TITLE: How much do I need to learn algebraic geometry to understand arithmetics over number fields
QUESTION [11 upvotes]: I am at the stage of learning. Mostly, I am attracted by algebraic number theory. Roughly speaking, I am interested in the rational points of algebraic varieties. I am little bit afraid to start to learn algebraic geometry, since I find the modern language used there is category theory (topos, sheaves, schemes,...). Here is my question:
Is it necessary to know the modern algebraic geometry (Grothendieck formalism) to explore the theory of rational points of algebraic varieties?
Concrete examples are welcome.

REPLY [4 votes]: I'm going to say no: it's not necessary to know the modern algebraic geometry to explore the theory of rational points of algebraic varieties.  Obviously, as pointed out in some of the answers, it's helpful, because modern algebraic geometry is used in many of the main results proved in the last century.
But Arithmetic geometry is a HUGE subject, and the techniques people use to prove new theorems, improve existing ones, etc. very often don't involve modern algebraic geometry.  Even more often, one can get away with just black-boxing some basic ideas from modern algebraic geometry, especially if you're trying to prove results about certain varieties that are well-understood geometrically.  (For example, you can read a ton of papers about elliptic curves without knowing more than classical algebraic geometry.)  In addition to scheme theory, étale cohomology, stacks, etc., there are a wealth of other concepts that are super important, such as (some of these have been mentioned by others) 

the circle method, exponential sums,...
zeta- and $L$-functions
heights
modular forms
geometry of numbers (counting lattice points)
group theory & representation theory (e.g. for studying Galois representations)
probability theory (e.g. many asymptotic problems in arithmetic geometry and number theory in general are modeled by random phenomena, which leads to powerful intuition.)
computation (like on a computer)
classical algebraic geometry (a good understanding of which can be disjoint from a good understanding of scheme theory -- you have to stay grounded!)
transcendence theory (big in unlikely intersections lately, in combination with tools from model theory...)
diophantine approximation (later proofs of Faltings' Theorem)
$p$-adic analysis, Berkovich spaces,...
dynamics (not just in "arithmetic dynamics," but also big in diophantine approximation)
ergodic theory
geometric group theory

Understanding everything is a tall order, but that's why we collaborate!  Progress in arithmetic geometry is made by combining ideas from all of these areas and more.  Don't be afraid of what you don't know.