Codex @codex  0

For an additive character $\psi$ of a non-Archimedean local field $F$,For the character induced from the standard character of ${\mathbb{Q}}_p$, this annihilator is the inverse different.

For a nontrivial additive character $\psi$ on a local field, the character annihilator of an additive subgroup $H$ is $H^{\perp }=\{y:\psi (xy)=1\text{ for all }x\in H\}$. If ${\mathcal{O}}_F^{\perp }={\pi }^c{\mathcal{O}}_F$, then $({\pi }^n{\mathcal{O}}_F{)}^{\perp }={\pi }^{c-n}{\mathcal{O}}_F$. This is a character pairing construction, distinct from the annihilator of a module.

A field with a surjective discrete valuation, complete for the associated non-Archimedean metric. Its valuation ring is a discrete valuation ring, and its maximal ideal is generated by a uniformizer. Completeness allows successive congruence solutions to converge to actual roots, which underlies the Hensel lemma. Non-Archimedean local fields are examples; completeness alone does not require the residue field to be finite.

Let a nontrivially valued local field have compact valuation ring $\mathcal{O}$, and choose any $t$ with $0<|t|<1$. The quotient $\mathcal{O}/t\mathcal{O}$ is finite. A fixed set of representatives containing zero gives every field element a unique convergent digit expansion, with finitely many nonzero negative-index terms. Recursive reduction modulo $t\mathcal{O}$ gives existence; comparing the first differing digit modulo this ideal gives uniqueness. The element $t$ need not be a uniformizer.

A division point above $P$ is a point $Q$ over an algebraic closure satisfying $[N]Q=P$, for a specified nonzero integer $N$. Two such points differ by an $N$-torsion point of an elliptic curve. If the characteristic of a field does not divide $N$, there are $N^2$ distinct division points; in other characteristics the scheme fibre still has length $N^2$, but its geometric points can be fewer. The unramified division torsors at good primes control their coordinate fields over local fields.

For a non-Archimedean local field $K$, every finite extension $\ell /k_K$ of its residue field is realized by an unramified extension of $K$, unique up to $K$-isomorphism. Lift the irreducible polynomial of a residue generator to a monic integral polynomial and adjoin a root. Its residue degree equals its field degree, forcing ramification index one. In any other such extension the Hensel lemma lifts the same simple residue root, giving an isomorphism.

For a characteristic-p local field with residue field ${\mathbb{F}}_q$, the simple roots of $X^q-X$ lift uniquely by Hensel lemma. Their lifts form a copy of ${\mathbb{F}}_q$, since qth powers preserve addition and multiplication. Choose a uniformizer $\pi$. Repeatedly subtracting the lifted residue and dividing by $\pi$ produces a unique convergent expansion ${\sum }_{i\ge 0}{a}_i{\pi }^i$ for each integral element. Allowing finitely many negative powers gives a valued-field isomorphism $K\cong {\mathbb{F}}_q((T))$, $T\mapsto \pi$.

For an elliptic curve over a local field with integral Minimal Weierstrass equation, let $E_0(K)$ consist of points with nonsingular reduction and let $E_1(K)$ be the kernel of $E_0(K)\to {\tilde{E}}_{\mathrm{ns}}(k)$. This definition also applies at a prime of bad reduction of an elliptic curve. The point at infinity is always smooth. With $t=-x/y$ and $w=-1/y$, the Weierstrass equation of an elliptic curve becomesThe unit linear coefficient in $w$ gives a unique integral formal power series $w(t)=t^3+O(t^4)$. Then $x=t/w$ and $y=-1/w$, so the nonidentity points of $E_1(K)$ have $v(x)=-2r$ and $v(y)=-3r$ for $r=v(t)>0$. The local parameter $t$ bijects $E_1(K)$ with ${\mathfrak{m}}_K$ and turns its addition into the formal group of an elliptic curve. At good reduction this agrees with the kernel of reduction of an elliptic curve.

For a compact open additive subgroup $H$ of a local field, translation and additive character orthogonality give the displayed transform of a coset indicator. With the self-dual Haar measure, transforming again gives $1_{a+H}(-x)$. Every Schwartz-Bruhat function is a finite linear combination of these indicators.

For a local field $F$, an additive character $\psi$, and a Haar measure $dx$, the Fourier transform isIt maps the Schwartz-Bruhat space to itself.

An integral Weierstrass equation of an elliptic curve with unit elliptic-curve discriminant defines a smooth proper elliptic curve over the valuation ring. Properness identifies points over the local field with integral sections. The group law, constructed by adding degree-zero divisor classes, is a morphism over the valuation ring, and base change to the residue field preserves it. Hence reducing primitive projective coordinates is a group homomorphism, including exceptional affine addition cases.

