Ranks and Parity of Ranks of Curves and Abelian Surfaces: MA842 at BU Spring 2019

Theorem 1.1. Descent theorem (see Thm. VIII 3.1).

Let \(A\) be an abelian group, suppose that there exists a function

with the following properties:

1. Let \(Q \in A\) then there is a constant \(c_1\) depending on \(Q\) and \(A\) such that
   \begin{equation*} h(P+Q) = 2h(P) + c_1,\,\forall P \in A\text{.} \end{equation*}
2. There is an integer \(m \ge 2\) and a constant \(c_2\) depending on \(A\) s.t.
   \begin{equation*} h(mP) \ge m^2 h(P) - c_2,\,\forall P\in A\text{.} \end{equation*}
3. For every constant \(c_3\text{,}\) the set
   \begin{equation*} \{P\in A : h(P) \le c_3\} \end{equation*}
   is finite.

suppose further that for the \(m\) in \(2.\) we have \(A/mA\) is finite. Then \(A\) is finitely generated.

Theorem 1.4. Weak Mordell-Weil.

Let \(K\) be a number field \(E/K\) an elliptic curve, \(m \ge 2\) then

Lemma 1.5.

Let \(L/K\) be a finite Galois extension, if

is finite then so is

Proposition 1.7.

\(\kappa\) is well defined, bilinear, the kernel in the first argument is \(m E(K)\) and in the second argument is \(G_{\overline K/L}\) where \(L = K(\lb m \rb \inv E(K))\) is the compositum of all fields \(\kappa(x(Q), y(Q))\) as \(Q\) ranges over all the points of \(E(\overline K)\) s.t. \(m Q \in E(K)\text{.}\)

Hence the Kummer pairing induces a perfect bilinear pairing

i.e. the map

is an isomorphism.

Proposition 1.9.

Let \(E/K\) be an elliptic curve

for \(P \in E(K)\) have \(\frac 12 P \in E(\overline K)\) s.t. \(\frac 12 P \oplus \frac 12 P = P\text{.}\)

