Final answer:
The prosperity of Athens declined sharply at the start of the Peloponnesian War around 431 BCE, characterized by military disasters and plagues, culminating in Athens' surrender in 404 BCE.
Step-by-step explanation:
The war in which the prosperity of Athens took a sharp turn for the worse around 431 BCE is known as the Peloponnesian War. This conflict was an internal struggle between Athens and Sparta that saw the height of Greek culture but also led to significant woes for the city-states involved. The war saw terrible events for Athens, such as the disastrous expedition against Syracuse in 413 BCE, where the Athenians suffered a crushing defeat. This, along with a Plague, weakened their military and economic reaches. Ultimately, the Peloponnesian War concluded with the siege of Athens and the destruction of its navy, leading to Athens' surrender in 404 BCE.