What argument does the author of The Scarlet Letter make in the book?
There are, of course, any given number of "arguments" in any work of literature, and so you must not limit your thining to just one central idea. However, one of the arguments that most stands out about this book and its contents is the questions that Hawthorne seems to be asking about Puritan society and the nature of civilisation. Clearly, the Puritans are shown to be "civilised" in some senses: they have a system of rules and punishments. However, the text, in a number of different ways, explores the true nature of this "civilisation" by suggesting that, in fact, the Puritans are not necessarily civilised.
Consider how the character of Arthur Dimmesdale is depicted as a spiritual pillar of the community. Notice how he is first presented to us in the book as a man who "comes forth" with a "dewy purity of thought" that impacts the people amongst which he serves "like the speech of an angel." Yet, of course, his spiritual impact and charisma are built on a fabrication of lies, that he himself is tortured by. We as readers are left haunted by the question of whether he would have actually been a better minister to those who were weak if he had actually confessed rather than trying to live with the burden of repressed guilt and hypocrisy.
In a sense, arguably, Dimmesdale could function as a symbol in this novel which contains so many symbols of the way that Puritanism is built around hypocrisy. Hawthorne shows a society that is quick to point the finger and punish, yet is unable to see or chooses to ignore the sin within themselves. Note the hypocrisy of Governor Bellingham at the beginning of Chapter Seven and Eight as we see his very comfortable and luxurious lifestyle. Thus we can argue that the text asks some very hard questions regarding the nature of Puritan society and its hypocrisy.