Atkin–Lehner theory for Drinfeld modular forms and applications

Valentino, Maria

doi:10.1007/s11139-021-00465-0

Cited by 3 publications

(8 citation statements)

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“…As already mentioned in the introduction, in the positive characteristic setting, we do not have an analogue of the Petersson inner product; therefore, we need a different approach. In [4, Section 3], we defined oldforms and newforms of level t. The definition has been generalised in [6,13], but Dalal and Kumar in [9, Section 4.3] pointed out the existence of a twisted Eisenstein form both new and old (for our definition) when the level is pq (q another prime different from p).…”

Section: Hecke Operatorsmentioning

confidence: 99%

“…In [4, Section 3], we defined oldforms and newforms of level t . The definition has been generalised in [6, 13], but Dalal and Kumar in [9, Section 4.3] pointed out the existence of a twisted Eisenstein form both new and old (for our definition) when the level is ( another prime different from ). Since we shall only work with levels one and , we can still rely on our original definition, which we now recall.…”

Section: Setting and Notationmentioning

confidence: 99%

“…The present paper mainly deals with a function field counterpart of the above results. It comes after a series of papers [3,4,6,13], in which we investigated:…”

Section: Introductionmentioning

confidence: 99%

“…The present paper mainly deals with a function field counterpart of the above results. It comes after a series of papers [3, 4, 6, 13], in which we investigated: diagonalisability of Hecke operators; injectivity of Hecke operators; newforms and oldforms, for the Drinfeld modular forms . Let , respectively , be the space of Drinfeld cusp forms of weight k , type m and level one, respectively level , where is a prime ideal of and q a power of a fixed prime (see Section 2 for details of the definitions and notation appearing in this introduction).…”

Section: Introductionmentioning

confidence: 99%

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Hecke Operators and Drinfeld Cusp Forms of Level

Bandini

Valentino²

2022

Bull. Aust. Math. Soc.

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We use a linear algebra interpretation of the action of Hecke operators on Drinfeld cusp forms to prove that when the dimension of the $\mathbb {C}_\infty $ -vector space $S_{k,m}(\mathrm {{GL}}_2(\mathbb {F}_q[t]))$ is one, the Hecke operator $\mathbf {T}_t$ is injective on $S_{k,m}(\mathrm {{GL}}_2(\mathbb {F}_q[t]))$ and $S_{k,m}(\Gamma _0(t))$ is a direct sum of oldforms and newforms.

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Section: Hecke Operatorsmentioning

confidence: 99%

Section: Setting and Notationmentioning

confidence: 99%

“…The present paper mainly deals with a function field counterpart of the above results. It comes after a series of papers [3,4,6,13], in which we investigated:…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hecke Operators and Drinfeld Cusp Forms of Level

Bandini

Valentino²

2022

Bull. Aust. Math. Soc.

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“…Bandini and Valentino [3][4][5]19]) defined p-oldforms and p-newforms, and studied some of their properties. In [3], they defined T-oldforms S T-old k,l (Γ 0 (T)) and T-newforms S T-new k,l (Γ 0 (T)) for p = (T).…”

Section: Introductionmentioning

confidence: 99%

Notes on Atkin–lehner Theory for Drinfeld Modular Forms

Dalal¹,

Kumar

2022

Bull. Aust. Math. Soc.

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We settle a part of the conjecture by Bandini and Valentino [‘On the structure and slopes of Drinfeld cusp forms’, Exp. Math.31(2) (2022), 637–651] for $S_{k,l}(\Gamma _0(T))$ when $\mathrm {dim}\ S_{k,l}(\mathrm {GL}_2(A))\leq 2$ . We frame and check the conjecture for primes $\mathfrak {p}$ and higher levels $\mathfrak {p}\mathfrak {m}$ , and show that a part of the conjecture for level $\mathfrak {p} \mathfrak {m}$ does not hold if $\mathfrak {m}\ne A$ and $(k,l)=(2,1)$ .

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Fourier coefficients and slopes of Drinfeld modular forms

Bandini,

Valentino

2023

Int. J. Number Theory

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Let [Formula: see text] be a Drinfeld modular form of level [Formula: see text] which is an eigenform for the Hecke operator [Formula: see text] ([Formula: see text] a prime of [Formula: see text]). We study the relations between the Fourier coefficients of [Formula: see text] and the [Formula: see text]-adic valuation of its eigenvalue (slope). We use formulas for some of the Fourier coefficients of [Formula: see text] to provide bounds and estimates on the slopes and, in particular, to find necessary conditions for “large” slopes, whose existence is closely connected with conjectures on oldforms and newforms.

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Atkin–Lehner theory for Drinfeld modular forms and applications

Cited by 3 publications

References 10 publications

Hecke Operators and Drinfeld Cusp Forms of Level

Hecke Operators and Drinfeld Cusp Forms of Level

Notes on Atkin–lehner Theory for Drinfeld Modular Forms

Fourier coefficients and slopes of Drinfeld modular forms

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