• 60s, Mazur’s Remarks on the Alexander polynomial: primes are analogous to knots.
• Impetus: Thomson 1867, On vortex atoms.
• Idea: topological distinction of knots may explain distinct atoms, and links for molecules.
• P. G. Tait (and Kirkman? Around 1880) tabulates many knots.
• Rigorous tools to distinguish: Wirtinger, Dehn, Alexander in late 1800s/early 1990s.
• Dehn (1910), Wirtinger (1905): a presentation of the knot group \(\pi_1({\mathbb{R}}^3\setminus K)\) in terms of generators and relations.
• Alexander (1927): Topological invariants of knots and links, observes difficulties distinguishing group presentations, introduces the simpler Alexander polynomial.
  • E.g. for the trefoil, \(p(x) = 1 - x + x^2\).
  • He gives a simple algorithm computing it as the determinant of a matrix, and shows how it can be derived from the knot group.
  • Succeeded in distinguishing many tabulated knots (but not all).
  • See SnapPy! You can fly around the hyperbolic space of a knot complement, and compute a large number of invariants.

• NT: consider solving \(x^2+y^2=z^2\).
  • Factor \(x^2+y^2 = (x-iy)(x+iy)\) and write \(x\pm iy = (m\pm in)^2\) to obtain \(x^2=m^2-n^2, y=2mn, z = m^2+n^2\).
  • This yields all solutions, using unique factorization in \({\mathbb{Z}}[i]\).
• Consider \(x^p+y^p=z^p\) (Fermat)
  • To solve: factor \(z^p = \prod_k x+\zeta_p^k y\).
• Lame (1847): an incorrect proof that falsely assumes \(L_p \mathrel{\vcenter{:}}={\mathbb{Z}}[\zeta_p]\) is a UFD, since \({ \operatorname{cl}} (L_p) \neq 1\) for \(p=23\) (shown by Kummer) and factorization fails.
• Kummer fixes this proof assuming a weaker condition: \(p\notdivides { \operatorname{cl}} (L_p)\), i.e. this is regular.
  • This fails for \(p=37\).
  • Kummer proves FLT fails for regular primes.
• Kummer (1850) shows \(p\) is irregular iff \(p\) divides on of the first \(p-3\) coefficients of \(x\over e^x-1\).
• Iwasawa (59, 62) gives a polynomial \(P_p\) whose degree quantifies the irregularity of \(p\); \(p\) is irregularity iff \(P_p = 1\) is constant.
  • This definition uses a Galois group \(G_p\).

Mazur (63) observes similarity between the Alexander polynomial and Iwasawa’s polynomial. Both are pure group theory, and the construction arises from action of \(G^{\operatorname{ab}}\curvearrowright[G, G]^{\operatorname{ab}}\) and taking a characteristic polynomial. Analogies:

• \(G_p\) is similar to \(\pi_1(S^3\setminus K)\),
• the linking number is similar to the quadratic residue \({ \left( {p} \over {q} \right) }\).
• Symmetry of linking numbers is analogous to quadratic reciprocity.
• \(\operatorname{Spec}{\mathbb{Z}}/p{\mathbb{Z}}\hookrightarrow\operatorname{Spec}{\mathbb{Z}}\) is like a knot in a 3-manifold.
• \(\operatorname{Spec}{\mathbb{Z}}\) is like \(S^3\), with primes corresponding to embedded knots.

Analogies (partially) due to Mazur, guided by results of Artin and Tate.

Defining \(G_p\): take all rings over \(k\) with some finite rank \(r\) which admit exactly \(r\) embeddings in \({ \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu }\), take their union, take ring automorphisms. Constructing such rings: find polynomials \(p\) with \({\operatorname{disc}}_p = p^k\) for some \(k\). Problem: \(G_p\) is very mysterious! Probably finitely generated but infinite, the only element we can produce is complex conjugation, \(z\mapsto \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu\). Tate (62) shows that \(G_p\) cohomologically looks like a knot group, currently the strongest formal analogy.

Idea: compare statistical properties of random knots vs random primes.

