Galois Representations, Fontaine–Mazur, and the Langlands¶
Job: develop the Galois representations from geometry (Tate module / étale cohomology), the Fontaine–Mazur conjecture (geometric Galois reps), the modularity / Langlands / Serre, the \(p\)-adic Hodge theory (Fontaine / crystalline / semistable / prismatic), and the Tate–Shafarevich / Selmer / Cassels–Tate — with the modularity of \(X_0(11)\) / the Fontaine–Mazur / the geometric Galois rep as the worked example.
Core¶
Galois representations from geometry¶
The Galois representation \(\rho_{X,\ell}:\mathrm{Gal}(\overline{K}/K)\to\mathrm{GL}(H^i_{\mathrm{\acute{e}t}}(X_{\overline{K}},\mathbb{Q}_\ell))\) (the geometric Galois rep, the étale cohomology the source). The Tate module \(T_\ell(E)=\lim E[\ell^n]\) (the \(\ell\)-adic rep of the elliptic curve, the \(H^1\)).
| Galois rep / Tate module | Description |
|---|---|
| \(\rho_{X,\ell}\) | \(\mathrm{Gal}(\overline{K}/K)\to\mathrm{GL}(H^i_{\mathrm{\acute{e}t}})\) (the geometric / the étale cohomology) |
| Tate module \(T_\ell(E)\) | \(\lim E[\ell^n]\) (the \(\ell\)-adic rep of the elliptic curve, the \(H^1\)) |
| Properties | the unramified / the crystalline / the semisimple (the Fontaine–Mazur) |
The Fontaine–Mazur conjecture and the geometric Galois reps¶
Conjecture (Fontaine–Mazur). The Galois rep \(\rho:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_n(\overline{\mathbb{Q}}_\ell)\) that is unramified outside finitely many primes and de Rham at \(\ell\) is geometric (arises from geometry / the motives).
| Fontaine–Mazur / geometric / Fontaine | Description |
|---|---|
| Fontaine–Mazur | the unramified + de Rham \(\Rightarrow\) geometric (arises from the geometry / the motives) |
| Geometric | the comes from the geometry / the étale cohomology (the Fontaine–Mazur the converse) |
| Fontaine \(p\)-adic Hodge | the crystalline / the semistable / the de Rham, the Fontaine rings \(B_{\mathrm{cris}},B_{\mathrm{st}},B_{\mathrm{dR}}\) |
The modularity and the Langlands¶
| Modularity / Langlands / Serre | Description |
|---|---|
| Modularity (Wiles) | \(\rho_{E,\ell}\cong\rho_{f,\ell}\) for a modular form \(f\) (the Wiles / the Taylor–Wiles) |
| Langlands / automorphy | the Galois rep the automorphic (the langlands, the global Langlands) |
| Serre's modularity (Khare–Wintenberger) | the mod \(\ell\) Galois reps the modular mod \(\ell\) (the Serre / the Khare–Wintenberger) |
The \(p\)-adic Hodge and the comparison¶
| \(p\)-adic Hodge / comparison / prismatic | Description |
|---|---|
| \(p\)-adic Hodge (Fontaine) | \(H^i_{\mathrm{\acute{e}t}}\otimes B_{\mathrm{dR}}\cong H^i_{\mathrm{dR}}\otimes B_{\mathrm{dR}}\) (the \(p\)-adic comparison) |
| Crystalline / semistable | the good / the semistable reduction, the \(p\)-adic Hodge the local |
| Prismatic (Bhatt–Scholze) | the prismatic cohomology the generalization (the prismatic) |
The Tate–Shafarevich and the Selmer¶
| Sha / Selmer / Cassels–Tate | Description |
|---|---|
| Tate–Shafarevich \(\mathrm{Sha}(E/K)\) | \(\ker(H^1(K,E)\to\prod_v H^1(K_v,E))\) (the locally trivial torsors, the Hasse failure) |
| Selmer group \(\mathrm{Sel}^{(n)}(E/K)\) | the \(E(K)/nE(K)\), the descent, the rank / the BSD |
| Cassels–Tate pairing | the duality on \(\mathrm{Sha}\) (the Cassels / the Tate, the BSD) |
See it¶
Modularity of the elliptic curve \(X_0(11)\) and the Fontaine–Mazur / the geometric Galois rep. The elliptic curve \(E=X_0(11)\) (the conductor 11, the modular curve / the modular form \(f\) of weight 2 level 11). The Galois rep \(\rho_{E,\ell}:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_2(\mathbb{Q}_\ell)\) (the Tate module, the \(H^1_{\mathrm{\acute{e}t}}\), the rank-2 \(\ell\)-adic rep). The Fontaine–Mazur: the \(\rho_{E,\ell}\) is unramified outside \(\{11,\ell\}\) and crystalline at \(\ell\) (the good reduction at \(\ell\ne11\), the crystalline at \(\ell\) by the Fontaine), hence geometric (arises from the elliptic curve / the étale cohomology, the Fontaine–Mazur the converse). The modularity: \(\rho_{E,\ell}\cong\rho_{f,\ell}\) (the Galois rep of the modular form \(f\), the Wiles / the modularity theorem). The \(p\)-adic Hodge: at \(\ell\), the \(\rho_{E,\ell}|_{\mathrm{Gal}(\overline{\mathbb{Q}}_\ell/\mathbb{Q}_\ell)}\) is crystalline (the good reduction, the Fontaine \(B_{\mathrm{cris}}\), the comparison \(H^1_{\mathrm{\acute{e}t}}\otimes B_{\mathrm{cris}}\cong H^1_{\mathrm{cris}}\)). The key point: the Galois rep of \(X_0(11)\) the geometric (Fontaine–Mazur) the modular (Wiles) — the arithmetic geometry the Galois / the modularity / the \(p\)-adic Hodge the one object.
