arXiv subjects

Igor E Shparlinski

Publications and source records attributed to Igor E Shparlinski.

3 recordsLinked to original sources

arXiv2011

Character sums with division polynomials

We obtain nontrivial estimates of quadratic character sums of division polynomials $Ψ_n(P)$, $n=1,2, ...$, evaluated at a given point $P$ on an elliptic curve over a finite field of $q$ elements. Our bounds are nontrivial if the order of $P$ is at least $q^{1/2 + ε}$ for some fixed $ε> 0$. This work is motivated by an open question about statistical indistinguishability of some cryptographically relevant sequences which has recently been brought up by K. Lauter and the second author.

math.NT↗

arXiv2006

Arithmetic properties of the Ramanujan function

We study some arithmetic properties of the Ramanujan function $τ(n)$, such as the largest prime divisor $P(τ(n))$ and the number of distinct prime divisors $ω(τ(n))$ of $τ(n)$ for various sequences of $n$. In particular, we show that \hbox{$P(τ(n)) \geq (\log n)^{33/31 + o(1)}$} for infinitely many $n$, and \begin{equation*} P(τ(p)τ(p^2)τ(p^3)) > (1+o(1))\frac{\log\log p\log\log\log p} {\log\log\log\log p} \end{equation*} for every prime $p$ with \hbox{$τ(p)\neq 0$}.

math.NT↗

arXiv2004

Prime divisors of sequences associated to elliptic curves

We consider the primes which divide the denominator of the x-coordinate of a sequence of rational points on an elliptic curve. It is expected that for every sufficiently large value of the index, each term should be divisible by a primitive prime divisor, one that has not appeared in any earlier term. Proofs of this are known in only a few cases. Weaker results in the general direction are given, using a strong form of Siegel's Theorem and some congruence arguments. Our main result is applied to the study of prime divisors of Somos sequences.

math.NT↗