In a landmark decision this month, a federal judge spared Google from a breakup proposed by the government to rein in the company’s monopoly in search.

Now, the company wants a second win.

Starting on Monday, Judge Leonie M. Brinkema will hear arguments in the U.S. District Court for the Eastern District of Virginia on how to fix Google’s monopoly over some online advertising technology. The judge ruled this year that Google had broken the law to protect its dominance over the largely invisible system of technology that places ads on pages across the web. The hearing is expected to last for two to three weeks.

The Justice Department has said it wants Google to sell the technology it uses to connect buyers and sellers of ad space, among other potential remedies. Google has instead proposed measures including changing its ad auction bidding system in ways that would benefit publishers.

Judge Brinkema’s ultimate decision could disrupt the company’s $3.1 trillion business and help rewrite the playbook for tech dominance in the modern internet era.

She has already indicated she might do so, asking during a May hearing whether forcing the sale of one of Google’s advertising software products could resolve her concerns. The case also aligns more closely with past tough antitrust decisions, which could make it easier for the judge to take an aggressive tack, legal experts said.

The hearing comes after a conservative antitrust decision by Judge Amit P. Mehta of the U.S. District Court for the District of Columbia this month in the search monopoly case, in which he ruled Google’s search business could essentially continue operating as is. The company must share its search results with rivals and restrict payments that ensure its search engine gets prime placement in web browsers and on smartphones. But Judge Mehta did not approve the government’s request that Google be forced to sell its popular Chrome web browser, alongside other harsher penalties.