Check¶
Show that the Galois rep \(\rho_{E,\ell}\) of the elliptic curve \(E=X_0(11)\) is unramified outside \(\{11,\ell\}\) and crystalline at \(\ell\), and explain the Fontaine–Mazur.
The Galois rep \(\rho_{E,\ell}:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_2(\mathbb{Q}_\ell)\) (the Tate module \(T_\ell(E)=\lim E[\ell^n]\), the \(H^1_{\mathrm{\acute{e}t}}\)). The unramified outside \(\{11,\ell\}\): the \(E\) has the good reduction at \(p\ne11,\ell\) (the Néron–Ogg–Shafarevich: the good reduction \(\Leftrightarrow\) the inertia \(I_p\) acts trivially on \(T_\ell\), the unramified at \(p\ne11,\ell\)). The crystalline at \(\ell\): the \(E\) has the good reduction at \(\ell\ne11\) (the crystalline by the Fontaine: the good reduction \(\Rightarrow\) the crystalline, the \(\rho|_{\mathrm{Gal}(\overline{\mathbb{Q}}_\ell/\mathbb{Q}_\ell)}\) the crystalline). The Fontaine–Mazur: the \(\rho_{E,\ell}\) unramified outside \(\{11,\ell\}\) + crystalline (de Rham) at \(\ell\) \(\Rightarrow\) geometric (arises from the elliptic curve \(E\) / the étale cohomology, the Fontaine–Mazur the converse — the rep the geometric, the Fontaine–Mazur the criterion).
Explain the Tate–Shafarevich group and the Cassels–Tate pairing, and the role in the BSD.
The Tate–Shafarevich \(\mathrm{Sha}(E/K)=\ker(H^1(K,E)\to\prod_v H^1(K_v,E))\) (the locally trivial torsors under \(E\), the principal homogeneous spaces \(E\)-torsors locally trivial at all \(v\) but globally nontrivial — the Hasse failure). The Cassels–Tate pairing: the \(\mathrm{Sha}\times\mathrm{Sha}\to\mathbb{Q}/\mathbb{Z}\) (the duality, the Cassels / the Tate, the perfect pairing on \(\mathrm{Sha}/\mathrm{Sha}_{\mathrm{div}}\), the nondegenerate). The BSD: the \(|\mathrm{Sha}|\) in the refined BSD (the leading coefficient \(=|\mathrm{Sha}|\,\Omega\,R\,\prod c_p/|E(K)_{\mathrm{tors}}|^2\)), the Cassels–Tate the duality the input (the \(|\mathrm{Sha}|\) the square / the finiteness the BSD). The role: the Sha the Hasse failure / the locally trivial torsors, the Cassels–Tate the duality, the BSD the \(|\mathrm{Sha}|\) the analytic (the finiteness the open, the BSD the \(|\mathrm{Sha}|\)).
Pitfalls¶
- The Fontaine–Mazur the geometric (the unramified + de Rham \(\Rightarrow\) geometric, the conjecture — the partial, not the full). The Fontaine–Mazur the conjecture (the unramified + de Rham \(\Rightarrow\) geometric), the partial the proven cases, not the full.
- The modularity the Wiles (the elliptic / the rank-2, the general / the higher-dimensional the partial, the Fontaine–Mazur the converse). The modularity theorem the elliptic / the rank-2, the higher-dimensional the partial, the Fontaine–Mazur the converse (the geometric \(\Rightarrow\) the modular).
- The Serre / the Khare–Wintenberger the mod \(\ell\) modularity (the mod \(\ell\) Galois reps the modular mod \(\ell\)), the Serre the conjecture, the Khare–Wintenberger the proof.
- The \(p\)-adic Hodge the Fontaine (the crystalline / semistable / de Rham the local, the prismatic the general / the Bhatt–Scholze the recent). The Fontaine the local (the \(B_{\mathrm{cris}},B_{\mathrm{st}},B_{\mathrm{dR}}\), the local), the prismatic the general (the Bhatt–Scholze, the prismatic, the recent).
- The Sha the conjecturally finite (the BSD the \(|\mathrm{Sha}|\) the analytic, the finiteness the open). The \(|\mathrm{Sha}|\) the BSD (the analytic), the finiteness the open (the Cassels–Tate the duality, the BSD the square / the finiteness).
Refs¶
- Fontaine–Mazur, "Geometric Galois representations" (1995); the Fontaine–Mazur conjecture
- Wiles, "Modular elliptic curves and FLT" (Annals 1995); Khare–Wintenberger, "Serre's modularity conjecture" (2009)
- Fontaine, "Il n'y a pas de variété abélienne sur \(\mathbb{Z}\)" (Invent. 1985); the \(p\)-adic Hodge / the rings
- Cassels, "Arithmetic on curves of genus 1" (the Cassels–Tate pairing); Tate, the Sha / the BSD
Next¶
AG index · Foundations · Mordell–Weil & BSD · Diophantine & Arakelov · Étale & Weil · Langlands · Prismatic cohomology · SAG / chromatic