As President Biden welcomes India’s prime minister to the White House, the two leaders will be looking for more than a fine vegetarian meal and a night of glitzy entertainment.

Under the guise of pomp and pageantry, state visits are a chance for presidents to push foreign dignitaries to align with American interests. They can be a way to celebrate old, ironclad alliances. And with lengthy guest lists, multicourse meals and top-flight entertainment, they are much-coveted invites in Washington.

“These are not just dinners,” said Matthew Costello, a senior historian for the White House Historical Association. “There’s a lot more that goes into them in terms of planning, in terms of invitations, and a lot is geopolitics, a lot is foreign policy.”

When President Dwight D. Eisenhower invited Nikita Khrushchev, the leader of the Soviet Union, to the White House in 1959, he was focused on thawing Cold War tensions after the launch of Sputnik. Before President Barack Obama hosted President Xi Jinping of China, the two countries negotiated for weeks over an arms control accord for cyberspace. President Ulysses S. Grant held the first state dinner for King David Kalakaua of Hawaii to strengthen trade.

The dinners can also provide a window into the regions the United States is prioritizing — and the ones being neglected.

European and Latin American nations have received the most state dinner invitations, while sub-Saharan African and Southeast Asian nations have received the fewest, according to a study by the Center for Global Development that tracked 40 years of state visits from the Carter to Obama administrations.

Out of 160 dinners, just 15 were with guests from sub-Saharan Africa, the study found.

“To be a foreign leader and not get the state dinner, you feel snubbed,” said Douglas Brinkley, a presidential historian. “It’s often the smaller countries in the world who don’t get them, but when you’re dealing with big power players like India, it’s a must.”

The invitation to Prime Minister Narendra Modi of India was not without controversy. Mr. Biden has made the global struggle between democracy and autocracy a key part of his foreign policy, but Mr. Modi’s government has cracked down on dissent in ways that have raised fears of authoritarianism.

Still, the White House views the world’s most populous nation as a potentially welcome addition to its coalition against Russia’s invasion of Ukraine, as well as a crucial player in its growing economic competition with China.

The other nations whose leaders received the official invitation to dine with Mr. Biden — France and South Korea — have also been partners in Mr. Biden’s effort to confront Russia.

The state dinner can sometimes be a means of smoothing over hiccups among allies.

Mr. Biden hosted President Emmanuel Macron of France for the first state visit of his administration, more than a year after the two nations feuded over a deal to provide Australians with nuclear-powered submarines. Mr. Biden invited Anthony Albanese, the prime minister of Australia, for a state visit after he canceled a trip there in May because of negotiations over the debt ceiling with congressional Republicans.

“There are multiple times we see presidents using these visits to not just describe immediate concerns, but also to talk through short-term and long-term solutions,” Mr. Costello said.

Domestic politics often hang over the dinner, as well.

Julianna Smoot, Mr. Obama’s social secretary from 2010 to 2011, said she made sure to invite the often-feuding majority and minority leaders of the Senate, Harry Reid and Mitch McConnell, to state dinners for a rare détente. Governors and mayors who had previously expressed support or campaigned for the president were likely to make the list. And the primary donors of presidential campaigns could expect an invitation, particularly if they had business connections to the visiting nation.

“They didn’t become donors in politics because they were slouches,” Ms. Smoot said. “A lot of them do international work and have an interest” in attending the dinner.

The prospect of strengthening political partnerships overseas and within U.S. borders was usually enough to get a quick response from invitees.

“You’re supposed to say yes,” Ms. Smoot said of responding to the invitations, “unless there’s a death in the family.”