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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 $\mathrm{Spec}\mathbb{Z}$ 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.

Hensel lifting improves a solution of a polynomial congruence modulo a prime power to one modulo a higher power. For $f\in \mathbb{Z}[X]$, suppose $f(a)\equiv 0(\mathrm{mod}{p}^k)$ and $f^{\prime }(a)\not\equiv 0(\mathrm{mod}p)$. A lift $a+t{p}^k$ satisfiesThe invertible derivative therefore determines a unique $t$ modulo $p$.

For $f(X)=X^2-2$, the root $3$ modulo seven lifts to $10$ modulo $49$, since $3+7=10$ and $10^2-2=98$. Repetition gives a root in the p-adic integers. The simple-root hypothesis ensures uniqueness; when the derivative vanishes modulo $p$, there may be multiple lifts or none.