Which areas of arithmetic algebraic geometry can be learned as "black boxes" and are there any references where they are treated as such? - MathOverflow

Which areas of arithmetic algebraic geometry can be learned as "black boxes" and are there any references where they are treated as such?

2011-02-15T19:20:24Z

In Matthew Emerton's comment on Terry Tao's blog, he speaks about learning etale cohomology or the theory of Neron models as "black boxes". By this he means that you can learn what the theory is about and how to use it, without going into the detailed proofs of why they can be used.

Which theories (e.g. etale cohomology) can be learned as black boxes?

And where would one go (e.g. find lecture notes) to learn something like that?

Notes on something like this would ideally give you an idea of what is going on, give examples, and most importantly illustrate how they would be used to solve problems. I am mainly interested in arithmetic algebraic geometry and algebraic number theory, so I would especially like to know about "black boxes" in this direction, though "black boxes" in other areas might also be worth knowing about.

Answer by known google for Which areas of arithmetic algebraic geometry can be learned as "black boxes" and are there any references where they are treated as such?

2011-02-16T03:20:25Z

I have found Groebner bases to be incredibly useful for testing concrete ideas about varieties, and although I did spend time learning how they work, all I've ever really needed to know is how to interpret the results of a Groebner Basis calculation, and how to choose monomial orders that will produce useful answers. I don't actually know how the most efficient algorithms and their implementations (Fauguère F4 and F5) arrive at an answer---the textbook algorithms are painfully slow for complicated calculations.

Answer by Georges Elencwajg for Which areas of arithmetic algebraic geometry can be learned as "black boxes" and are there any references where they are treated as such?

2011-02-16T10:29:03Z

I find Hodge theory pretty scary stuff with its compact inclusions of Sobolev spaces, pseudodifferential operators and parametrixes for elliptic differential operators. However it is very easy to use the results of Hodge theory as emanating from a black box. I remember how exhilarated I was by the argument that a Hopf surface, homeomorphic to $S^1 \times S^3$, could not be Kähler, and much less projective, just because its first Betti number is $b_1=1$. Whereas by Hodge theory a compact Kähler manifold $X$ has betti numbers $b_q(X)$ which are even whenever $q$ is odd.