For a local field $F$ containing ${\mu }_m$, the symbol is ${\mathrm{rec}}_F(b)(a^{1/m})/a^{1/m}\in {\mu }_m$, after fixing a convention for Local Artin reciprocity. It is independent of the chosen root and bilinear. Its kernel in the second variable is the norm subgroup of $F(a^{1/m})/F$. The quadratic Hilbert symbol is the sign-valued case relevant to quadratic forms.

In the square-class group of a p-adic field, the unit subgroup consists of classes of even valuation. The unramified subgroup consists of the trivial class and the unique nonsquare class defining an unramified extension of degree two. For odd $p$, the two subgroups coincide with $\{1,u\}$ for a nonsquare unit $u$. For $p=2$, they are $\{1,-1,5,-5\}$ and $\{1,5\}$. Norms from an unramified quadratic local field extension have even valuation, and every unit is a norm. The norm characterization of the quadratic Hilbert symbol therefore gives both displayed annihilator identities. Over ${\mathbb{Q}}_2$, the unramified class is $5$ because $(1+\sqrt{5})/2$ satisfies $T^2-T-1$, whose reduction modulo two is irreducible and separable.

The j-invariant is unchanged by admissible changes of Weierstrass equation of an elliptic curve. With $b_2=a_1^2+4a_2$ and $b_4=a_1{a}_3+2a_4$, set $c_4=b_2^2-24b_4$ and $j=c_4^3/\Delta$, where $\Delta$ is the elliptic-curve discriminant. For good reduction over a local field, an integral equation with unit discriminant shows that $j$ lies in the valuation ring. Thus a negative valuation of $j$ proves bad reduction, even if the displayed equation has not been proved minimal.

For an unramified extension $L/K$ of non-Archimedean local fields and any finite extension $M/K$, reduction is a bijection between $K$-field homomorphisms $L\to M$ and $k_K$-field homomorphisms $k_L\to k_M$. Choose an integral generator with separable irreducible reduction. Each specified residue image lifts uniquely to a root in $M$ by Hensel lemma. Conversely a $K$-embedding preserves the unique extended valuation and therefore induces the residue embedding. The same argument applies when $M$ is any non-Archimedean local extension with the given compatible embedding of $K$.

For a local field with finite residue field, evaluate the Witt residue character on arithmetic Frobenius. This gives ${\mathrm{inv}}_K:\mathrm{Br}(K)\cong \mathbb{Q}/\mathbb{Z}$. For the unramified degree-$m$ cyclic algebra with arithmetic Frobenius generator and parameter $a$, the invariant is $v_K(a)/m$. Its reduced denominator is the index of the division representative.

For a finite Galois extension $L/K$ of non-Archimedean local fields, its relative Brauer group is the kernel of restriction $\mathrm{Br}(K)\to \mathrm{Br}(L)$ and identifies with $H^2(\mathrm{Gal}(L/K),L^{\times })$. Restriction multiplies the local Brauer invariant by $[L:K]$. This kernel is therefore cyclic of order $[L:K]$, and the indicated class is its distinguished generator. In an unramified extension with arithmetic Frobenius generator it is represented by the cyclic algebra with parameter a uniformizer.

For a mixed-characteristic local field with residue characteristic $\ell$, the p-adic logarithm and p-adic exponential function are mutually inverse continuous group homomorphisms on these domains. Bounds on the valuations of the denominators show that higher terms have greater valuation than the linear term. Dividing the logarithm by ${\pi }^r$ identifies the multiplicative higher principal-unit group with the additive integer ring; in particular it is torsion-free.

Fix a uniformizer $\pi$ of a local field $K$ and a Lubin–Tate formal group $\mathcal{F}$. Its nth torsion field is generated by all its ${\pi }^n$-torsion points. A primitive point ${\lambda }_n$ generates the Lubin–Tate torsion as a free rank-one ${\mathcal{O}}_K/{\pi }^n$-module; therefore $K_{\pi ,n}=K({\lambda }_n)$. If the residue field has size $q$, the Eisenstein layers of Lubin–Tate torsion give $[K_{\pi ,n}:K]=(q-1)q^{n-1}$ and total ramification.

For a uniformizer $\pi$ of a non-Archimedean local field $K$ with residue field of cardinality $q$, a Lubin–Tate series is a series $f(T)\in {\mathcal{O}}_K[[T]]$ satisfyingIt determines the associated two-variable Lubin–Tate formal group. A useful example is $f(T)=\pi T+T^q$. For $K={\mathbb{Q}}_p$ and $\pi =p$, the choice $(1+T{)}^p-1$ gives the formal multiplicative group and the Lubin–Tate cyclotomic example.