Note: for \(L\) a field, \begin{align*} { {L}_{\scriptscriptstyle \square} } \mathrel{\vcenter{:}}=\left\{{a^2 {~\mathrel{\Big\vert}~}a\in L}\right\} .\end{align*}

Recall that any symplectic vector space is isomorphic to \(\qty{{\mathbb{R}}^{2g}, J\mathrel{\vcenter{:}}={ \begin{bmatrix} {0} & {\operatorname{id}_g} \\ {- \operatorname{id}_g} & {0} \end{bmatrix} }}\). Write \begin{align*} {\operatorname{SP}}(V) = \left\{{g: V\to V{~\mathrel{\Big\vert}~}{\left\langle {gx},~{gy} \right\rangle} = {\left\langle {x},~{y} \right\rangle}}\right\} = \left\{{A\in \operatorname{GL}_{2g}({\mathbb{R}}) {~\mathrel{\Big\vert}~}A^tJA = J}\right\} .\end{align*} Note \begin{align*} {\operatorname{SP}}_2({\mathbb{R}}) = \left\{{M\in \operatorname{GL}_2({\mathbb{R}}) {~\mathrel{\Big\vert}~}\operatorname{det}M = 1}\right\} ,\end{align*} and \({\operatorname{SP}}_{2g}({\mathbb{R}})\) is connected but \({\operatorname{SP}}_{2g}({\mathbb{R}}) \simeq{\mathbb{R}}^2\setminus\left\{{0}\right\}\). Let \(p: G\to {\operatorname{SP}}_{2g}({\mathbb{R}})\) be the universal cover; since the base is a topological group, picking any \(p^{-1}(1)\) yields an essentially unique group structure on \(G\) by analytically continuing the group law. In fact \(G\) is a Lie group and \(\ker p \cong {\mathbb{Z}}\in Z(G)\) is central, so there is a SES \begin{align*} {\mathbb{Z}}\to G\to {\operatorname{SP}}_{2g}({\mathbb{R}}) .\end{align*} Note that there are not faithful finite-dimensional reps of \(G\), but there are infinite-dimensional reps important to quantization in physics.

A theorem of Meyer on surfaces: let \(\Sigma\) be a (compact closed oriented smooth) surface and \(\rho: \pi_1 \Sigma\to {\operatorname{SP}}_{2g}{\mathbb{R}}\); this corresponds to a local system, so form the twisted cohomology \(H^1(\Sigma; \rho)\). Using the cup product and symplectic pairing, one can produce a symmetric pairing \begin{align*} L: H^1(\Sigma;\rho){ {}^{ \scriptstyle\otimes_{{\mathbb{R}}}^{2} } }\to {\mathbb{R}} .\end{align*} There is an isomorphism \((H^1, L)\cong ({\mathbb{R}}^{p+q}, \operatorname{diag}_{p}(-1) \oplus \operatorname{diag}_q(1))\), so \(\operatorname{sig}(\Sigma, \rho) \mathrel{\vcenter{:}}=\operatorname{sig}L \mathrel{\vcenter{:}}= p-q\) is an interesting invariant.

Recall that 4-manifolds \(M\) also admit a signature on \(H^2(M; {\mathbb{R}})\). Let \(M\to\Sigma\) be surface bundle over a surface – if \(M\) is a product, \(\operatorname{sig}M = 0\). Chern-Hirzebruch-Serre: there is a monodromy morphism \(\pi_1\Sigma\to {\operatorname{SP}}_{2g}{\mathbb{R}}\) which is trivial if \(\operatorname{sig}M = 0\). In fact \(\operatorname{sig}(M) = \operatorname{sig}(\Sigma, \rho)\), so this situation naturally arises for fibered 4-manifolds.

Recall \(\pi_1 \Sigma_g = \left\langle{a_i,b_i {~\mathrel{\Big\vert}~}\prod_i [a_i, b_i] = e}\right\rangle\). Consider trying to form a lift:

Link to Diagram

Note \(m\mathrel{\vcenter{:}}=\prod [\tilde\rho a_i, \tilde\rho b_i] \in \ker p \cong {\mathbb{Z}}\), and it turns out that \begin{align*} \operatorname{sig}(\Sigma, \rho) = p-q = 4m .\end{align*} The fact that \(\ker p \in Z(G)\) was essential in making sure this doesn’t depend on choices. Note that this only determines \(m\) up to sign!

