Why are they meeting?
Mr. Abe wants to find out how serious Mr. Trump was during his campaign when he repeatedly criticized Japan on trade issues and for not paying more of the cost of its own defense.
And since the Japanese establishment was expecting Hillary Clinton to win the presidency, Mr. Abe wants to assure Mr. Trump that he is willing and eager to work with him.
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What is Mr. Abe most worried about?
The prime minister is concerned about Mr. Trump's commitment to protecting Japan.
The United States is Japan's most important ally, and is legally obligated to defend it against attack. There are around 50,000 American troops stationed in Japan, a powerful deterrent against the rising threat of North Korea, and an increasingly assertive China.
During the campaign, President-elect Trump suggested he might withdraw American troops.
As president, could he do that?
Not without resistance from members of Congress, officials at the State Department, diplomats and military advisers. The United States-Japan alliance has been in place since the end of World War II, and has a lot of backing on both sides of the Pacific Ocean.
Can Japan pay the United States more for defense?
Maybe. There are signs that Japan might be willing to pay or do more, but many experts contend American troops are not only defending Japan but the United States' own interests in Asia.
The Pentagon is budgeted to spend about $5.5 billion to support troops and bases on Okinawa and elsewhere around Japan this year. Japan is set to spend an additional $1.8 billion to support the bases, in addition to at least $4 billion on related expenses, including compensation for the communities that host the bases and funds for relocating American troops.
Under its Constitution, which was written by American occupying forces after World War II, Japan can keep an army for defensive purposes. But Mr. Abe has said he wants to revise the Constitution and expand the military. In August, his government requested the latest in a series of increases in military spending.
Would Japan pursue nuclear weapons?
Japan has committed to the Nuclear Nonproliferation Treaty, and Mr. Abe will probably remind Mr. Trump of that.
On the campaign trail, Mr. Trump decried the cost of providing a so-called nuclear umbrella to allies such as Japan and South Korea that do not have their own nuclear arms. In March, Mr. Trump told The New York Times that he was open to those countries building their own nuclear arsenals, but on Twitter this week he denied having said so.
Mr. Abe has reaffirmed his country's commitment not to develop atomic weapons. Any move to do so would face severe public opposition in Japan, the only country that has suffered a nuclear attack.
Will Mr. Abe press his case on trade and the Trans-Pacific Partnership?
He has been one of the biggest enthusiasts for the Trans-Pacific Partnership — or TPP, as it is known — though Japan was the last of a dozen Pacific Rim countries to join the sweeping trade pact. Mr. Trump repeatedly criticized the TPP during the presidential campaign.
Mr. Abe made politically painful concessions on agricultural imports to close the deal and get Japanese manufacturers tariff-free access to export markets in the United States and elsewhere. The trade partnership could bolster the limp economic growth rate in his country and serve as a counterweight to China, the region's fast-rising superpower.
With Mr. Trump’s election, isn’t TPP dead?
Mr. Abe acknowledged after Mr. Trump's victory that the Trans-Pacific Partnership was in a "very difficult situation." But his government still plans to have Parliament ratify the agreement, and he has not given up on selling it to Mr. Trump.
Mr. Abe may also seek to soften Mr. Trump's angry campaign stance, in which he accused Japan of crushing the United States on trade, and manipulating its currency to gain an economic advantage.
Mr. Trump says he's for free trade in principle, but wants a "better deal" for the United States. What that means, for now, remains a mystery.