Mahler measures
Boyd and Deninger have discovered experimentally in 1997:
m x +
1
x
+ y +
1
y
+ 1

 ?
= L0
(E, 0) =
15
4π2
L(E, 2)
where E is the elliptic curve with affine equation x + 1
x + y + 1
y + 1 = 0.
Boyd also found families of identities, for example
m x +
1
x
+ y +
1
y
+ k

 ?
= L0
(Ek , 0) (k ∈ Z\{0, ±4}).
Only finitely many identities are proven: |k| ∈ {1, 2, 3, 5, 8, 12, 16}.
The proof requires Ek to be parametrised by modular units.
More precisely, we need ϕ: X1(Nk ) → Ek such that ϕ∗
(x) and ϕ∗
(y) are
modular units.
m(Pk )
Jensen
=
Z
γ
η(x, y) =
Z
e
γ
η(ϕ∗
(x), ϕ∗
(y))
Rogers-
Zudilin
= L0
(fk , 0).
2