Toward generalizing, the above central extension determines a class \(b\in H^2({\operatorname{SP}}_{2g}({\mathbb{R}}); {\mathbb{Z}})\) and \(p^* b\in H^2( \Sigma; {\mathbb{Z}})\). Using the pairing against the fundamental class \([\Sigma]\) yields \begin{align*} \operatorname{sig}(\Sigma, \rho) = 4 \int_\Sigma p^* b .\end{align*}

When replacing \({\mathbb{R}}\) with \(L\), replace \({\mathbb{Z}}\) by \(W_L\), the Witt group of quadratic forms over \(L\).

Consider now varying the situation in a 1-dimensional family – take \(\Sigma\to M\to S^1\) a surface bundle over a circle with fibers \(\Sigma_t\), each yielding \(\rho_t: \pi_1\Sigma_t \to {\operatorname{SP}}_{2g}({\mathbb{R}})\) for \(t\in S^1\). Each \(t\) yields a quadratic vector space \(H^1(\Sigma_t,\rho_t)\) of signature \((p, q)\). Monodromy yields an element \(m\in \mathop{\mathrm{Aut}}H^1(\Sigma_0, \rho_0) \subseteq {\operatorname{O}}_{p, q}\). If \(p,q>0\) then \({\sharp}\pi_0{\operatorname{O}}_{p, q} = 4\) – which connected component does this land in? How to separate the components: there is a determinant map \begin{align*} \operatorname{det}: {\operatorname{O}}_{p, q} \to \left\{{\pm 1}\right\} .\end{align*} There is a spinor norm \begin{align*} {\mathrm{spinornorm}}: {\operatorname{O}}_{p, q} &\to \left\{{\pm 1}\right\} = {\mathbb{R}}^{\times}/{ {{\mathbb{R}}}_{\scriptscriptstyle \square} }^{\times}\\ \mathrm{reflection}_v &\mapsto {\left\langle {v},~{v} \right\rangle} .\end{align*} This works with \({\mathbb{R}}\) replaced by \(L\), using the fact that \({\operatorname{O}}_{p, q}\) is generated by reflections. For \((V, L)\) a symmetric space, one gets \begin{align*} \operatorname{det}: {\operatorname{O}}(V) &\to \left\{{\pm 1}\right\} \end{align*} and \begin{align*} {\mathrm{spinornorm}}: {\operatorname{O}}(V) &\to L^{\times}/{ {L}_{\scriptscriptstyle \square} }^{\times} .\end{align*}

For \(X\in{\mathsf{sm}}\mathop{\mathrm{proj}}{\mathsf{Var}}_{/ {{\mathbb{F}}_q}}\) and \(\rho: \pi_1 X\to {\operatorname{SP}}_{2g} k\) for \(k\) finite containing \(\sqrt{q}\), there is an associated \(L{\hbox{-}}\)function \(L(X, \rho; T)\in k(T)\) where \(T\approx q^{-s}\). There is a functional equation relating \(T\rightleftharpoons{1\over qT}\). Evaluate at the center of symmetry \(T = {1\over\sqrt q}\) to define \(L(X, \rho) \mathrel{\vcenter{:}}= L(X, \rho; 1/\sqrt q)\).

Interpretation: \(L(X,\rho) = ab^2\) where \(a\) is simple and \(b\) is complicated. BSD gives a conjectural formula for this which includes a lot of squares. So there is a cohomological obstruction to the existence of a square root of \(L(X, \rho)\).

On the proof: try to pass validity from one known example \((X,\rho)\) to other examples \((X', \rho')\) with \(X_{/ {{\mathbb{F}}_q}} , X'_{/ {{\mathbb{F}}_q'}}\) and images in \({\operatorname{SP}}_{2g}(k), {\operatorname{SP}}_{2g}(k')\) respectively.

• Pass from \(q\) to \(q'\) using a moduli space of pairs \((X,\rho)\) where a topological theorem controls the fiber over \({\mathbb{C}}\).
• Pass from \(k\) to \(k'\) using compatible local systems from NT.

Final comparison:

• For 2-manifolds, \(\operatorname{sig}(\Sigma, \rho) = \int_\Sigma p^* b\).
• For 3-manifolds, \({\mathrm{RT}}(M,\rho) = \int_M p^* c\).
• Symplectic \(L{\hbox{-}}\)functions: ?