Boyd and Rodriguez Villegas's Mahler measure identity for a conductor-15 elliptic curve

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Let m(P)m(P) denote the Mahler measure of a Laurent polynomial PP. Let EE be the elliptic curve

E:(1+x)(1+y)(1+1x)(1+1y)=1,E: (1+x)(1+y)(1+\frac{1}{x})(1+\frac{1}{y})=1,

which has conductor 1515. Boyd and Rodriguez Villegas's conjecture. The Mahler measure satisfies

m((1+x)(1+y)+z)=−2L′(E,−1).m((1+x)(1+y)+z)=-2L'(E,-1).

This identity is presented as an example of the expected relationship between Mahler measures of exact polynomials and special values of LL-functions, motivated by Beilinson's conjectures and work of Boyd and Rodriguez Villegas. The question mark in the source indicates that the equality is conjectural.

References

Primary source

François Brunault, “On the Mahler measure of (1+x)(1+y)+z”, arXiv:2305.02992 (2023).

Additional references

2 papers in this index state this conjecture (2020–2023). The statement above is taken from the most recent of them; the others are arXiv:2009.07614.

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