= Arithmetic geometry
Arithmetic geometry studies <polynomial equations> using both <algebraic geometry> and <number theory>. The coefficient arithmetic matters: a geometric object may have real points while having no integer or rational points of the required kind.
For example, $x^2+y^2=3$ has real solutions, but no solutions in the <integers>. A square is congruent to zero or one modulo four, so a sum of two squares cannot be congruent to three. Such <polynomial congruences> modulo a prime or prime power can therefore obstruct the existence of global arithmetic solutions.
In the language of <schemes>, an object over $\operatorname{Spec}\mathbb Z$ has fibres over prime numbers that are varieties over <finite fields>, together with a generic fibre over $\mathbb Q$. Comparing these fibres links the geometry of curves and <abelian varieties> to their arithmetic. Local solvability need not guarantee global solvability; the passage between these questions is a central source of structure and difficulty.
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