scholarly journals Linear independence in linear systems on elliptic curves

2021 ◽  
Vol 96 (2) ◽  
pp. 199-213
Author(s):  
Bradley Brock ◽  
Bruce Jordan ◽  
Bjorn Poonen ◽  
Anthony Scholl ◽  
Joseph Wetherell
Keyword(s):  
Elliptic Curves ◽  
Linear Systems ◽  
Linear Independence

scholarly journals Chebyshev’s bias for analytic L-functions

2019 ◽  
Vol 169 (1) ◽  
pp. 103-140 ◽  
Author(s):  
LUCILE DEVIN
Keyword(s):  
Elliptic Curves ◽  
Prime Number ◽  
General Framework ◽  
Linear Independence ◽  
Logarithmic Density ◽  
Independence Properties

AbstractWe discuss the generalizations of the concept of Chebyshev’s bias from two perspectives. First, we give a general framework for the study of prime number races and Chebyshev’s bias attached to general L-functions satisfying natural analytic hypotheses. This extends the cases previously considered by several authors and involving, among others, Dirichlet L-functions and Hasse–Weil L-functions of elliptic curves over Q. This also applies to new Chebyshev’s bias phenomena that were beyond the reach of the previously known cases. In addition, we weaken the required hypotheses such as GRH or linear independence properties of zeros of L-functions. In particular, we establish the existence of the logarithmic density of the set $ \{x \ge 2:\sum\nolimits_{p \le x} {\lambda _f}(p) \ge 0\}$ for coefficients (λf(p)) of general L-functions conditionally on a much weaker hypothesis than was previously known.

ELLIPTIC CURVES AND -ADIC ELLIPTIC TRANSCENDENCE

2021 ◽  
pp. 1-6
Author(s):  
DUC HIEP PHAM
Keyword(s):  
Elliptic Curves ◽  
Short Proof ◽  
Linear Independence ◽  
Elliptic Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Algebraic Numbers ◽  
Algebraic Dependence ◽  
Necessary And Sufficient

Abstract We prove a necessary and sufficient condition for isogenous elliptic curves based on the algebraic dependence of p-adic elliptic functions. As a consequence, we give a short proof of the p-adic analogue of Schneider’s theorem on the linear independence of p-adic elliptic logarithms of algebraic points on two nonisogenous elliptic curves defined over the field of algebraic numbers.

Elliptic Curves

1997 ◽  
Author(s):  
Henry McKean ◽  
Victor Moll
Keyword(s):  
Elliptic Curves
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