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 $$\Sha$$.