A condition for the rationality of certain elliptic modular forms over primes dividing the level

Andrea Mori

Displaying similar documents to “A condition for the rationality of certain elliptic modular forms over primes dividing the level”

Classical and overconvergent modular forms of higher level

Robert F. Coleman (1997)

Journal de théorie des nombres de Bordeaux

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We define the notion overconvergent modular forms on $Γ_{1} (N p^{n})$ where $p$ is a prime, $N$ and $n$ are positive integers and $N$ is prime to $p$ . We show that an overconvergent eigenform on $Γ_{1} (N p^{n})$ of weight $k$ whose $U_{p}$ -eigenvalue has valuation strictly less than $k - 1$ is classical.

An integrality criterion for elliptic modular forms

Andrea Mori (1990)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Let $f$ be an elliptic modular form level of N. We present a criterion for the integrality of $f$ at primes not dividing N. The result is in terms of the values at CM points of the forms obtained applying to $f$ the iterates of the Maaß differential operators.

Overconvergent modular forms

Vincent Pilloni (2013)

Annales de l’institut Fourier

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We give a geometric definition of overconvergent modular forms of any $p$ -adic weight. As an application, we reprove Coleman’s theory of $p$ -adic families of modular forms and reconstruct the eigencurve of Coleman and Mazur without using the Eisenstein family.

Dimensions of spaces of modular forms for $Γ_{H} (N)$

Jordi Quer (2010)

Acta Arithmetica

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Gauss–Manin connections for $p$ -adic families of nearly overconvergent modular forms

Robert Harron, Liang Xiao (2014)

Annales de l’institut Fourier

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We interpolate the Gauss–Manin connection in $p$ -adic families of nearly overconvergent modular forms. This gives a family of Maass–Shimura type differential operators from the space of nearly overconvergent modular forms of type $r$ to the space of nearly overconvergent modular forms of type $r + 1$ with $p$ -adic weight shifted by $2$ . Our construction is purely geometric, using Andreatta–Iovita–Stevens and Pilloni’s geometric construction of eigencurves, and should thus generalize to higher rank...

Bounds on sup-norms of half-integral weight modular forms

Eren Mehmet Kıral (2014)

Acta Arithmetica

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Bounding sup-norms of modular forms in terms of the level has been the focus of much recent study. In this work the sup-norm of a half-integral weight cusp form is bounded in terms of the level: we prove that $| | y^{κ / 2} {f ̃ | |}_{\infty} ≪_{ε, κ} N^{1 / 2 - 1 / 18 + ε} | | y^{κ / 2} f ̃ {| |}_{L^{2}}$ for a modular form f̃ of level 4N and weight κ, a half-integer.

Rational torsion points on Jacobians of modular curves

Hwajong Yoo (2016)

Acta Arithmetica

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Let p be a prime greater than 3. Consider the modular curve X₀(3p) over ℚ and its Jacobian variety J₀(3p) over ℚ. Let (3p) and (3p) be the group of rational torsion points on J₀(3p) and the cuspidal group of J₀(3p), respectively. We prove that the 3-primary subgroups of (3p) and (3p) coincide unless p ≡ 1 (mod 9) and $3^{(p - 1) / 3} \equiv 1 (m o d p)$ .

Computing modular degrees using $L$ -functions

Christophe Delaunay (2003)

Journal de théorie des nombres de Bordeaux

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We give an algorithm to compute the modular degree of an elliptic curve defined over $ℚ$ . Our method is based on the computation of the special value at $s = 2$ of the symmetric square of the $L$ -function attached to the elliptic curve. This method is quite efficient and easy to implement.

Hecke operators in half-integral weight

Soma Purkait (2014)

Journal de Théorie des Nombres de Bordeaux

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In [], Shimura introduced modular forms of half-integral weight, their Hecke algebras and their relation to integral weight modular forms via the Shimura correspondence. For modular forms of integral weight, Sturm’s bounds give generators of the Hecke algebra as a module. We also have well-known recursion formulae for the operators $T_{p^{ℓ}}$ with $p$ prime. It is the purpose of this paper to prove analogous results in the half-integral weight setting. We also give an explicit formula for how operators...