Elizabeth Athaide ; Emma Cardwell ; Christy Thompson - CLASS NUMBER FORMULAS FOR CERTAIN BIQUADRATIC FIELDS

hrj:12573 - Hardy-Ramanujan Journal, February 28, 2024, Volume 46 - 2023 - https://doi.org/10.46298/hrj.2024.12573

Authors: Elizabeth Athaide 1; Emma Cardwell 2; Christy Thompson 3

  • 1 Department of Mathematics [MIT]
  • 2 Department of Mathematics [Cambridge]
  • 3 Department of Mathematics [Stanford]

We consider the class numbers of imaginary quadratic extensions $F(\sqrt{-p})$, for certain primes $p$, of totally real quadratic fields $F$ which have class number one. Using seminal work of Shintani, we obtain two elementary class number formulas for many such fields. The first expresses the class number as an alternating sum of terms that we generate from the coefficients of the power series expansions of two simple rational functions that depend on the arithmetic of $F$ and $p$. The second makes use of expansions of $1/p$, where $p$ is a prime such that $p \equiv 3 \pmod{4}$ and $p$ remains inert in $F$. More precisely, for a generator $\varepsilon_F$ of the totally positive unit group of $\mathcal{O}_F$, the base-$\varepsilon_{F}$ expansion of $1/p$ has period length $\ell_{F,p}$, and our second class number formula expresses the class number as a finite sum over disjoint cosets of size $\ell_{F,p}$.

Volume: Volume 46 - 2023
Published on: February 28, 2024
Accepted on: February 28, 2024
Submitted on: November 20, 2023
Keywords: [MATH]Mathematics [math]
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  • REU Site: Algebra and Number Theory at Emory University; Funder: National Science Foundation; Code: 2002265
  • REU Site: Arithmetic Geometry, Number Theory, and Representation Theory at the University of Virginia; Funder: National Science Foundation; Code: 2147273

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