Mordell–Weil, Heights, and the Birch–Swinnerton-Dyer Conjecture¶

Job: develop the Mordell–Weil theorem / canonical height / descent / Selmer, the BSD conjecture (rank = $L$-order, refined BSD / Tamagawa / regulator / Sha), the Gross–Zagier / Kolyvagin partial BSD, and the Hasse–Weil $L$-function / modularity / $p$-adic BSD — with the BSD for $y^2=x^3-x$ (rank 0) / the Heegner point (rank 1) as the worked example.

Core¶

The Mordell–Weil theorem and the canonical height¶

Theorem (Mordell–Weil). $A(K)\cong A(K)_{\mathrm{tors}}\oplus\mathbb{Z}^r$ (the finitely-generated abelian group for an abelian variety $A$ over a number field $K$).

The canonical height $\hat{h}:A(K)\to\mathbb{R}_{\ge0}$ (the Néron–Tate quadratic form, $\hat{h}(P)=\lim_{n\to\infty}h(nP)/n^2$, the kernel the torsion). The descent: the Selmer group $\mathrm{Sel}^{(n)}(A/K)=\ker(H^1(K,A[n])\to H^1(K,A))$ controls $A(K)/nA(K)$, the weak Mordell–Weil $|A(K)/nA(K)|<\infty$, the descent / the rank computation.

Mordell–Weil / height / descent	Description
Mordell–Weil	$A(K)\cong A(K)_{\mathrm{tors}}\oplus\mathbb{Z}^r$ (the finitely-generated)
Canonical height $\hat{h}$	the Néron–Tate quadratic ($\hat{h}(P)=\lim h(nP)/n^2$, the kernel the torsion)
Descent / Selmer	$\mathrm{Sel}^{(n)}=\ker(H^1(K,A[n])\to H^1(K,A))$ controls $A(K)/nA(K)$
Weak Mordell–Weil	$

The Birch–Swinnerton-Dyer conjecture¶

Conjecture (BSD). For $E/K$ an elliptic curve, $\mathrm{rank}\,E(K)=\mathrm{ord}_{s=1}L(E,s)$ (the rank = the order of vanishing of the $L$-function at $s=1$).

Conjecture (refined BSD). The leading coefficient: $$ \lim_{s\to1}(s-1)^{-r}L(E,s)=\frac{|\mathrm{Sha}|\,\Omega_E\,R_E\,\prod c_p}{|E(K)_{\mathrm{tors}}|^2}, $$ the Tamagawa numbers $c_p$, the regulator $R_E$, the Tate–Shafarevich group $|\mathrm{Sha}|$.

BSD / refined	Description
BSD (rank)	$\mathrm{rank}\,E(K)=\mathrm{ord}_{s=1}L(E,s)$ (the rank = the $L$-order)
Refined BSD	the leading coefficient $=
Tamagawa / regulator / Sha	the $c_p$ the Tamagawa, $R_E$ the regulator, $
Analytic rank	$\mathrm{ord}_{s=1}L(E,s)$ the analytic rank (the BSD the algebraic = the analytic)

The Gross–Zagier / Kolyvagin and the partial BSD¶

Theorem (Gross–Zagier). The Heegner point / the $L'(E,1)\ne0$ the rank 1: the Gross–Zagier the height of the Heegner point the $L'$, the analytic rank 1 = the algebraic rank 1.

Theorem (Kolyvagin). The Euler system / the Heegner point the non-torsion $\Rightarrow$ the rank 1 and $|\mathrm{Sha}|<\infty$.

Gross–Zagier / Kolyvagin	Description
Gross–Zagier	the Heegner point / the $L'(E,1)=c\cdot\hat{h}(P)$ (the height the $L'$, the rank 1)
Kolyvagin	the Euler system / the Heegner non-torsion $\Rightarrow$ rank 1 and $
Partial BSD	the Gross–Zagier / Kolyvagin the rank $\le1$, the full BSD open

The $L$-function and the modularity¶

Hasse–Weil $L$ / modularity / Iwasawa	Description
Hasse–Weil $L$-function	$L(E/K,s)=\prod_v L_v(E/K,s)$ (the Euler product, the Frobenius at the good primes, the bad / the conductor)
Modularity	the Wiles–Taylor–Wiles / the Breuil–Conrad–Diamond–Taylor: $L(E,s)=L(f,s)$ for a modular form $f$ of weight 2 (the modularity theorem)
Iwasawa / $p$-adic BSD	the Mazur / the Greenberg, the $p$-adic $L$-function / the $p$-adic BSD

See it¶

BSD for the rank-0 elliptic curve $y^2=x^3-x$ and the Heegner point for rank 1. The curve $E:y^2=x^3-x$ (the congruent number curve, the conductor 32, the analytic rank 0, the $L(E,1)\ne0$): the BSD rank 0 ($E(\mathbb{Q})=\{(0,0),(0,1),(0,-1),\mathcal{O}\}$ the rank 0, the $L(E,1)\ne0$, the Kolyvagin / the rank 0). The Heegner point: for a rank-1 curve (e.g. $E:y^2=x^3-157^2 x$ the congruent number 157, the rank 1), the Heegner point $P\in E(\mathbb{Q})$ (the CM point on $X_0(N)$ / the modular parametrization), the Gross–Zagier: $L'(E,1)=c\cdot\hat{h}(P)$ (the $L'$ the canonical height of the Heegner point), the rank 1 (the $L'(E,1)\ne0$ the non-torsion $P$, the Gross–Zagier / the Kolyvagin). The key point: the BSD the rank = the $L$-order, the rank 0 the $L\ne0$ (the Kolyvagin), the rank 1 the $L'\ne0$ (the Gross–Zagier Heegner) — the analytic / the algebraic the bridge.

