# Tamagawa numbers

Let $$K$$ be a finite extension of $$\mathbb{Q}_p$$ with absolute Galois group $$G_K$$. Let $$A$$ be an abelian variety defined over $$K$$. The (geometric) Tamagawa number is defined as the order of the quotient $$c(A/K)=A(K)/A_0(K)$$ where $$A_0(K)$$ denotes the $$K$$-points of $$A$$ that are sent to the identity component of the special fiber of the Néron model of $$A$$ via the natural reduction map.

On the other hand, following Fontaine and Perrin-Riou (in "Cohomologie galoisienne et valeurs de fonctions L", §4), starting with the $$p$$-adic Tate module $$T_p=T_p(A)$$, one may define the ($$p$$-adic) Tamagawa number $${\rm Tam}_\omega(T_p)$$. The latter depends on a choice of a generator $$\omega\in\det(t_{V_p})^*$$, where $$V_p=T_p\otimes\mathbb Q_p$$ and $$t_{V_p}=(B_{\rm dR}/B_{\rm dR}^+\otimes V_p)^{G_K}$$.

As suggested by the vocabulary, can one expect a direct relation between $$c(A/K)$$ and $${\rm Tam}_\omega(T_p)$$ to hold? Is there a proof of this?

The number $${\rm Tam}_\omega(T_p)$$ is a power of $$p$$, so one might expect it to be comparable with the $$p$$-part of $$c(A/K)$$. However, one would need to have an almost cannonical choice for $$\omega$$.

Denote by $$\Phi$$ the quotient of $$\mathcal A^\vee$$, the special fiber of the smooth (but not necessarily proper) model of the dual abelian variety $$A^\vee$$, by the connected component of $$0$$ of $$\mathcal A^\vee$$. Then it is indeed true that the Tamagawa number $$c_p$$ as defined in $$L$$-functions and Tamagawa number of motives (Bloch, Kato) or in Autour des conjectures de Bloch-Kato (Fontaine, Perrin-Riou) is then equal up to a $$p$$-adic unit to $$|\Phi[p^\infty]|$$, but this is not quite obvious.
• First, relate $$\Phi$$ to the formal group of $$A^\vee$$, and then through the $$\log$$ map to $$\operatorname{Lie}_{\mathbb Z_p}A^{\vee}$$ (which is also the tangent space of the formal group with values in $$\mathbb Z_p$$).
• Then notice that this induces isomorphisms between the determinant of $$\operatorname{Lie}_{\mathbb Z_p}A^{\vee}$$ and the determinant of the first Bloch-Kato cohomology group $$H^1_f(\mathbb Q_p,T_pA)$$ of the $$p$$-adic Tate module of $$A$$. This isomorphism involves finite corrective terms which come from the fact that the formal group is a subgroup with finite index of $$H^1_f(\mathbb Q_p,T_pA)$$ (and similarly for the image of the $$\log$$ with respect to $$\operatorname{Lie}_{\mathbb Z_p}A^{\vee}$$).
• Now the Tamagawa number $$c_p$$ arises as an alternate product of these corrective terms and of the Euler factor at $$p$$ (because the exponential map of Bloch-Kato coincide in this case with $$\log$$).
• Finally, relate this alternate product to $$\Phi$$. This involves the classical description of $$\Phi$$ in terms of the Neron model as well as a study of the reduction of $$A^{\vee}$$.
One thing that I would mention is that, as you say, the computations of the proof above involve in several steps the choice of a $$\mathbb Z_p$$-basis, but as this plays a role both in the normalization of maps with target $$\operatorname{Lie}A^\vee$$ as well as in the definition of $$c_p$$, the resulting computation does not depend on the choice of the basis (up to a $$p$$-adic unit, which plays no role in the Tamagawa number conjectures of Bloch-Kato).