The Bloch-Kato Tamagawa Number Conjecture

Here, τ(G) is roughly the volume of G(AK)/G(K) with respect to Haar measures on G(Kv) for all places v, where we have to use L-functions to make the product measure converge, and also have to restrict to measuring some “compact part” of G(Kv), by taking the kernel of all |χ|v, χ : G→ Gm. Now, let’s formulate a version of this for abelian varieties. For simplicity, assume that E is an elliptic curve over Q, with E(Q) finite. There is a Neron model E for E, with Neron form ω. This induces a measure on E(Qp): one way to formulate this is that the map log = ∫ ω : E(Zp)→ Lie(E)Qp (multiply till you land in “kernel of reduction” E(pZp), then evaluate power series) induces a measure on E(Zp) by declaring that it preserves measure and that Lie(E)Zp has volume 1. A calculation shows that vol(E(Qp)) = |Ẽ (Fp)| p · |Φp(Fp)|, where Φp is the component groupscheme. Define cp = |Φp(Fp)|, the Tamagawa factor at p. Also, vol(E(R)) = ∫ E(R) ω is the real period. The product ∏ p vol(E(Qp)) does not converge. However, note that L(E, 1) = ∏ p det(1 − p−1f |((VlE))v) = ∏ p det(1−f |(Vl(E))v) = ∏