# Journal of the Ramanujan Mathematical Society

Volume 39, Issue 1, March 2024 pp. 79–90.

Greenberg's generalized conjecture and pairings
of p-units in the 4p-cyclotomic field

**Authors**:
Hiroki Sumida-Takahashi, Naoki Furuya and Kodai Kitano

**Author institution:**
Department of Mathematical Sciences, Tokushima University, Minamijosanjima-cho 2-1, Tokushima 770-8506, Japan.

**Summary: **
We study pairings of p-units in the 4p-cyclotomic field, following results on
the p-cyclotomic field by McCallumâ€“Sharifi. We compute zeros of the parings
for each odd prime number p < 2{16} = 65, 536, which give a sufficient
condition for Greenberg's generalized conjecture. We also explain rare zeros
which do not appear in the p-cyclotomic field.

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