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CM points, class numbers, and the Mahler measures of
x
3
+
y
3
+
1
β
k
x
y
x^3+y^3+1-kxy
x
3
+
y
3
+
1
β
k
x
y
Authors
Xuejun Guo
Zhengyu Tao
Publication date
19 October 2023
Publisher
View
on
arXiv
Abstract
We study the Mahler measures of the polynomial family
Q
k
(
x
,
y
)
=
x
3
+
y
3
+
1
β
k
x
y
Q_k(x,y) = x^3+y^3+1-kxy
Q
k
β
(
x
,
y
)
=
x
3
+
y
3
+
1
β
k
x
y
using the method previously developed by the authors. An algorithm is implemented to search for CM points with class numbers
β©½
3
\leqslant 3
β©½
3
, we employ these points to derive interesting formulas that link the Mahler measures of
Q
k
(
x
,
y
)
Q_k(x,y)
Q
k
β
(
x
,
y
)
to
L
L
L
-values of modular forms. As a by-product, three conjectural identities of Samart are confirmed. For
k
=
729
Β±
405
3
3
k=\sqrt[3]{729\pm405\sqrt{3}}
k
=
3
729
Β±
405
3
β
β
, we also prove an equality that expresses a
2
Γ
2
2\times 2
2
Γ
2
determinant with entries the Mahler measures of
Q
k
(
x
,
y
)
Q_k(x,y)
Q
k
β
(
x
,
y
)
as some multiple of the
L
L
L
-value of two isogenous elliptic curves over
Q
(
3
)
\mathbb{Q}(\sqrt{3})
Q
(
3
β
)
.Comment: 17 pages, 2 tables. Comments are welcom
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oai:arXiv.org:2310.12510
Last time updated on 06/01/2024