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 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

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

scholarly journals Universal Equivalence and Majority of Probabilistic Programs over Finite Fields

2022 ◽  
Vol 23 (1) ◽  
pp. 1-42
Author(s):  
Gilles Barthe ◽  
Charlie Jacomme ◽  
Steve Kremer
Keyword(s):  
Programming Language ◽  
Finite Fields ◽  
Differential Privacy ◽  
Zeta Functions ◽  
Equivalence Problem ◽  
Linear Recurrence ◽  
Universal Equivalence ◽  
Local Zeta Functions ◽  
Recurrence Sequences ◽  
Probabilistic Programs

We study decidability problems for equivalence of probabilistic programs for a core probabilistic programming language over finite fields of fixed characteristic. The programming language supports uniform sampling, addition, multiplication, and conditionals and thus is sufficiently expressive to encode Boolean and arithmetic circuits. We consider two variants of equivalence: The first one considers an interpretation over the finite field F q , while the second one, which we call universal equivalence, verifies equivalence over all extensions F q k of F q . The universal variant typically arises in provable cryptography when one wishes to prove equivalence for any length of bitstrings, i.e., elements of F 2 k for any k . While the first problem is obviously decidable, we establish its exact complexity, which lies in the counting hierarchy. To show decidability and a doubly exponential upper bound of the universal variant, we rely on results from algorithmic number theory and the possibility to compare local zeta functions associated to given polynomials. We then devise a general way to draw links between the universal probabilistic problems and widely studied problems on linear recurrence sequences. Finally, we study several variants of the equivalence problem, including a problem we call majority, motivated by differential privacy. We also define and provide some insights about program indistinguishability, proving that it is decidable for programs always returning 0 or 1.

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