1. \(K(\frac 12 P) /K\) is a Galois extension and \(\Gal{K(\frac 12 P)}{K} = C_2 \times C_2\) from Lemma 1.
2. \begin{equation*} \phi_P \colon \absgal{K} \to E(K)[2] \end{equation*}
   \begin{equation*} g \mapsto Q^\sigma - Q = g(\frac 12 P) - \frac 12 P \end{equation*}
   is well defined and has kernel \(\Gal{K}{K(\frac 12 P)}\text{.}\)
3. \begin{equation*} \phi \colon E(K) / 2E(K) \to \Hom_{cts}(\absgal{K}, E(K)[2]) \end{equation*}
   \begin{equation*} P \mapsto \phi_P \end{equation*}
   is well defined and injective. Now \(\phi_P\) is continuous by \(2.\) and so
   \begin{equation*} \phi_{P\oplus Q} (g) = g(\frac 12 (P\oplus Q)) - (\frac 12 P\oplus \frac 12 Q) \end{equation*}
   \begin{equation*} = g(\frac 12 P) \oplus g(\frac 12Q) - \frac 12 P \ominus \frac 12 Q \end{equation*}
   \begin{equation*} = \phi_P(g) \oplus \phi_Q(g) \end{equation*}
   a homomorphism.
   \begin{equation*} \phi_{2Q}(g) = g(\frac 12 2 Q)) - \frac12 2(Q) = g(Q) - Q = 0 \end{equation*}
   for all \(g\in \absgal K\) if \(Q \in E(K)\) so this is well defined. For injectivity:
   \begin{equation*} \phi_P(g) = 0 \implies g(\frac 12 P) = \frac 12 P \forall g \in \absgal K \end{equation*}
   \begin{equation*} \implies \frac 12 P \in E(K) \implies P \in 2 E(K) \end{equation*}
   which gives injectivity.
4. \begin{equation*} \eta \colon \Hom_{cts}(\absgal K , E(K) [2] ) \to K^\times/{K^\times}^2 \times K^\times/{K^\times}^2 \times K^\times/{K^\times}^2 \end{equation*}
   \begin{equation*} \psi \mapsto\psi_\alpha ,\psi_\beta, \psi_\gamma \end{equation*}
   \begin{equation*} \psi(g )\in \{0, (\alpha,0)\} \subseteq E(K) \iff g \in \Gal{\overline K }{K(\sqrt {\psi_\alpha})} \end{equation*}
   then \(\eta\) is an injective homomorphism. It is an isomorphism to the subgroup of triples \(a,b,c\) s.t. \(abc \in {K^\times}^2\text{.}\) Proof:
   \begin{equation*} \Hom_{cts}(\absgal K, C_2) \simeq K^\times/{K^\times}^2 \end{equation*}
   with \(\psi\) s.t. \(\ker \psi = \Gal{\overline K}{ K\sqrt d} \leftrightarrow d\text{.}\) It is an isomorphism:
   \begin{equation*} \ker \psi_i = \Gal{\overline K}{ K(\sqrt {d_i})},\,i =1,2 \end{equation*}
   \begin{equation*} \ker \psi_1 \psi_2 = \Gal{\overline K}{ K(\sqrt {d_1d_2})} \end{equation*}
   Now apply this to \(E(K) \lb 2\rb = C_2 \times C_2\) to get an isomorphism to \(K^\times / {K^\times}^2 \times K^\times / {K^\times}^2 \text{.}\) Record this third homomorphism to get \(\eta\text{.}\)
5. If \(P = (x_0, y_0) \in E(K)\) then
   \begin{equation*} \eta(\phi_P) = (x_0 - \alpha, x_0 - \beta, x_0 -\gamma)\text{.} \end{equation*}
   Proof sketch: If
   \begin{equation*} E\colon y^2 = x^3 + Ax^2 + Bx \end{equation*}
   then for \(Q = (x_0, y_0) \in E(K)\text{.}\)
   \begin{equation*} 2Q = \left(\left( \frac {x_0 - B}{2y_0}\right)^2 , \ldots\right) \end{equation*}
   Hence if \(2Q = P = (x_1, y_1)\) then \(\sqrt{x_1} \in K(\frac 12 P)\text{.}\) So if
   \begin{equation*} E \colon y^2 = (x-\alpha)(x-\beta)(x-\gamma) \end{equation*}
   then
   \begin{equation*} P = (x_2, y_2) \end{equation*}
   then
   \begin{equation*} \sqrt{x_2 - \alpha} , \sqrt{x_2 - \beta} , \sqrt{x_2 - \gamma} \in K(\frac 12 P) \end{equation*}
   \begin{equation*} K(\sqrt{x_2 - \alpha}) , K(\sqrt{x_2 - \beta}) , K(\sqrt{x_2 - \gamma}) \subseteq K(\frac 12 P) \end{equation*}
   \begin{equation*} \implies K(\frac 12 P) = K(\sqrt{x_2 - \alpha},\sqrt{x_2 - \beta} , \sqrt{x_2 - \gamma}) \end{equation*}

Theorem 1.12. Complete 2-decent.

Let \(K\) be a field of characteristic 0 and

The map

replacing \(x_0 -\alpha\) with \((x_0 - \beta) (x_0 - \gamma)\) if 0.

Triples \((a,b,c)\) that lie in the image satisfy \(abc \in {K^\times}^2\text{.}\) A triple \(a,b,c\) with \(abc \in {K^\times}^2\) lies in the image iff it is in the image of \(E(K)\lb 2\rb\) or

is soluble with \(z_i \in K^\times\text{.}\) In which case

iii) If \(K\) is a number field and \((a,b,c)\) is in the image then

only ramifies at primes dividing \(2(\alpha - \beta)(\alpha - \gamma)(\beta - \gamma)\text{.}\)

Lemma 1.14.

Let \(n \ge 1\)

1. \begin{equation*} \psi \colon E(K)/ nE(K) \to \{ K \subseteq F \subseteq \overline K\} \end{equation*}
   \begin{equation*} P\mapsto K(\frac 1n P, E[n]) \end{equation*}
   is well defined.
2. \(K(\frac 1n P, E\lb n \rb ) / K\) only ramifies at \(\ideal p | n \Delta_E\text{.}\)
3. \begin{equation*} \Gal{K(\frac 1n P, E [n] )}{K} \le \ZZ/n \times \ZZ/n \end{equation*}
4. There are only finitely many fields satisfying 2. and 3. so \(\image \psi\) is finite.

