After reading this blog post, I learned the BSD conjectural formula for the coefficient of the leading term $a_0$ of the L-function of an elliptic curve $E$, namely
$$
a_0 \stackrel{?}{=} \frac{\Omega_E\cdot Reg_E \cdot \prod_p c_p \cdot \#Sha(E/\mathbb{Q})}{(\# E_{tors}(\mathbb{Q}))^2}
$$
All the terms are defined, if people are interested and don't already know, at the above link. Now the factors in the numerator here come in several flavours: the real period $\Omega_E$ arises after looking at the curve over $\mathbb{R}$; the numbers $c_p$ are 1 for all but finitely many primes $p$, and come from looking at the curve over $p$-adic numbers, hence completions of $\mathbb{Q}$; $Reg_E$, the regulator, is the volume of a certain torus (not the curve itself!) given by comparing the rational and real points of $E$; the Sha group arises from comparing Galois cohomology over $\mathbb{Q}$ with all its completions at finite primes. Clearly the Archimedean place and the non-Archimedean ones behave differently, but one can often unify them in certain formalisms. (If one is willing to split the denominator, and invert the regulator, then it is a product of three ratios, each of which is something like (something about a completion)/(some sort of volume), but this just *extremely* flaky and ignorant, and best ignored)

My question is this: can we write this product (or perhaps the whole quotient) more uniformly via reinterpreting various terms in more abstract ways via places?

Please note I'm **not** trying to *do* anything with such a formulation, I'm just curious if it is known.

EDIT: the factor $\Omega_E/\# E_{tors}(\mathbb{Q})$ is the volume of the stack given by the action groupoid $E(\mathbb{Q})\otimes\mathbb{R}//E(\mathbb{Q})$. Do the other terms measure other geometric objects, such that the whole thing is the measure of some adelic object?

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