CONGRUENCES AMONG POWER SERIES COEFFICIENTS OF MODULAR FORMS

Many authors have investigated the congruence relations among the coefficients of power series expansions of modular forms f in modular functions t. In a recent paper, R. Osburn and B. Sahu examine several power series expansions and prove that the coefficients exhibit congruence relations similar to the congruences satisfied by the Apéry numbers associated with the irrationality of ζ(3). We show that many of the examples of Osburn and Sahu are members of infinite families.

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Power series expansions of modular forms and p-adic interpolation of the square roots of Rankin–Selberg special values

International Journal of Number Theory ◽

10.1142/s1793042122500300 ◽

2021 ◽

pp. 1-41

Author(s):

Andrea Mori

Keyword(s):

Power Series ◽

Modular Forms ◽

Level Structure ◽

Automorphic Representation ◽

Base Change ◽

Series Expansions ◽

Square Roots ◽

Infinite Type ◽

Special Values ◽

Power Series Expansions

Let [Formula: see text] be a newform of even weight [Formula: see text] for [Formula: see text], where [Formula: see text] is a possibly split indefinite quaternion algebra over [Formula: see text]. Let [Formula: see text] be a quadratic imaginary field splitting [Formula: see text] and [Formula: see text] an odd prime split in [Formula: see text]. We extend our theory of [Formula: see text]-adic measures attached to the power series expansions of [Formula: see text] around the Galois orbit of the CM point corresponding to an embedding [Formula: see text] to forms with any nebentypus and to [Formula: see text] dividing the level of [Formula: see text]. For the latter we restrict our considerations to CM points corresponding to test objects endowed with an arithmetic [Formula: see text]-level structure. Also, we restrict these [Formula: see text]-adic measures to [Formula: see text] and compute the corresponding Euler factor in the formula for the [Formula: see text]-adic interpolation of the “square roots”of the Rankin–Selberg special values [Formula: see text], where [Formula: see text] is the base change to [Formula: see text] of the automorphic representation of [Formula: see text] associated, up to Jacquet-Langland correspondence, to [Formula: see text] and [Formula: see text] is a compatible family of grössencharacters of [Formula: see text] with infinite type [Formula: see text].

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Computing Power Series Expansions of Modular Forms

Contributions in Mathematical and Computational Sciences - Computations with Modular Forms ◽

10.1007/978-3-319-03847-6_13 ◽

2014 ◽

pp. 331-361 ◽

Cited By ~ 6

Author(s):

John Voight ◽

John Willis

Keyword(s):

Power Series ◽

Modular Forms ◽

Series Expansions ◽

Computing Power ◽

Power Series Expansions

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Power series expansions for monogenic functions

Complex Variables Theory and Application An International Journal ◽

10.1080/17476938908814340 ◽

1989 ◽

Vol 11 (3-4) ◽

pp. 215-222 ◽

Cited By ~ 2

Author(s):

F. Sommen

Keyword(s):

Power Series ◽

Series Expansions ◽

Monogenic Functions ◽

Power Series Expansions

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Parallel Software to Offset the Cost of Higher Precision

ACM SIGAda Ada Letters ◽

10.1145/3463478.3463483 ◽

2021 ◽

Vol 40 (2) ◽

pp. 59-64

Author(s):

Jan Verschelde

Keyword(s):

Power Series ◽

Parallel Algorithms ◽

Series Expansions ◽

Use Case ◽

Double Precision ◽

Algebraic Space ◽

Space Curves ◽

Computational Overhead ◽

Power Series Expansions ◽

The Cost

Hardware double precision is often insufficient to solve large scientific problems accurately. Computing in higher precision defined by software causes significant computational overhead. The application of parallel algorithms compensates for this overhead. Newton's method to develop power series expansions of algebraic space curves is the use case for this application.

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