تقرير
Mahler measure of a non-reciprocal family of elliptic curves
| العنوان: | Mahler measure of a non-reciprocal family of elliptic curves |
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| المؤلفون: | Samart, Detchat |
| المصدر: | Q. J. Math. 74 (2023), 1187--1208 |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Number Theory, 11R06, 11F67, 33C75 |
| الوصف: | In this article, we study the logarithmic Mahler measure of the one-parameter family \[Q_\alpha=y^2+(x^2-\alpha x)y+x,\] denoted by $m(Q_\alpha)$. The zero loci of $Q_\alpha$ generically define elliptic curves $E_\alpha$ which are $3$-isogenous to the family of Hessian elliptic curves. We are particularly interested in the case $\alpha\in (-1,3)$, which has not been considered in the literature due to certain subtleties. For $\alpha$ in this interval, we establish a hypergeometric formula for the (modified) Mahler measure of $Q_\alpha$, denoted by $\tilde{n}(\alpha).$ This formula coincides, up to a constant factor, with the known formula for $m(Q_\alpha)$ with $|\alpha|$ sufficiently large. In addition, we verify numerically that if $\alpha^3$ is an integer, then $\tilde{n}(\alpha)$ is a rational multiple of $L'(E_\alpha,0)$. A proof of this identity for $\alpha=2$, which is corresponding to an elliptic curve of conductor $19$, is given. Comment: Corrigendum: sign error in Theorem 1 and miscalculation in the proof of Lemma 9 fixed |
| نوع الوثيقة: | Working Paper |
| DOI: | 10.1093/qmath/haad016 |
| URL الوصول: | http://arxiv.org/abs/2301.05390 |
| رقم الأكسشن: | edsarx.2301.05390 |
| قاعدة البيانات: | arXiv |
| DOI: | 10.1093/qmath/haad016 |
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