Theorem 1.17.

If

is an exact sequence of \(G\)-modules then

Lemma 1.18.

1. \(\psi\) is finite-to-one (gives Mordell-Weil)
2. Let
   \begin{equation*} \phi_P \colon G_K \to E[n] \end{equation*}
   \begin{equation*} \phi_P (gh) = \phi_P(g) + g \phi_P(h) \end{equation*}
   is a skew (or crossed) homomorphism. If \((\frac 1n P ) '\) is another choice of \(\frac 1n P\) and \(\phi_P'\) is the corresponding skew homomorphism, then
   \begin{equation*} \phi_P - \phi_P ' \end{equation*}
   is of the form
   \begin{equation*} g\mapsto T \ominus gT \end{equation*}
   where \(T \in E\lb n\rb\text{.}\)
3. \(\phi_P\) factors through
   \begin{equation*} \Gal{K(\frac 1n P, E\lb n \rb)}{K}\text{.} \end{equation*}
4. \begin{equation*} \phi\colon E(K) / nE(K) \to Z/B \end{equation*}
   \begin{equation*} P \mapsto \phi_P \end{equation*}
   is an injective homomorphism. Where
   \begin{equation*} Z = \{\text{skew homs } G_K \to E[n]\} \end{equation*}
   \begin{equation*} B = \{\text{skew homs } G_K \to E[n] \text{ of the form }g\mapsto T\ominus gT,\,T\in E\lb n\rb\}\text{.} \end{equation*}

Claim 1.19.

corresponding to principal homogeneous spaces for \(E\) (genus 1 curves whose jacobian is \(E\))

Theorem 1.21.

The twists of \(C/K\) up to \(K\)-isomorphism are in 1-1 correspondence with elements of

where

Proposition 1.25.

Let \(E/K\) and \(C/K\) be a principal homogeneous space for \(E/K\text{.}\) Fix a point \(p_0 \in C\) and define a map

1. \(\theta\) is an isomorphism over \(K(p_0)\text{.}\) In particular \(C/K\) is a twist of \(E/K\text{.}\)
2. \(\forall p,q \in C\)
   \begin{equation*} q- p = \theta\inv(q) - \theta \inv (p)\text{.} \end{equation*}
3. \(\theta\) is a morphism over \(K\text{.}\)

Proposition 1.27.

Let \(C/K\) be a PHS for \(E/K\) then \(C/K\) is in the trivial class \(\iff C(K) \ne \emptyset\text{.}\)

Theorem 1.28.

Let \(E/K\) then there is a natural bijection after fixing \(p_0 \in C\)

Theorem 1.33.

There is an exact sequence

1. \begin{equation*} 0 \to E(K)/mE(K) \to \Sel^m(E/K) \to \Sha(E/K)[m] \to 0 \end{equation*}
2. \(\Sel^m(E/K)\) is finite.

Theorem 1.36.

The sign in the functional equation of \(L(E,s)\) is equal to the global root number of \(E\text{.}\)

This is defined by

the local root numbers defined in terms of the local galois representations. Non-trivial to understand, but manageable.

Conjecture 1.37. Parity conjecture.

Theorem 1.39.

Assume \(\Sha\) is finite, let \(\phi \colon E \to E'\) be an isogeny whose degree is not divisible by \(\characteristic(K)\text{,}\) then

Lemma 1.42.

Assume \(\Sha\) is finite, let

be a \(K\)-rational isogeny of degree \(d\text{.}\)

Write \(n = \rk_E = \rk_{E'}\text{.}\) Pick a basis \(\Lambda = \langle P_1, \ldots, P_n\rangle\) for

write \(\Lambda'\) for a basis of \(E'(K)/\tors\text{.}\) Write \(\phi^\vee \colon E' \to E\) for the dual isogeny s.t. \(\phi\phi^\vee = \lb d\rb\text{.}\)

using the following fact

Then

Conjecture 1.43. \(p\)-parity.

Theorem 1.46.

Let \(E/K\) be an elliptic curve, \(K\) a number field, if \(\Psi \) is s.t. \(\Psi \Psi^\vee = \lb p \rb\) then