From the classical to the noncommutative Iwasawa theory (for totally real number fields)

Kakde, Mahesh

doi:10.1017/cbo9780511984440.005

Non-Abelian Fundamental Groups and Iwasawa Theory 2011

DOI: 10.1017/cbo9780511984440.005

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From the classical to the noncommutative Iwasawa theory (for totally real number fields)

Mahesh Kakde

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2024

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SPANNING TREES IN $\mathbb {Z}$ -COVERS OF A FINITE GRAPH AND MAHLER MEASURES

PENGO,

VALLIÈRES

2024

J. Aust. Math. Soc.

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Using the special value at $u=1$ of Artin–Ihara L-functions, we associate to every $\mathbb {Z}$ -cover of a finite connected graph a polynomial, which we call the Ihara polynomial. We show that the number of spanning trees for the finite intermediate graphs of such a cover can be expressed in terms of the Pierce–Lehmer sequence associated to a factor of the Ihara polynomial. This allows us to express the asymptotic growth of the number of spanning trees in terms of the Mahler measure of this polynomial. Specialising to the situation where the base graph is a bouquet or the dumbbell graph gives us back previous results in the literature for circulant and I-graphs (including the generalised Petersen graphs). We also express the p-adic valuation of the number of spanning trees of the finite intermediate graphs in terms of the p-adic Mahler measure of the Ihara polynomial. When applied to a particular $\mathbb {Z}$ -cover, our result gives us back Lengyel’s calculation of the p-adic valuations of Fibonacci numbers.

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SPANNING TREES IN $\mathbb {Z}$ -COVERS OF A FINITE GRAPH AND MAHLER MEASURES

PENGO,

VALLIÈRES

2024

J. Aust. Math. Soc.

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From the classical to the noncommutative Iwasawa theory (for totally real number fields)

Cited by 1 publication

References 31 publications

SPANNING TREES IN $\mathbb {Z}$ -COVERS OF A FINITE GRAPH AND MAHLER MEASURES

SPANNING TREES IN $\mathbb {Z}$ -COVERS OF A FINITE GRAPH AND MAHLER MEASURES

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