Using Continued Fractions to Compute Iwasawa Lambda Invariants of\n Imaginary Quadratic Number Fields

2014/03/16 by Jordan Schettler, Schettler, Jordan
Mathematics · #History and Theory of Mathematics #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.1403.3946

Abstract

Let \ℓ>3 be a prime such that \ℓ \≡ 3 pmod4 and\n\ℚ(\√(\ℓ)) has class number 1. Then Hirzebruch and Zagier\nnoticed that the class number of \ℚ(\√(-\ℓ)) can be expressed as\nh(-\ℓ) = (1/3)(b₁+b₂ + \⋯ + b_m) - m where the b_i are partial\nquotients in the `minus' continued fraction expansion \√(\ℓ) = [[b₀;\n\b₁, b₂, \…, b_m]]. For an odd prime p \≠ \ℓ, we prove\nan analogous formula using these b_i which computes the sum of Iwasawa lambda\ninvariants \λ_p(-\ℓ)+\λ_p(-4) of \ℚ(\√(-\ℓ)) and\n\ℚ(\√(-1)). In the case that p is inert in\n\ℚ(\√(-\ℓ)), the formula pleasantly simplifies under some\nadditional technical assumptions.\n

Citations

• Computation of Iwasawa Lambda invariants for imaginary quadratic fields

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