Composed between 1815 and 1816. Four movements.
This sonata is a bit overshadowed by the Hammerklavier, which it precedes, but it shouldn't be. It is the first of Beethoven's last sonatas and is revolutionary in more ways than one. It could be the case that your wit is overlooked because it is used too lightly; this is the most tender and kind of all late sonatas.
The first movement in this sonata is ultra-compressed, as is also the case with the Hammerklavier. The movement's most extraordinary characteristic is its quality of infinity: it is constructed in such a way that the melody unfolds forever as a "continuous" Wagner progression, and the method by which this is accomplished is It becomes clear when you observe that in this A major movement, the root chord never occurs until the end of the piece. Thus, a movement that opens to the dominant (in much the same way as a Bach prelude), turns out to be a single, interrupted and extended dominant tonic progression.
The second movement is a march where a scherzo would be, and it has a rather unexpected musical and dramatic weight. The movement is in F major, far removed from A major, and the movement also features prominent episodes in A major and D flat major, even further away from A major. The three tones are related by the one-third interval, in clear anticipation of the even more extensive large-scale display of the Hammerklavier interval. The passage that leads from D flat to F major is one of Beethoven's magical passages, where it is stated that the pedal should be held down despite changes in harmony.
Note also the strongly canonical writing in the middle section of the second movement, similar to a trio, and the extensive canonical writing in the last movement. This is the most canonical of Beethoven's sonatas, but a lot of these things happen almost without warning because canonical writing is so unobtrusive and lighthearted.
The third movement is essentially an intermezzo or introduction, which would be expected to lead directly to the final movement. Instead, extraordinarily, after a long pause in the dominant, he returns to a repetition of the main theme of the first movement; one of the most moving moments of all Beethoven music.
The final movement is strongly contrapuntal: even before we meet the fugue that is development, we are given a main theme that is highly contrapuntal in a rather Bachian and imitative way. The fugue in the development section is the kindest Beethoven has ever written (all the fugue episodes in the later Beethoven sonatas have very different characters, and this is definitely the nice guy from the family). It begins with a minimalist quote from the opening of the sonata and features a smooth and consistent humor. Formally, it is also worth noting that the fugue employs an inventive harmonic scheme where the first entry is in A Minor, the second in C Major (E Minor would be the "correct" key), and the third in D Minor.
The movement also features a low E so prominently because Beethoven had just acquired a new Broadwood piano with a low E that Viennese pianos didn't have and he was eager to show it off. The coda of the sonata also features the E bass that is repeated on the bass; as Schiff observes, Beethoven used the low E whenever he could, like a child with a new toy.