Ribet, Class groups and Galois representations

Reference: Ribet, “Class Groups and Galois Representations”. https://math.berkeley.edu/~ribet/herbrand.pdf

Alternate definition of Unsorted/class group : the group of fractional ideals.
- Defined as \({\mathbf{Z}}[\operatorname{mSpec}{\mathcal{O}}_K]\) (the free \({\mathbf{Z}}{\hbox{-}}\)module on maximal ideals) modulo principal fractional ideal
What is the maximal unramified extension, i.e. the Hilbert class field?
The Artin map from class field theory : \(\operatorname{Cl} (K) \xrightarrow{\sim} { \mathsf{Gal}} (H/K)\) where \({\mathfrak{p}}\mapsto \operatorname{Frob}_{{\mathfrak{p}}}\).
Set \(G_k \coloneqq{ \mathsf{Gal}} ({ \mkern 1.5mu\overline{\mkern-1.5mu \mathbf{Q} \mkern-1.5mu}\mkern 1.5mu }/K)\), then \(\operatorname{Cl} (K)\) is a quotient of \(G_k^{{\mathsf{Ab}}}\).
- Equivalently, \(\operatorname{Cl} (K) {}^{ \vee }\leq G_k {}^{ \vee }\) where \(({-}) {}^{ \vee }\coloneqq\mathop{\mathrm{Hom}}_{{\mathsf{Top}}{\mathsf{Grp}}}({-}, {\mathbf{C}}^{\times})\). I.e. take continuous characters.
Open question: are there infinitely many quadratic fields \(K\) for which \(\operatorname{Cl} (K) = 0\)
Dedekind zeta function

\begin{align*} \zeta_K \coloneqq\prod_{{\mathfrak{p}}\in \operatorname{mSpec}{\mathcal{O}}_K}(1 - N({\mathfrak{p}})^{-s} )^{-1} .\end{align*}

Note: guessing about the indexing set here. Original source just indexes over \({\mathfrak{p}}\)…
\(\mathop{\mathrm{Res}}_{s=1} \zeta_K\) involves \(h_k \coloneqq{\sharp} \operatorname{Cl} (K)\).
Serre stresses: use functional equation to look at \(s=0\) instead of \(s=1\)! Leads to cleaner/simpler formulas.
Kummer theory proved FLT for exponent \(p\) for regular primes, i.e. \(\gcd(h_K, p) = 1\).
- Kummer’s criterion: \(p\) is regular iff \(p\) divides none of the numerators of some Bernoulli numbers.
What is the Teichmuller character?
See Birch and Swinnerton-Dyer conjecture
- Define the Tate-Shafarevich group as \(\Sha(E/{\mathbf{Q}})\) for an elliptic curve, then

\begin{align*} {\sharp}\Sha(E/{\mathbf{Q}}) \underset{?}{=} \qty{ L(E, 1) \over \Omega} \qty{ {\sharp}(E({\mathbf{Q}}))^2 \over \prod_\ell w_\ell} ,\end{align*}

where \(\Omega\) is a period and \(w_\ell\) are the local Tamagawa numbers.

Need lower bounds of sizes of class groups. Might be able to use elliptic curves, congruences between Eisenstein series and cusp forms on \(\operatorname{U}(2, 2)\) – can obtain 4-dimensional Galois representations that lead to nontrivial elements of \(\Sha\).

Ribet, “Class groups and Galois representations”