Vectorial Drinfeld modular forms over Tate algebras

Pellarin, Federico; Perkins, Rudolph

doi:10.1142/s1793042118501063

Cited by 11 publications

(12 citation statements)

References 27 publications

(46 reference statements)

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“…One should compare the results of this section with the congruences obtained by Petrov [25] and by C. Vincent [29] and to those originally found by Gekeler [10]. A similar congruence is shown for Gekeler's false Eisenstein series E in [24,Theorem 4.13].…”

Section: 3supporting

confidence: 75%

“…Thanks. This paper began as an attempt to generalize, by first principles, certain congruences found by Pellarin and the author in [24]. The focus shifted after G. Böckle heard of the constructions in this paper and directed the author to the intriguing paper [17] of Goss.…”

Section: Böckle's Eichler-shimura Relation and Goss' Questionsmentioning

confidence: 99%

“…Example: Derived Eisenstein Series. In §4.4.3 of [24], many forms in the generalized eigenspaces described above were constructed via vectorial modular forms over Tate algebras. For example, letting…”

Section: Böckle's Eichler-shimura Relation and Goss' Questionsmentioning

confidence: 99%

“…3. Finally, there is a canonical choice of representative for the space of single cuspidal forms modulo double cuspidal forms of weight two and type 1 for Γ 0 (p); see also [24]. We recall the false Eisenstein series of Gekeler normalized via its A-expansion E(z) := a∈A+ au(az), 1 We make a brief historical note.…”

Section: Introductionmentioning

confidence: 99%

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Twisting eigensystems of Drinfeld Hecke eigenforms by characters

Perkins¹

2017

J. Théor. Nombres Bordeaux

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We address some questions posed by Goss related to the modularity of Drinfeld modules of rank 1 defined over the field of rational functions in one variable with coefficients in a finite field.For each positive characteristic valued Dirichlet character, we introduce certain projection operators on spaces of Drinfeld modular forms with character of a given weight and type such that when applied to a Hecke eigenform return a Hecke eigenform whose eigensystem has been twisted by the given Dirichlet character. Unlike the classical case, however, the effect on Goss' u-expansions for these eigenforms -and even on Petrov's A-expansions -is more complicated than a simple twisting of the u− (or A−) expansion coefficients by the given character.We also introduce Eisenstein series with character for irreducible levels p and show that they and their Fricke transforms are Hecke eigenforms with a new type of A-expansion and A-expansion in the sense of Petrov, respectively. We prove congruences between certain cuspforms in Petrov's special family and the Eisenstein series and their Fricke transforms introduced here, and we show that in each weight there are as many linearly independent Eisenstein series with character as there are cusps for Γ 1 (p).

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Section: 3supporting

confidence: 75%

Section: Böckle's Eichler-shimura Relation and Goss' Questionsmentioning

confidence: 99%

Section: Böckle's Eichler-shimura Relation and Goss' Questionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Twisting eigensystems of Drinfeld Hecke eigenforms by characters

Perkins¹

2017

J. Théor. Nombres Bordeaux

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“…The submodule Sol(∆) = H is also used at other places, e.g. for studying periods, quasi-periods and logarithms [11], vectorial Drinfeld modular forms [18] or Drinfeld modules over Tate algebras [4,Def. 6.4].…”

Section: 22] -As Well As a Generalization Ofmentioning

confidence: 99%

Periods of t-modules as special values

Maurischat

2022

Journal of Number Theory

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In this article we show that all periods of uniformizable t-modules (resp. their coordinates) can be obtained via specializing a rigid analytic trivialization of a related dual t-motive at t = θ. The proof is even constructive. The central object in the construction is a subset H of the Tate algebra points of E which turns out to be isomorphic to the period lattice of E via kind of generating series in one direction and residues in the other. This isomorphism even holds for arbitrary t-modules E, even non-abelian ones.

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From the Carlitz Exponential to Drinfeld Modular Forms

Pellarin

2020

Lecture Notes in Mathematics

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This paper contains the written notes of a course the author gave at the VIASM of Hanoi in the Summer 2018. It provides an elementary introduction to the analytic naive theory of Drinfeld modular forms for the simplest 'Drinfeld modular group' GL 2 (Fq[θ]) also providing some perspectives of development, notably in the direction of the theory of vector modular forms with values in certain ultrametric Banach algebras. Contents 1. Introduction 1 2. Rings and fields 3 3. Drinfeld modules and uniformisation 11 4. The Carlitz module and its exponential 16 5. Topology of the Drinfeld upper-half plane 26 6. Some quotient spaces 37 7. Drinfeld modular forms 43 8. Eisenstein series with values in Banach algebras 50 9. Modular forms with values in Banach algebras 54 References 61 Definition 2.3. A valued field which is locally compact is called a local field. Note that R and C, with their euclidean topology, are locally compact, but not valued. Some authors define local fields as locally compact topological field for a non-discrete topology. Then, they distinguish between the non-Archimedean (or ultrametric) local fields, which are the valued ones, and the Archimedean local fields: R and C. An important property is the following. Any valued local field L of characteristic 0 is isomorphic to a finite extension of the field of p-adic numbers Q p for some p, while any local field L of characteristic p > 0 is isomorphic to a local field F q ((π)), and with q = p e for some integer e > 0. We say that π is an uniformiser. Note that |L × | = |π| Z and |π| < 1. The proof of this result is a not too difficult deduction from the following well known fact: a locally compact topological vector space over a non-trivial locally compact field has finite dimension. 2.2. Valued rings and fields for modular forms. Let C be a smooth, projective, geometrically irreducible curve over F q , together with a closed point ∞ ∈ C. We set R = A := H 0 (C \ {∞}, O C). This is the F q-algebra of the rational functions over C which are regular everywhere except, perhaps, at ∞. The choice of ∞ determines an equivalence class of valuations | • | ∞ on A in the following way. Let d ∞ be the degree of ∞, that is, the degree of the extension F of F q generated by ∞ (which is also equal to the least integer d > 0 such that τ d (∞) = ∞, where τ is the geometric Frobenius endomorphism). Then, for any a ∈ A, the degree deg(a) := dim Fq (A/aA) is a multiple −v ∞ (a)d ∞ of d ∞ and we set |a| ∞ = c −v∞(a) for c > 1, which is easily seen to be a valuation. It is well known that A is an arithmetic Dedekind domain with A × = F ×. In addition

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Vectorial Drinfeld modular forms over Tate algebras

Cited by 11 publications

References 27 publications

Twisting eigensystems of Drinfeld Hecke eigenforms by characters

Twisting eigensystems of Drinfeld Hecke eigenforms by characters

Periods of t-modules as special values

From the Carlitz Exponential to Drinfeld Modular Forms

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