In the first and third of the temptations of Jesus that are recorded in Luke, interestingly enough, Satan’s challenge is prefaced by the words “If you are the Son of God” (Luke 4:3,9). So Satan’s challenges to Jesus ask to be understood and come within the context of whether or not He is the Son of God.
These temptations, and the written record of these temptations, reflect upon a time at which a messiah is expected. In popular comprehension, this messiah would be the son of God in that he would be Israel’s king, but in some circles he would also be the Son of God, in that many believed it to be the case that the Creator God would take human flesh upon Himself in order to personally intervene on behalf of His people, establishing Israel’s rule over the nations. As Israel’s king therefore, this would mean that the Creator God, through His son, was going to establish His own physically present rule over the whole of the world.
So when Satan tempts Jesus, it appears that what He is tempting Jesus to do is to take a shortcut to acclimation as king. He could turn stone into bread (an allusion to Israel receiving manna in the wilderness), and with such a miraculous demonstration He prove Himself to be the messiah and therefore immediately hailed as king. Such would be a reasonable expectation on the part of Satan, because as would come to be seen, when Jesus multiplies the loaves and the fishes, the people attempted to come and make Him king by force, which caused Jesus to quickly withdraw into isolation.
It would seem that Jesus knew that there was only one way for Him to ascend to the throne, and ultimately only one way in which He would be recognized as the Messiah (along with all that would entail according to the expectations of a great number of His fellow citizens), and that was going to be the path of suffering on an altogether unlikely cross. That, presumably, is why Jesus refutes Satan’s insistence to worship him in exchange for earthly rule, and why He also rejects the idea of putting His God to the test by casting Himself from the Temple for the purpose of forcing a miraculous rescue that would have the likely effect of an immediate elevation to the throne of Israel. Apart from that, Jesus may not have been altogether confident that the covenant God would come to His rescue if he flung Himself from the pinnacle of the Temple, as such would not have been an exilic suffering in the mold of Israel into which their God would enter in to provide deliverance.
What does this have to do with Adam? Well, where Adam failed, Jesus succeeded. When the serpent tempted Adam with being like the Creator God, Adam acceded to the temptation. When Satan tempted Jesus (the second Adam) with the opportunity to show Himself forth as God ( being “like God” for all practical purposes), Jesus refused. Subtle to be sure, but here in the wilderness Jesus destroyed a work of the devil. By holding on to the fate of suffering, love was put on display.
To further reinforce the congruity between Adam (son) and Jesus (Son) as presented through Jesus’ experience of wilderness temptations, Mark adds that Jesus was “with the wild animals” (1:13c). Adam, according to the Genesis narrative, was given “rule over the fish of the sea and the birds of the air, over the cattle, and over all the earth, and over all the creatures that move on the earth” (1:26b). Additionally, the animals that had been created were brought before Adam “to see what he would name then, and whatever the man called each living creature, that was its name. So the man named all the animals, the birds of the air, and the living creatures of the field” (2:19b-20a). Adam, the son of God before Jesus, was very much with the wild animals. Perhaps this is why Mark, drawing from the son of God tradition that would have included Adam, makes what seems like a rather out of place insertion about Jesus being with the wild animals during His wilderness trials?