The Ly$\alpha$ Luminosity Function and Cosmic Reionization at $z \sim$
  7.0: a Tale of Two LAGER Fields
  We present the largest-ever sample of 79 Ly$\alpha$ emitters (LAEs) at
$z\sim$ 7.0 selected in the COSMOS and CDFS fields of the LAGER project (the
Lyman Alpha Galaxies in the Epoch of Reionization). Our newly amassed ultradeep
narrowband exposure and deeper/wider broadband images have more than doubled
the number of LAEs in COSMOS, and we have selected 30 LAEs in the second field
CDFS. We detect two large-scale LAE-overdense regions in the COSMOS that are
likely protoclusters at the highest redshift to date. We perform injection and
recovery simulations to derive the sample incompleteness. We show significant
incompleteness comes from blending with foreground sources, which however has
not been corrected in LAE luminosity functions in {the} literature. The bright
end bump in the Ly$\alpha$ luminosity function in COSMOS is confirmed with 6 (2
newly selected) luminous LAEs (L$_{Ly\alpha}$ $>$ 10$^{43.3}$ erg s$^{-1}$).
Interestingly, the bump is absent in CDFS, in which only one luminous LAE is
detected. Meanwhile, the faint end luminosity functions from the two fields
well agree with each other. The 6 luminous LAEs in COSMOS coincide with 2
LAE-overdense regions, while such regions are not seen in CDFS. The bright-end
luminosity function bump could be attributed to ionized bubbles in a patchy
reionization. It appears associated with cosmic overdensities, thus supports an
inside-out reionization topology at $z$ $\sim$ 7.0, i.e., the high density
peaks were ionized earlier compared to the voids. An average neutral hydrogen
fraction of $x_{HI}$ $\sim$ 0.2 -- 0.4 is derived at $z\sim$ 7.0 based on the
cosmic evolution of the Ly$\alpha$ luminosity function.
