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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 groups.

Gauss's ₂F₁ hypergeometric function and the congruent number elliptic curve

Ahmad El-Guindy, Ken Ono (2010)

Acta Arithmetica

Genera of arithmetic Fuchsian groups

Stefan Johansson (1998)

Acta Arithmetica

Genera of congruence subgroups in Q-quaternion algebras.

David A. Cox, Walter R. Parry (1984)

Journal für die reine und angewandte Mathematik

Generalization of the symplectic modular group

S. Kolmer (1966)

Acta Arithmetica

Generalizations of Dedekind sums and their reciprocity laws

Yumiko Nagasaka, Kaori Ota, Chizuru Sekine (2003)

Acta Arithmetica

Generalized continued fractions and orbits under the action of Hecke triangle groups

Elise Hanson, Adam Merberg, Christopher Towse, Elena Yudovina (2008)

Acta Arithmetica

Generalized Dedekind sums.

L. Carlitz (1964)

Mathematische Zeitschrift

Generalized Dedekind sums, correlation of L-series, and the Galois module of cot(π/N)cot(mπ/N)

Kurt Girstmair (2003)

Acta Arithmetica

Generalized divisor problem for new forms of higher level

Krishnarjun Krishnamoorthy (2022)

Czechoslovak Mathematical Journal

Suppose that $f$ is a primitive Hecke eigenform or a Mass cusp form for $Γ_{0} (N)$ with normalized eigenvalues $λ_{f} (n)$ and let $X > 1$ be a real number. We consider the sum $𝒮_{k} (X) : = \sum_{n < X} \sum_{n = n_{1}, n_{2}, ..., n_{k}} λ_{f} (n_{1}) λ_{f} (n_{2}) ... λ_{f} (n_{k})$ and show that $𝒮_{k} (X) ≪_{f, ϵ} X^{1 - 3 / (2 (k + 3)) + ϵ}$ for every $k \geq 1$ and $ϵ > 0$ . The same problem was considered for the case $N = 1$ , that is for the full modular group in Lü (2012) and Kanemitsu et al. (2002). We consider the problem in a more general setting and obtain bounds which are better than those obtained by the classical result of Landau (1915) for $k \geq 5$ . Since the result is valid for arbitrary...

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Galois representations, Hecke operators, and the $mod- p$ cohomology of $GL (3, ℤ)$ with twisted coefficients.

Galois representations into GL2(Zp[[X]]) attached to ordinary cusp forms.

Galois representations modulo p and cohomology of Hilbert modular varieties

Galois representations of dihedral type over ℚₚ

Galois representations of octahedral type and 2-coverings of elliptic curves.

Gaps Between Consecutive Zeros of the Riemann Zeta-Function on the Critical Line.

Gauss Sums and Elliptic Functions: II. The Quartic Sum.

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

Gauss's ₂F₁ hypergeometric function and the congruent number elliptic curve

Genera of arithmetic Fuchsian groups

Genera of congruence subgroups in Q-quaternion algebras.

Generalization of the symplectic modular group

Generalizations of Dedekind sums and their reciprocity laws

Generalized continued fractions and orbits under the action of Hecke triangle groups

Generalized Dedekind sums.

Generalized Dedekind sums, correlation of L-series, and the Galois module of cot(π/N)cot(mπ/N)

Generalized divisor problem for new forms of higher level

Generalized Eisenstein series and modified Dedekind sums.

Generalized Eta-Functions and Certain Ray Class Invariants of Real Quadratic Fields.

Generalized Hecke groups and Hecke polygons.

Currently displaying 621 – 640 of 2107