On Jan. 31, 2018, the day Senator Robert Menendez was formally cleared of bribery charges that had dogged him for nearly three years in New Jersey, he got a text from Nadine Arslanian, a woman he would soon begin to date and later marry.

“Now re-election!!!!” Ms. Arslanian wrote.

“Yes!” Mr. Menendez replied before asking, “Are you around on Friday?”

She was. After a dinner date at a New Jersey restaurant, it was Ms. Arslanian’s turn to send a text with a question: “What is your international position?”

Mr. Menendez, 70, responded that he was the “ranking member” on the Senate Foreign Relations Committee — “which means senior Democrat.”

The text exchanges, along with emails and recordings of voice mail messages and other exhibits, were part of hours of evidence that federal prosecutors presented on Tuesday, in the third week of Mr. Menendez’s corruption trial in Manhattan.

Prosecutors used the volley of communications to begin to lay out an origin story of not only a romantic relationship but also what they claim was a burgeoning, five-year bribery conspiracy.