Check¶

Show that the canonical height $\hat{h}(P)=\lim_{n\to\infty}h(nP)/n^2$ is a quadratic form with kernel the torsion, and explain the descent.

The canonical height $\hat{h}(P)=\lim_{n\to\infty}h(nP)/n^2$ (the Néron–Tate, the limit of the naive height $h(nP)$ divided by $n^2$). The quadratic: $\hat{h}(P+Q)+\hat{h}(P-Q)=2\hat{h}(P)+2\hat{h}(Q)$ (the parallelogram law, the Néron–Tate). The kernel: $\hat{h}(P)=0$ iff $P$ torsion (the height the positive definite modulo the torsion, the kernel the torsion). The descent: the Selmer group $\mathrm{Sel}^{(n)}(A/K)=\ker(H^1(K,A[n])\to H^1(K,A))$ (the Kummer sequence $0\to A[n]\to A\xrightarrow{n}A\to0$ the $H^1$), controls $A(K)/nA(K)$ (the Selmer the finite, the $A(K)/nA(K)$ the finite quotient). The weak Mordell–Weil: $|A(K)/nA(K)|<\infty$ (the Selmer finite $\Rightarrow$ the quotient finite), the descent / the rank computation (the rank $r$ from the Selmer / the descent).

Explain the refined BSD: the leading coefficient $=|\mathrm{Sha}|\,\Omega_E\,R_E\,\prod c_p/|E(K)_{\mathrm{tors}}|^2$, and identify the terms.

The refined BSD: the leading coefficient $\lim_{s\to1}(s-1)^{-r}L(E,s)=\frac{|\mathrm{Sha}|\,\Omega_E\,R_E\,\prod c_p}{|E(K)_{\mathrm{tors}}|^2}$. The terms: $|\mathrm{Sha}|$ the Tate–Shafarevich group (the locally trivial torsors, the Hasse failure, the conjecturally finite), $\Omega_E$ the real / complex period (the Néron differential the integral), $R_E$ the regulator (the canonical height pairing of the generators, $\det(\hat{h}(P_i,P_j))$), $c_p$ the Tamagawa numbers (the local factors, the component groups at the bad primes), $|E(K)_{\mathrm{tors}}|$ the torsion (the denominator). The refined BSD the deep: the leading coefficient the arithmetic / the geometric invariants (the Sha, the period, the regulator, the Tamagawa, the torsion), the full BSD the open (the partial the Gross–Zagier / Kolyvagin the rank $\le1$).

Pitfalls¶

The BSD is a conjecture (the rank = the $L$-order, the refined — the open, the partial the Gross–Zagier / Kolyvagin the rank $\le1$). Do not state the BSD as a theorem; state the proven cases (the rank $\le1$, the Gross–Zagier / Kolyvagin).
The Sha the conjecturally finite (the BSD the $|\mathrm{Sha}|$ the analytic, the finiteness the open). The $|\mathrm{Sha}|$ in the refined BSD the conjectural (the finiteness the open, the analytic the BSD).
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 Wiles / the BCDT) the elliptic / the rank-2, the higher-dimensional the partial.
The $p$-adic BSD / the Iwasawa (the Mazur / the Greenberg) the $p$-adic $L$ / the $p$-adic BSD, the $p$-adic the analog, not the direct (the $p$-adic the different $L$-function).

Refs¶

Silverman, The Arithmetic of Elliptic Curves / Advanced Topics; Hindry–Silverman, Diophantine Geometry
Gross–Zagier, "Heegner points and derivatives of $L$-series" (Invent. 1986); Kolyvagin, "Euler systems" (1990)
Wiles, "Modular elliptic curves and Fermat's Last Theorem" (Annals 1995); Breuil–Conrad–Diamond–Taylor (2001)
Mazur, "On the arithmetic of elliptic curves" (the $p$-adic BSD); Greenberg, the Iwasawa / the $p$-adic BSD

Next¶

AG index · Foundations · Diophantine & Arakelov · Étale & Weil · Galois Reps & Fontaine–Mazur · Analytic NT / L-functions · Langlands

Hasse–Weil \(L\) / modularity / Iwasawa	Description
Hasse–Weil \(L\)-function	\(L(E/K,s)=\prod_v L_v(E/K,s)\) (the Euler product, the Frobenius at the good primes, the bad / the conductor)
Modularity	the Wiles–Taylor–Wiles / the Breuil–Conrad–Diamond–Taylor: \(L(E,s)=L(f,s)\) for a modular form \(f\) of weight 2 (the modularity theorem)
Iwasawa / \(p\)-adic BSD	the Mazur / the Greenberg, the \(p\)-adic \(L\)-function / the \(p\)-adic BSD