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For a non-Archimedean local field $F$, the Schwartz-Bruhat space $\mathcal{S}(F)$ consists of the locally constant, compactly supported complex-valued functions on $F$.

If an elliptic curve has good reduction over a local field $K$, its integral projective model is smooth. Every point over the residue field lifts by the Hensel lemma, applied in a chart with a unit partial derivative. Thus reduction gives an exact sequence $0\to E_1(K)\to E(K)\to \tilde{E}(k)\to 0$. The kernel of reduction of an elliptic curve is the maximal ideal of the valuation ring, with the formal group of an elliptic curve as its group operation.

For a finite unramified extension of non-Archimedean local fields, the field norm on residue units is surjective: on a cyclic group of order $q^n-1$ it is exponentiation by $(q^n-1)/(q-1)$ and has image of order $q-1$. For a common uniformizer $\pi$ and $r\ge 1$, expansion of the conjugate product givesThe field trace of a finite finite field extension is onto: its polynomial $X+X^q+\cdots +X^{q^{n-1}}$ is nonzero of degree below $q^n$, so defines a nonzero linear map to the base field. Lift a residue norm and correct successively with elements of $1+{\pi }^r{\mathcal{O}}_L$. The infinite product converges by completeness, and its norm is the prescribed unit. This argument also works in positive characteristic and when $p$ divides $n$.

The biquadratic morphism for elliptic sums and differences has no common zero, so the local coefficient norms giveuniformly in $P,Q$. Upper bounds use the fixed coefficients. Lower bounds follow from the absence of common zeros over every local field; the constants are $1$ outside finitely many places, where the integral morphism still has no base point after reduction.

For the binary quadratic with roots $u,v\in {\mathbb{P}}^1(K)$, the projective height of its coefficient vector is $h(u)+h(v)+O(1)$. At nonarchimedean places the coefficient maximum norm of a product of linear polynomials is the product of their maximum norms by the multiplicativity of the nonarchimedean polynomial norm. At archimedean places the same norms are comparable up to a fixed factor for degree two. This also includes roots at infinity. Combining the comparisons yields $h_x(P+Q)+h_x(P-Q)=2h_x(P)+2h_x(Q)+O(1)$. After dividing the identity for $2^{n}P,2^{n}Q$ by $4^n$ and taking limits, it gives the exact height parallelogram identity for the canonical height of an elliptic curve.

For an integral element of a finite extension of non-Archimedean local fields, reduce the multiplication endomorphism on ${\mathcal{O}}_L$ modulo a base uniformizer. Its filtration by the powers of the maximal ideal has $e$ successive quotients, each the residue field ${\kappa }_L$. Multiplication by $a$ induces multiplication by $\bar{a}$ on each quotient. Additivity of the matrix trace through the filtration proves the formula. If the residue characteristic divides the ramification index, the trace of every integral element lies in the base maximal ideal; a Galois extension is not required.

For a finite Galois extension of non-Archimedean local fields, this is a homomorphism from the inertia group to $k_L^{\times }$ with kernel the wild inertia group. Inertia acts trivially on residue units, which proves multiplicativity and independence of the chosen uniformizer. Passing to the maximal unramified subextension reduces the kernel calculation to a totally ramified extension. Hence it embeds the tame inertia quotient into the cyclic multiplicative group of the residue field.

Let $E/K$ have good reduction over a local field of residue characteristic $p$, with $p\nmid n$. All solutions of $nQ=P$, for all $P\in E(K)$ together, lie in a single finite unramified extension of degree at most $nf$, where $f=[K(E[n]):K]$. First lift a solution over the algebraic closure of the residue field to a finite unramified extension; the multiplication isomorphism of a formal group law uniquely corrects the error in $E_1$. This also puts $E[n]$ in the maximal unramified extension. If $\sigma$ is its Frobenius generator, ${\sigma }^f$ fixes $E[n]$. For $nQ=P$, set $T={\sigma }^{f}Q-Q\in E[n]$. Then ${\sigma }^{nf}Q=Q+nT=Q$, uniformly in $P$.

For an unramified extension of non-Archimedean local fields, let $\varphi$ generate its cyclic Galois group. The ordinary Hilbert theorem 90 supplies $b\in L^{\times }$ with $x=\varphi (b)/b$. A common uniformizer $\pi \in K$ gives $y={\pi }^{-v_L(b)}b\in {\mathcal{O}}_L^{\times }$, retaining the same quotient because $\varphi$ fixes $\pi$. The reverse implication follows by telescoping the field norm.

For a finite extension of local fields with normalized discrete valuations, the field norm satisfies the displayed equality. The extended absolute values on a field of all conjugates agree, so $|N(x){|}_K=|x{|}_L^{[L:K]}$. Using $[L:K]=e(L/K)f(L/K)$ and $v_L{|}_K=e(L/K)v_K$ gives the equality. Thus the field norm is a unit precisely when the original element is a unit.

If $L/K$ is an unramified extension and $M/K$ is a totally ramified extension of non-Archimedean local fields inside a fixed algebraic closure, then $LM/L$ is totally ramified of degree $[M:K]$. A uniformizer of $M$ has an Eisenstein polynomial over $K$. Since a uniformizer of $K$ remains a uniformizer in $L$, that polynomial remains Eisenstein over $L$ and generates $LM$.

At a prime of good reduction of an elliptic curve with residue characteristic not dividing $m$, multiplication by $m$ extends to a finite étale map of the smooth proper elliptic group scheme over the valuation ring. A local rational point extends to a section by properness. Its division fibre is therefore a finite étale torsor, so the Kummer map of an elliptic curve gives an unramified cohomology class. This bounds ramification in the Weak Mordell-Weil theorem.

A direct proof describes the coordinates, not only the cohomology class. Reduce $P$ by properness and choose a point $\overline{Q}$ with $[m]\overline{Q}=\tilde{P}$ over a finite extension of the residue field. Over the corresponding unramified extension of the complete local field, lift $\overline{Q}$ by the Hensel lemma. The error $[m]Q_0-P$ is in the kernel of reduction of an elliptic curve; prime-to-residue-characteristic multiplication on a formal group corrects it uniquely. There is no $m$-torsion in that kernel, even after a ramified field extension, so every division point with that reduction equals this unramified lift. Consequently all division points of an elliptic curve above $P$ have unramified coordinate fields. The torsion case is discussed in J. S. Milne's Elliptic Curves, Chapter V, Proposition 8.1.

For non-Archimedean local fields, let $L/K$ be an unramified extension of degree $f$, and let $E/K$ be finite. The degree-$f$ unramified extension of $E$ has a residue field containing the degree-$f$ residue extension of $K$. Lift an integral residue generator of $L$ into that field by Hensel lemma. This embeds $L$ into an unramified extension of $E$, so $LE/E$ is an unramified subextension. Applying this argument to each completion of a number field at a prime ideal proves the finite-prime assertion for number fields, even when $E/K$ is ramified.

For an unramified extension of local fields $L/K$, an integral element whose residue generates $k_L/k_K$ generates $L/K$: its degree is at least the residue degree $f=[L:K]$. The reduced powers $1,\bar{\alpha },\dots ,{\bar{\alpha }}^{f-1}$ are a basis of the residue field extension. If $x=\sum c_j{\alpha }^j$, factor out a coefficient of least valuation. Reduction of the remaining sum is nonzero by that residue basis, so $v_L(x)={\min }_j{v}_K(c_j)$. Consequently $x$ is integral exactly when all $c_j$ are integral.