scholarly journals Regularization of p-adic string amplitudes, and multivariate local zeta functions

2018 ◽  
Vol 109 (5) ◽  
pp. 1167-1204 ◽  
Author(s):  
Miriam Bocardo-Gaspar ◽  
H. García-Compeán ◽  
W. A. Zúñiga-Galindo
Keyword(s):  
Zeta Functions ◽  
Local Zeta Functions ◽  
String Amplitudes

scholarly journals Meromorphic continuation of Koba-Nielsen string amplitudes

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
M. Bocardo-Gaspar ◽  
Willem Veys ◽  
W. A. Zúñiga-Galindo
Keyword(s):  
Characteristic Zero ◽  
Zeta Functions ◽  
Open String ◽  
Local Fields ◽  
Meromorphic Continuation ◽  
Partition Functions ◽  
Interesting Connection ◽  
Bona Fide ◽  
Local Zeta Functions ◽  
String Amplitudes

Abstract In this article, we establish in a rigorous mathematical way that Koba-Nielsen amplitudes defined on any local field of characteristic zero are bona fide integrals that admit meromorphic continuations in the kinematic parameters. Our approach allows us to study in a uniform way open and closed Koba-Nielsen amplitudes over arbitrary local fields of characteristic zero. In the regularization process we use techniques of local zeta functions and embedded resolution of singularities. As an application we present the regularization of p-adic open string amplitudes with Chan-Paton factors and constant B-field. Finally, all the local zeta functions studied here are partition functions of certain 1D log-Coulomb gases, which shows an interesting connection between Koba-Nielsen amplitudes and statistical mechanics.

scholarly journals Local Zeta Functions and Koba–Nielsen String Amplitudes

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 967
Author(s):  
Miriam Bocardo-Gaspar ◽  
Hugo García-Compeán ◽  
Edgar Y. López ◽  
Wilson A. Zúñiga-Galindo
Keyword(s):  
Zeta Functions ◽  
Local Fields ◽  
Tree Level ◽  
Direct Connection ◽  
Kinematic Parameters ◽  
Crossing Symmetry ◽  
New Type ◽  
Feynman Amplitudes ◽  
Local Zeta Functions ◽  
String Amplitudes

This article is a survey of our recent work on the connections between Koba–Nielsen amplitudes and local zeta functions (in the sense of Gel’fand, Weil, Igusa, Sato, Bernstein, Denef, Loeser, etc.). Our research program is motivated by the fact that the p-adic strings seem to be related in some interesting ways with ordinary strings. p-Adic string amplitudes share desired characteristics with their Archimedean counterparts, such as crossing symmetry and invariance under Möbius transformations. A direct connection between p-adic amplitudes and the Archimedean ones is through the limit p→1. Gerasimov and Shatashvili studied the limit p→1 of the p-adic effective action introduced by Brekke, Freund, Olson and Witten. They showed that this limit gives rise to a boundary string field theory, which was previously proposed by Witten in the context of background independent string theory. Explicit computations in the cases of 4 and 5 points show that the Feynman amplitudes at the tree level of the Gerasimov–Shatashvili Lagrangian are related to the limit p→1 of the p-adic Koba–Nielsen amplitudes. At a mathematical level, this phenomenon is deeply connected with the topological zeta functions introduced by Denef and Loeser. A Koba–Nielsen amplitude is just a new type of local zeta function, which can be studied using embedded resolution of singularities. In this way, one shows the existence of a meromorphic continuations for the Koba–Nielsen amplitudes as functions of the kinematic parameters. The Koba–Nielsen local zeta functions are algebraic-geometric integrals that can be defined over arbitrary local fields (for instance R, C, Qp, Fp((T))), and it is completely natural to expect connections between these objects. The limit p tends to one of the Koba–Nielsen amplitudes give rise to new amplitudes which we have called Denef–Loeser amplitudes. Throughout the article, we have emphasized the explicit calculations in the cases of 4 and 5 points.

scholarly journals Igusa’s Local Zeta Functions and Exponential Sums for Arithmetically Non Degenerate Polynomials

2018 ◽  
Vol 30 (1) ◽  
pp. 331-354 ◽  
Author(s):  
Adriana Albarracín-Mantilla ◽  
Edwin León-Cardenal
Keyword(s):  
Exponential Sums ◽  
Zeta Functions ◽  
Local Zeta Functions

Holomorphy of local zeta functions for curves

1993 ◽  
Vol 295 (1) ◽  
pp. 635-641 ◽  
Author(s):  
Willem Veys
Keyword(s):  
Zeta Functions ◽  
Local Zeta Functions

scholarly journals Computing -series of geometrically hyperelliptic curves of genus three

2016 ◽  
Vol 19 (A) ◽  
pp. 220-234 ◽  
Author(s):  
David Harvey ◽  
Maike Massierer ◽  
Andrew V. Sutherland
Keyword(s):  
Quadratic Field ◽  
Algebraic Closure ◽  
Zeta Functions ◽  
Hyperelliptic Curves ◽  
Double Cover ◽  
Good Reduction ◽  
Usual Form ◽  
Local Zeta Functions ◽  
Curves Of Genus Three

Let$C/\mathbf{Q}$be a curve of genus three, given as a double cover of a plane conic. Such a curve is hyperelliptic over the algebraic closure of$\mathbf{Q}$, but may not have a hyperelliptic model of the usual form over$\mathbf{Q}$. We describe an algorithm that computes the local zeta functions of$C$at all odd primes of good reduction up to a prescribed bound$N$. The algorithm relies on an adaptation of the ‘accumulating remainder tree’ to matrices with entries in a quadratic field. We report on an implementation and compare its performance to previous algorithms for the ordinary hyperelliptic case.

scholarly journals Meromorphic continuations of local zeta functions and their applications to oscillating integrals

2013 ◽  
Vol 65 (2) ◽  
pp. 159-178 ◽  
Author(s):  
Toshihisa Okada ◽  
Kiyoshi Takeuchi
Keyword(s):  
Zeta Functions ◽  
Local Zeta Functions
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