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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

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.

GCD sums from Poisson integrals and systems of dilated functions

Christoph Aistleitner, István Berkes, Kristian Seip (2015)

Journal of the European Mathematical Society

Upper bounds for GCD sums of the form ∑ k , ℓ = 1 N ( gcd ( n k , n ℓ ) ) 2 α ( n k n ℓ ) α are established, where ( n k ) 1 ≤ k ≤ N is any sequence of distinct positive integers and 0 < α ≤ 1 ; the estimate for α = 1 / 2 solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for α = 1 / 2 . The method of proof is based on identifying the sum as a certain Poisson integral on a polydisc; as a byproduct, estimates for the largest eigenvalues of the associated GCD matrices are also found. The bounds for such GCD sums are used to establish...

General Dirichlet series, arithmetic convolution equations and Laplace transforms

Helge Glöckner, Lutz G. Lucht, Štefan Porubský (2009)

Studia Mathematica

In the earlier paper [Proc. Amer. Math. Soc. 135 (2007)], we studied solutions g: ℕ → ℂ to convolution equations of the form a d ∗ g ∗ d + a d - 1 ∗ g ∗ ( d - 1 ) + ⋯ + a ₁ ∗ g + a ₀ = 0 , where a ₀ , . . . , a d : ℕ → ℂ are given arithmetic functions associated with Dirichlet series which converge on some right half plane, and also g is required to be such a function. In this article, we extend our previous results to multidimensional general Dirichlet series of the form ∑ x ∈ X f ( x ) e - s x ( s ∈ ℂ k ), where X ⊆ [ 0 , ∞ ) k is an additive subsemigroup. If X is discrete and a certain solvability criterion is satisfied,...

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