Nilpotence order growth of recursion operators
in characteristic p

Medvedovsky, Anna

doi:10.2140/ant.2018.12.693

Cited by 5 publications

(11 citation statements)

References 17 publications

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“…In fact, Hecke operators appearing in the theory of modular forms are of this type. Surprising results concerning the dynamics of the operators T D for self-correspondences over finite fields have been obtained in a recent work by Medvedovsky [2018], and, applied to Hecke operators, those results provide a new and elementary proof of certain deep modularity result of Gouvêa and Mazur [1998]. We plan to come back to these questions on a subsequent work.…”

Section: Introductionmentioning

confidence: 78%

“…There are two obvious finite completes sets, the set of supersingular points and the sets of cusps. The complete set of supersingular points is étale (easy since l ̸ = p) and irreducible (this can be proved by direct analysis, or, as Krishnamoorthy notes in [Krishnamoorthy 2018], simply as a consequence of Proposition 3.2.5 below). The complete set of cusps may be reducible, and none of its irreducible components are étale (again a consequence of Proposition 3.2.5).…”

Section: Introductionmentioning

confidence: 99%

“…To the best of our knowledge, the question we have in mind has not been solved, and not even asked, except in some very particular cases by Pakovich (see [Pakovich 1995] and Remark 3.3.7 below). A partial exception is the recent article of Krishnamoorthy [2018], essentially the second chapter of his PhD thesis at Columbia University. Though the focus of the paper is different, as Krishnamoorthy is concerned with general correspondences rather than self-correspondences, we borrow several ideas and concepts from him, and we gladly acknowledge our debt to his work.…”

Section: Introductionmentioning

confidence: 99%

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On self-correspondences on curves

Bellaïche

2023

Alg. Number Th.

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À la toute fin du processus de révision, le comité éditorial d'Algebra & Number Theory a eu l'immense tristesse d'apprendre le décès de Joël Bellaïche. Joël faisait des maths comme il respirait, avec une immense joie. Les mathématiques n'étaient pas pour lui séparées de la vie, elles étaient la vie même -la vie vraiment vivante.At the very end of the reviewing process, the editorial board of Algebra & Number Theory learned with great sadness that Joël Bellaïche had passed away. Joël did math as he breathed, with immense joy. For him mathematics was not disconnected from life; it was life itself, and his way of living it to the fullest.We study the algebraic dynamics of self-correspondences on a curve. A self-correspondence on a (proper and smooth) curve C over an algebraically closed field is the data of another curve D and two nonconstant separable morphisms π 1 and π 2 from. We show that self-correspondences are divided into two classes: those that have only finitely many finite complete sets, and those for which C is a union of finite complete sets. The latter ones are called finitary, and happen only when deg π 1 = deg π 2 and have a trivial dynamics. For a nonfinitary self-correspondence in characteristic zero, we give a sharp bound for the number of étale finite complete sets. MSC2020: 14A10, 37E99.

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Section: Introductionmentioning

confidence: 78%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On self-correspondences on curves

Bellaïche

2023

Alg. Number Th.

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“…Note that A(1) t ∼ = F x, y if t is unobstructed in the sense of deformation theory. See [18] for p = 2, [2] for p ≥ 5, [13] for p = 3, and [12] for more discussion of p = 2, 3, 5, 7, 13. Question 2.…”

Section: Regularity Conditions On the Hecke Algebramentioning

confidence: 99%

Newforms mod 𝑝 in squarefree level with applications to Monsky’s Hecke-stable filtration

Deo

Medvedovsky

2019

Trans. Amer. Math. Soc. Ser. B

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We propose an algebraic definition of the space of-new mod-p modular forms for Γ 0 (N) in the case that 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.

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“…Note that A(1) t ∼ = F x, y if t is unobstructed in the sense of deformation theory. See for p = 2, [2] for p ≥ 5, [13] for p = 3, and [12] for more discussion of p = 2, 3, 5, 7, 13.…”

Section: Hecke-stable Filtrations Mod Pmentioning

confidence: 99%

Newforms mod p in squarefree level, with applications to Monsky's Hecke-stable filtration

Deo¹,

Medvedovsky²,

Ghitza³

2018

Preprint

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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.

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Nilpotence order growth of recursion operators in characteristic p

Cited by 5 publications

References 17 publications

On self-correspondences on curves

On self-correspondences on curves

Newforms mod 𝑝 in squarefree level with applications to Monsky’s Hecke-stable filtration

Newforms mod p in squarefree level, with applications to Monsky's Hecke-stable filtration

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