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Irmak Balçık

Publications and source records attributed to Irmak Balçık.

4 recordsLinked to original sources
arXiv2025

Average rank of elliptic curves over function fields

Let $q$ be a prime with $q \geq 5$. We show that the average rank of elliptic curves over a function field $\mathbb{F}_{q}(t)$, when ordered by naive height, is bounded above by $25/14 \approx 1.8$. Our result improves the previous upper bound of $2.3$ proven by Brumer. The upper bound obtained is less than $2$, which shows that a positive proportion of elliptic curves has either rank $0$ or $1$. The proof adapts the work of Young, which shows that under the assumption of the General Riemann Hypothesis for $L$-functions of elliptic curves, the average rank for the family of elliptic curves over the rational numbers is bounded above by $ 25/14 \approx 1.8$.

math.NT↗
arXiv2024

The Effect of Quadratic Base Change on Torsion of Elliptic Curves

Let $K$ be a quadratic number field and let $E$ be an elliptic curve defined over $K$ such that $E[2] \not\subseteq E(K).$ In this paper, we study the effect of quadratic base change on $E(K)_{\text{tor}}.$ Moreover, for a given elliptic curve $E/K$ with prescribed torsion group over $K,$ (no restriction on its $2$-torsion part) we describe a fast algorithm to find all quadratic extensions $L/K$ in which $E(K)_{\text{tor}} \subsetneq E(L)_{\text{tor}}$ and describe $E(L)_{\text{tor}}$ in each such case. In particular, we determine the growth of $E(K)_{\text{tor}}$ upon quadratic base change when $K$ is any quadratic cyclotomic field, which completes the earlier work of the second author.

math.NT↗