On the existence of square roots in certain rings of power series

1965 ◽  
Vol 158 (2) ◽  
pp. 82-89 ◽  
Author(s):  
William G. Brown
Keyword(s):  
Power Series ◽  
Square Roots ◽  
Rings Of Power Series

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

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].

scholarly journals POWER SERIES EXPANSIONS OF MODULAR FORMS AND THEIR INTERPOLATION PROPERTIES

2011 ◽  
Vol 07 (02) ◽  
pp. 529-577 ◽  
Author(s):  
ANDREA MORI
Keyword(s):  
Power Series ◽  
Modular Form ◽  
Modular Forms ◽  
Power Series Expansion ◽  
Base Change ◽  
Square Roots ◽  
Type K ◽  
Power Series Expansions ◽  
Expansion Coefficients ◽  
Definition Of

We define a power series expansion of an holomorphic modular form f in the p-adic neighborhood of a CM point x of type K for a split good prime p. The modularity group can be either a classical conguence group or a group of norm 1 elements in an order of an indefinite quaternion algebra. The expansion coefficients are shown to be closely related to the classical Maass operators and give p-adic information on the ring of definition of f. By letting the CM point x vary in its Galois orbit, the rth coefficients define a p-adic K×-modular form in the sense of Hida. By coupling this form with the p-adic avatars of algebraic Hecke characters belonging to a suitable family and using a Rankin–Selberg type formula due to Harris and Kudla along with some explicit computations of Watson and of Prasanna, we obtain in the even weight case a p-adic measure whose moments are essentially the square roots of a family of twisted special values of the automorphic L-function associated with the base change of f to K.

Computing complex zeros of polynomials and power series

2011 ◽  
Author(s):  
Jean-Marc Couveignes
Keyword(s):  
Power Series ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Zeros Of Polynomials ◽  
Complex Data ◽  
Square Roots ◽  
Complex Roots ◽  
Roots Of Polynomials ◽  
Complex Zeros

The purpose of this chapter is twofold. First, it will prove two theorems (5.3.1 and 5.4.2) about the complexity of computing complex roots of polynomials and zeros of power series. The existence of a deterministic polynomial time algorithm for these purposes plays an important role in this book. More important, it will also explain what it means to compute with real or complex data in polynomial time. The chapter first recalls basic definitions in computational complexity theory, it then deals with the problem of computing square roots. The more general problem of computing complex roots of polynomials is treated thereafter and, finally, the chapter studies the problem of finding zeros of a converging power series.

Undecidable wreath products and skew power series fields

1998 ◽  
Vol 63 (1) ◽  
pp. 237-246 ◽  
Author(s):  
Françoise Delon ◽  
Patrick Simonetta
Keyword(s):  
Power Series ◽  
Large Class ◽  
Division Ring ◽  
Wreath Products ◽  
Ordered Group ◽  
Division Rings ◽  
Skew Fields ◽  
Image Position ◽  
Rings Of Power Series

We prove the undecidability of a very large class of restricted and unrestricted wreath products (Theorem 1.2), and of some skew fields of power series (Section2). Both undecidabilities are obtained by interpreting some enrichments of twisted wreath products, which are themselves proved to be undecidable (Proposition 1.1).We consider division rings of power series in various languages:We show (Theorem 2.8) that every power series division ring k((B)), whose field of constants k is commutative and whose ordered group of exponents is noncommutative with a convex center, is undecidable in every extension of the language of rings where the valuation and the ordered group B are definable.For certain k and B we prove here the undecidability of the structurewhere X↾k((B))xB is the restriction of the multiplication to k((B)) Χ B,and γ is a given conjugation of k((B)). This shows that we cannot hope to improve our previous result, a sort of Ax-Kochen-Ershov principle for power series division rings, which ensures thatis decidable for every decidable solvable B.

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