The authenticity of the document itself does not seem to be in question - no one seems to deny that the map dates to 1513 and is the work of Piri Reis. See Description of the Map - Page 8 of the available online sample.
The question is regarding the authenticity and accuracy of the antarctic coastline depicted in that map. The nexus of the dispute can be found in your reference:
PIRI REIS MAP OF ANTARCTICA
In 1953, a Turkish naval officer sent the Piri Reis map to the U.S.
Navy Hydro graphic Bureau. To evaluate it, M.I. Walters, the Chief
Engineer of the Bureau, called for help Arlington H. Mallery, an
authority on ancient maps, who had previously worked with him. After a
long study, Mallery discovered the projection method used.
To check out the accuracy of the map, he made a grid and transferred
the Piri Reis map onto a globe: the map was totally accurate. He
stated that the only way to draw map of such accuracy was the aerial
surveying: but who, 6000 years ago, could have used airplanes to map
the earth? And this was Antartica ice free.
The Hydro graphic Office couldn’t believe what they saw: they were
even able to correct some errors in the present days maps! The
precision on determining the longitudinal coordinates, on the other
hand, shows that to draw the map it was necessary to use the spheroid
trigonometry, a process supposedly not know until the middle of 18th
century.
Hapgood has proved that the Piri Reis map is plotted out in plane
geometry, containing latitudes and longitudes at right angles in a
traditional “grid”; yet it is obviously copied from an earlier map
that was projected using spherical trigonometry! Not only did the
early map makers know that the Earth was round, but they had knowledge
of its true circumference to within 50 miles!
However, on the comments of that page we find the following:
Instead of concluding the map was likely made by ancients with
knowledge of spheroid trigonometry, one could also go with a more
mainstream explanation which reads way simpler: that this northern
coastline of Antarctica is not perfectly detailed at all. For example
Gregory C. McIntosh at
www.diegocuoghi.com/Piri_Reis/McIntosh/McIntosh_PiriReis.htm: "it was
an easy matter for Hapgood to move landmasses, adjust scales, alter
orientations, rearrange landforms, redraw coastlines, twist the
geographical depictions, and “correct errors” on the Piri Reis Map to
match his hypothetical source maps."
It seems like a clear example of manipulating data under the influence
of a strong bias. Only a cursory glance would show any fact-checker
that actually there are many differences in locations, labels and
orientations. The similarities do not appear to be significant enough
to justify the claims.
Although I'd like to let the evidence speak for itself, it needs
perhaps mentioning that Hapgood was also a spiritist involved with
mediums making contact with dead people and various gods. His works
needs a more cautious approach and I'd advice against using this topic
to make some statement on climate change. It opens one up to way too
much valid criticism.
More detail (and less bias...) here:
A more sober analysis of these claims was published by Gregory McIntosh, a historian of cartography, who examined the map in depth in his book The Piri Reis Map of 1513:
...As far as the accuracy of depiction of the supposed Antarctic coast
is concerned, there are two conspicuous errors. First, it is shown
hundreds of miles north of its proper location; second, the Drake
Passage is completely missing, with the Antarctic Peninsula presumably
conflated with the Argentine coast. The identification of this area of
the map with the frigid Antarctic coast is also difficult to reconcile
with the notes on the map which describe the region as having a warm
climate.
It should be kept in mind that maps of the period generally depicted a
large continent named Terra Australis Incognita of highly variable
shape and extent. This land was posited by Ptolemy as a counterbalance
to the extensive continental areas in the northern hemisphere; due to
a lack of exploration and various misunderstandings its existence was
not fully abandoned until circumnavigation of the area during the
second voyage of James Cook in the 1770s showed that if it existed, it
was much smaller than imagined previously. The first confirmed landing
on Antarctica was not until 1820, and the coastline of Queen Maud Land
did not see significant exploration before Norwegian expeditions began
in 1891.In 1513 Cape Horn had not yet been discovered, and indeed
Ferdinand Magellan's voyage of circumnavigation was not to set sail
for another six years. It is unclear whether the map maker saw South
America itself as part of the unknown southern lands (as shown in the
Atlas Miller) or whether (as Dutch thought) he drew what was then
known of the coast with substantial distortion, but in any case
serious scholarship holds that there is no reason to believe that the
map is the product of genuine knowledge of the Antarctic coast...
No one is claiming that the map itself is "an elaborate fake". The question is simply regarding the historical and geographical accuracy of the antarctic coastline it depicts, given that in 1513 such knowledge was not available from any traditional source: Is it indeed a correct depiction derived from some unknown ancient source, or the product of "artistic license", often taken by cartographers of the period regarding "Terra Incognita".
Based on contemporary scholarship, claiming that the map is undeniably an accurate representation of an ice-free antarctic coastline during any period when it would have been possible for humans to determine such is quite far-fetched.
