Last summer, a federal judge, Amit P. Mehta, delivered a landmark ruling that found Google had illegally dominated online search.

“Google is a monopolist, and it has acted as one to maintain its monopoly,” he wrote.

Now, Judge Mehta has to figure out how to fix Google’s monopoly. His decision will be informed by a three-week hearing, which starts Monday in U.S. District Court in Washington, where the government and Google will argue over how to restore competition in online search.

The Justice Department wants Judge Mehta to break up Google by forcing the company to sell its Chrome browser, which could cut off a source of information it uses to improve its internet search. The department also wants to make the Silicon Valley giant hand over some of its most precious data to rivals, which could help their competing products.

In contrast, Google has asked the judge to only slightly restrict its ability to make deals to give its search engine prime placement on browsers and in smartphones.

The outcome could reshape the $1.86 trillion company, which has become synonymous with how people look for information online. Google faces mounting challenges, including the prospect of a breakup in a second antitrust case after a different federal judge ruled on Thursday that it had illegally maintained a monopoly in some ad technology. That puts Google in perhaps its most vulnerable position since Larry Page and Sergey Brin created the business in 1998.

Its fate could shake up Silicon Valley as the biggest companies fight to own a new technology era defined by artificial intelligence. Any restrictions on Google could hamper its ability to compete with OpenAI, Microsoft, Meta and others as search is increasingly powered by A.I. tools that can summarize research and offer humanlike guidance.