A nilpotent group and its elliptic curve: Non-uniformity of local zeta functions of groups

2001 ◽  
Vol 126 (1) ◽  
pp. 269-288 ◽  
Author(s):  
Marcus du Sautoy
Keyword(s):  
Nilpotent Group ◽  
Elliptic Curve ◽  
Zeta Functions ◽  
Local Zeta Functions ◽  
Zeta Functions Of Groups

Stability results for local zeta functions of groups algebras, and modules

2017 ◽  
Vol 165 (3) ◽  
pp. 435-444 ◽  
Author(s):  
TOBIAS ROSSMANN
Keyword(s):  
Group Theory ◽  
Functional Equations ◽  
Zeta Functions ◽  
Local Zeta Functions ◽  
Single Formula ◽  
Almost All ◽  
Stability Results ◽  
Zeta Functions Of Groups ◽  
Local Base

AbstractVarious types of local zeta functions studied in asymptotic group theory admit two natural operations: (1) change the prime and (2) perform local base extensions. Often, the effects of both of the preceding operations can be expressed simultaneously in terms of a single formula, a statement made precise using what we call local maps of Denef type. We show that assuming the existence of such formulae, the behaviour of local zeta functions under variation of the prime in a set of density 1 in fact completely determines these functions for almost all primes and, moreover, it also determines their behaviour under local base extensions. We discuss applications to topological zeta functions, functional equations, and questions of uniformity.

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.

A Titchmarsh divisor problem for elliptic curves

2015 ◽  
Vol 160 (1) ◽  
pp. 167-189 ◽  
Author(s):  
PAUL POLLACK
Keyword(s):  
Elliptic Curve ◽  
Complex Multiplication ◽  
Zeta Functions ◽  
Dirichlet Character ◽  
Good Reduction ◽  
Heuristic Argument ◽  
Mean Values ◽  
Average Size ◽  
Image Position ◽  
Ordinary Reduction

AbstractLet E/Q be an elliptic curve with complex multiplication. We study the average size of τ(#E(Fp)) as p varies over primes of good ordinary reduction. We work out in detail the case of E: y2 = x3 − x, where we prove that $$\begin{equation} \sum_{\substack{p \leq x \\p \equiv 1\pmod{4}}} \tau(\#E({\bf{F}}_p)) \sim \left(\frac{5\pi}{16} \prod_{p > 2} \frac{p^4-\chi(p)}{p^2(p^2-1)}\right)x, \quad\text{as $x\to\infty$}. \end{equation}$$ Here χ is the nontrivial Dirichlet character modulo 4. The proof uses number field analogues of the Brun–Titchmarsh and Bombieri–Vinogradov theorems, along with a theorem of Wirsing on mean values of nonnegative multiplicative functions.Now suppose that E/Q is a non-CM elliptic curve. We conjecture that the sum of τ(#E(Fp)), taken over p ⩽ x of good reduction, is ~cEx for some cE > 0, and we give a heuristic argument suggesting the precise value of cE. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions, we prove that this sum is ≍Ex. The proof uses combinatorial ideas of Erdős.

Sign in / Sign up

Export Citation Format

Share Document