  Application of the supersymmetric localization method to theories on anti-de
Sitter spacetime has received recent interest, yet still remains as a
challenging problem. In this paper, we focus on (global) Euclidean AdS$_2$, on
which we consider an Abelian ${\cal N}=(2,2)$ theory and implement localization
computation to obtain the exact partition function. For comparison, we also
revisit the theory on S$^2$ and perform a parallel computation. We refine the
notion of equivariant supersymmetry and use appropriate functional integration
measure. For AdS$_2$ we choose a supersymmetric boundary condition which is
compatible with the principle of variation. To evaluate the 1-loop determinant
about the localization saddle, we use index theory and fixed point formula,
where we pay attention to the effect of zero modes and their superpartners. The
existence of fermionic superpartner of 1-form boundary zero modes is proven.
Obtaining the 1-loop determinant requires expansion of the index that presents
an ambiguity, which we resolve using boundary condition. The resulting
partition function reveals an overall dependence on the size of the background
manifold, AdS$_2$ as well as S$^2$, as a sum of two types of contributions: a
local one from local conformal anomaly through the index computation and a
global one coming from zero modes. This overall size dependence is confirmed by
the perturbative 1-loop evaluation using heat kernel method.
