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Where do I find a proof of the fact that over global function fields of characteristic $p>0$, finiteness of the Tate-Shafarevich group of an abelian variety is equivalent to finiteness of its $\ell$-primary torsion part for some (arbitrary) prime $\ell$ ($\ell = p$ allowed)?

What is the main idea?

One knows the Tate-Shafarevich group is torsion, and can write, for $K$ a function field and $A/K$ an abelian variety:

$$\text{Sha}(A/K) = \prod_{\ell}\text{Sha}(A/K)[\ell^{\infty}]$$ so I guess one first shows that $\text{Sha}(A/K)[\ell^{\infty}]$ is trivial for almost all $\ell$, but then?

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One shows that the finiteness of one $\ell$-primary component of Sha is equivalent to the Birch-Swinnerton-Dyer conjecture. $\mathrm{rk} A(K) = \mathrm{ord}_{s=1}L(A/K,s)$ is independent of $\ell$ and a non-zero rational number (the special $L$-value) has only finitely many prime divisors.

The references are:

  1. Peter Schneider, Zur Vermutung von Birch und Swinnerton-Dyer über globalen Funktionenkörpern, Math. Ann. 260 (1982), no. 4, 495–510 ($\ell \neq p$)

  2. Werner Bauer, On the conjecture of Birch and Swinnerton-Dyer for abelian varieties over function fields in characteristic $p>0$, Invent. Math. 108 (1992), no. 2, 263–287 ($\ell = p$, good reduction)

  3. Kazuya Kato and Fabien Trihan, On the conjectures of Birch and Swinnerton-Dyer in characteristic $p>0$, Invent. Math. 153 (2003), no. 3, 537–592 ($\ell = p$, general case)

The finiteness of Sha is e.g. known for constant Abelian varieties:

There are similar results for higher dimensional bases: https://www.timokeller.name/BSDI.pdf (only for good reduction and $\ell \neq p$, contains also e.g. the finiteness of the prime-to-$p$ part of Sha of supersingular Abelian schemes over special base schemes)

The rough idea is as follows: The Kummer sequence $0 \to \mathscr{A}[\ell^n] \to \mathscr{A} \to \mathscr{A} \to 0$ (for the étale topology if $\ell \neq p$ and otherwise for the syntomic topology) gives a short exact sequence $$0 \to A(K) \otimes \mathbf{Z}_\ell \to H^1(X,T_\ell\mathscr{A}) \to T_\ell Ш(\mathscr{A}/X) \to 0.$$ Since $Ш(\mathscr{A}/X)[\ell^\infty]$ is of cofinite type, the $\ell$-adic Tate module is trivial iff the $\ell$-primary component is finite. The Hochschild-Serre spectral sequence degenerates by $\mathrm{cd}(\Gamma) = 1$ giving $$0 \to H^0(\bar{X},T_\ell\mathscr{A})_\Gamma \to H^1(X,T_\ell\mathscr{A}) \to H^1(\bar{X},T_\ell\mathscr{A})^\Gamma \to 0.$$ Now relate $H^1(\bar{X},T_\ell\mathscr{A})^\Gamma$ to the special $L$-value using Lemma 3.2 from [Bayer-Neukirch, On values of zeta functions and $l$-adic Euler characteristics, Invent. Math. 50 (1978/79), no. 1, 35–64]. ($X$ a model of the function field $K$, $\mathscr{A}/X$ the Néron model of $A/K$ [if it exists], $k = \mathbf{F}_q$ the finite ground field, $\bar{X} = X \times_k \bar{k}$ and $\Gamma = G_k$)

Another nice reference: Douglas Ulmer, Elliptic curves over function fields. Arithmetic of L-functions, 211–280, IAS/Park City Math. Ser., 18, Amer. Math. Soc., Providence, RI, 2011.

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