo divided-power ideals. Our approach is combinatorial: we introduce a p-equivalence relation on partitions, and use it to prove that certain linear combinations of power-sum functions are p-integral. We also include a second proof, short and algebraic, suggested by an anonymous referee. As a corollary we refine the Brauer-Nesbitt theorem for a single linear operator, motivated by the study of Hecke modules of modular forms modulo p.
• Deo, S. V., & Medvedovsky, A. (2024). Mod-2 Hecke algebras of level 3 and 5. Selecta Mathematica, New Series, 30(Issue 5). doi:10.1007/s00029-024-00961-3
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  We use deformation theory to study the big Hecke algebra acting on mod-2 modular forms of prime level N and all weights, especially its local component at the trivial representation. For N=3,5, we prove that the maximal reduced quotient of this big Hecke algebra is isomorphic to the maximal reduced quotient of the corresponding universal deformation ring. Then we completely determine the structure of this big Hecke algebra. We also describe a natural grading on mod-p Hecke algebras, and prove an R=T theorem for the partially full version of our mod-2 Hecke algebra.
• Conti, A., Lang, J., & Medvedovsky, A. (2023). Big images of two-dimensional pseudorepresentations. Mathematische Annalen, 385(Issue 3-4). doi:10.1007/s00208-021-02345-w
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  Bellaïche has recently applied Pink-Lie theory to prove that, under mild conditions, the image of a continuous 2-dimensional pseudorepresentation ρ of a profinite group on a local pro-p domain A contains a nontrivial congruence subgroup of SL2(B) for a certain subring B of A. We enlarge Bellaïche’s ring and give this new B a conceptual interpretation both in terms of conjugate self-twists of ρ, symmetries that constrain its image, and in terms of the adjoint trace ring of ρ, which we show is both more natural and the optimal ring for these questions in general. Finally, we use our purely algebraic result to recover and extend a variety of arithmetic big-image results for GL2 Galois representations arising from elliptic, Hilbert, and Bianchi modular forms and p-adic Hida or Coleman families of elliptic and Hilbert modular forms.
• Deo, S. V., Medvedovsky, A., & Ghitza, A. (2019). NEWFORMS MOD p IN SQUAREFREE LEVEL WITH APPLICATIONS TO MONSKY’S HECKE-STABLE FILTRATION. Transactions of the American Mathematical Society Series B, 6(Issue). doi:10.1090/btran/35
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  We propose an algebraic definition of the space of L-new mod-p modular forms for Γ0(N L) in the case that L is prime to N, which naturally generalizes to a notion of newforms modulo p in squarefree level. We use this notion of newforms to interpret the Hecke algebras on the graded pieces of the space of mod-2 level-3 modular forms described by Paul Monsky. Along the way, we describe a renormalized version of the Atkin-Lehner involution: No longer an involution, it is an automorphism of the algebra of modular forms, even in characteristic p.
• Medvedovsky, A. (2018). Nilpotence order growth of recursion operators in characteristic p. Algebra and Number Theory, 12(Issue 3). doi:10.2140/ant.2018.12.693
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  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 also explain the motivating application: obtaining a lower bound for the Krull dimension of a local component of a big mod p Hecke algebra in the genus-zero case. We sketch the application for p D 2 and p D 3 in level one. The case p D 2 was first established in by Nicolas and Serre in 2012 using different methods.