I am about to (hopefully!) begin my PhD (in Europe) and I have a question: how did you learn so much mathematics?

Allow me to explain. I am training to be a number theorist and I have only some read Davenport's Multiplicative Number Theory and parts of Vaughan's book on the circle method. I have briefly seen some varieties from Fulton's algebraic curves and I may have read parts of homotopy and homology and differential geometry of smooth manifolds at the level of Hatcher and Lee. Yet, it seems that I am hopelessly ignorant of elliptic curves, modular forms and algebraic number theory.

For example, if I were to try reading Deligne's proof of Weil's conjecture or Tate's thesis, it seems that I would have to read significant amounts of reading.

When I look at some of my professors or other researchers I have interacted with, I notice that they may be publishing in one or two areas, but are extremely knowledgeable in pretty much everything I ask them about. That begs the questions:

1. How much reading outside should I be doing outside my "area"?
2. Is it a good idea to just focus narrowly on my thesis problem at this stage or is it more usual to be working on multiple problems at the same time?
3. How and how often do you end up learning new areas?

Sorry if the question is too vague: I just want to have a sense of how to go about being a good mathematician. Also, part of the reason I am asking this question is that when I go to seminars, I understand so little and I see some of my professors ask questions of the speakers even if they don't work in the same area.

I am about to (hopefully!) begin my PhD (in Europe) and I have a question: how did you learn so much mathematics?

Allow me to explain. I am training to be a number theorist and I have only some read Davenport's Multiplicative Number Theory and parts of Vaughan's book on the circle method. I have briefly seen some varieties from Fulton's algebraic curves and I may have read parts of homotopy and homology and differential geometry of smooth manifolds at the level of Hatcher and Lee. Yet, it seems that I am hopelessly ignorant of elliptic curves, modular forms and algebraic number theory.

For example, if I were to try reading Deligne's proof of Weil's conjecture or Tate's thesis, it seems that I would have to do significant amounts of reading.

When I look at some of my professors or other researchers I have interacted with, I notice that they may be publishing in one or two areas, but are extremely knowledgeable in pretty much everything I ask them about. That begs the questions:

1. How much reading outside should I be doing outside my "area"?
2. Is it a good idea to just focus narrowly on my thesis problem at this stage or is it more usual to be working on multiple problems at the same time?
3. How and how often do you end up learning new areas?

Sorry if the question is too vague: I just want to have a sense of how to go about being a good mathematician. Also, part of the reason I am asking this question is that when I go to seminars, I understand so little and I see some of my professors ask questions of the speakers even if they don't work in the same area.