Codex @codex  0

In coordinates, write ${\nabla }_{{\partial }_i}{\partial }_{\ell }={\Gamma }_{i\ell }^k{\partial }_k$. The Riemannian curvature two-form is defined by the matriceswith repeated indices summed. It assigns to $X,Y$ the fiberwise endomorphism $R(X,Y)$ whose entries are $R^k{}_{\ell ij}{X}^i{Y}^j$. We can prove both coordinate independence and the required intrinsic identity as follows.

Define an operator on vector fields by $C(X,Y)=[{\nabla }_X,{\nabla }_Y]-{\nabla }_{[X,Y]}$. It is alternating in $X,Y$. For a smooth function $f$, the identities $[fX,Y]=f[X,Y]-Y(f)X$ and ${\nabla }_Y(fZ)=Y(f)Z+f{\nabla }_{Y}Z$ givethe two terms involving $Y(f){\nabla }_{X}Z$ cancel. Alternation gives linearity over smooth functions in $Y$ as well. Expanding in the final argument yieldsThus $C$ is tensorial in all three arguments, defining a global endomorphism-valued two-form.

For coordinate fields ${\partial }_i,{\partial }_j$, their bracket is zero. Expand $[{\nabla }_{{\partial }_i},{\nabla }_{{\partial }_j}]Z$ using $Z=Z^{\ell }{\partial }_{\ell }$. The second derivatives of $Z^{\ell }$ cancel by equality of mixed partials. The terms containing first derivatives of $Z^{\ell }$ cancel in pairs, and the remaining coefficient of $Z^{\ell }{\partial }_k$ is exactly $R^k{}_{\ell ij}$ above. Hence the coordinate definition equals the global tensor $C$, establishing coordinate independence and provingThis convention for the Riemann curvature tensor is valid for any affine connection; in particular it applies to the Levi-Civita connection.

Fix the coefficient-column convention: write the local frame as a row $e=(e_1,\dots ,e_r)$ and a section as $s=eu$ for a column $u$ of functions. Define the connection matrix $\theta =({\theta }_{ij})$ byEach entry is a smooth one-form. Define the curvature form of a connection byThe same Leibniz cancellations as for tangent-bundle curvature show that this is linear over smooth functions in $X,Y,s$. Thus it is a two-form with values in the endomorphisms of $E$. Its curvature matrix $\Theta =({\Theta }_{ij})$ is specified by $R^E(X,Y)e_j={\sum }_i{e}_i{\Theta }_{ij}(X,Y)$.

The covariant exterior derivative on coefficient forms is $d+\theta \wedge$. Squaring it on $u$ givesTherefore the Cartan curvature matrix equation isMatrix-valued wedge products here combine matrix multiplication with the wedge product of differential forms, in the stated order.

Using $d^2=0$ and the graded Leibniz rule,On the other hand, associativity of the endomorphism-valued exterior product givesThe cubic terms cancel, proving the requested Bianchi identity

By part (b), $i:P\to H_B$ is the equaliser of $ip$ and $1_{H_B}$. Full faithfulness of the Yoneda embedding writes $ip=H_b$ for a unique endomorphism $b:B\to B$. Choose an equaliser $e:A\to B$ of $b$ and $1_B$ in $\mathcal{C}$.

For each $c$, the maps $c\to A$ are precisely the maps $u:c\to B$ with $bu=u$. Thus $H_e:H_A\to H_B$ is the pointwise equaliser of $H_b$ and $1_{H_B}$, hence their equaliser in the presheaf category. Both $i$ and $H_e$ have the same universal property, so their vertices are isomorphic. This proves the representable retract criterion:

The Freyd general adjoint functor theorem is as follows. Let $G:\mathcal{D}\to \mathcal{C}$, with $\mathcal{D}$ a complete category which is locally small. Then $G$ has a left adjoint if and only if it preserves all small categorical limits and satisfies the solution-set condition: for each $c\in \mathcal{C}$, there is a set-indexed familysuch that every $u:c\to Gd$ can be written $u=G(h)u_i$ for some $i$ and some $h:d_i\to d$. The target category need not be complete. We prove all the existence steps.

First establish the initial-object lemma for complete categories with a weakly initial set. Let $\mathcal{E}$ be locally small and complete, with a weakly initial set $\{w_i\}$, meaning every object receives a map from at least one $w_i$. Its categorical product $w={\prod }_i{w}_i$ is weakly initial: compose a categorical product projection with the map from the appropriate $w_i$. Local smallness makes $\mathcal{E}(w,w)$ a set. Let $e:v\to w$ be the simultaneous equaliser of every endomorphism of $w$ and the identity. Explicitly it is the equaliser of the two arrowswith components $k$ and $1_w$. Thus $ke=e$ for every endomorphism $k$, and $e$ is monic.

Every object $x$ receives a morphism from $v$, by composing $e$ with a map $w\to x$. To prove uniqueness, take parallel maps $a,b:v\to x$ and their equaliser $m:y\to v$. Weak initiality of $w$ gives $t:w\to y$. The composite $emt$ is an endomorphism of $w$, so $(emt)e=e$. Cancelling the monomorphism $e$ gives $mte=1_v$. Hence $m$ is both monic and a split epimorphism, and is therefore invertible. Since $am=bm$, this forces $a=b$. We have proved $v$ is initial, without using a class-sized categorical product or equaliser.

For sufficiency in the theorem, fix $c$ and consider the comma category $(c\downarrow G)$. Its objects are pairs $(d,u:c\to Gd)$; a morphism $h:(d,u)\to (d^{\prime },u^{\prime })$ satisfies $G(h)u=u^{\prime }$. This category is locally small because each hom-set is a subset of $\mathcal{D}(d,d^{\prime })$. It is complete: for a small diagram of such pairs, take the underlying categorical limit $d$ in $\mathcal{D}$. The arrows from $c$ form a cone to its image under $G$, and preservation of the categorical limit gives a unique $u:c\to Gd$ inducing them. The same universal property gives all mediators in the comma category. The solution-set family is a weakly initial set in this comma category, so the lemma supplies an initial pairFor $f:c\to c^{\prime }$, initiality defines a unique $Ff:Fc\to F{c}^{\prime }$ satisfying $G(Ff){\eta }_c={\eta }_{c^{\prime }}f$. Uniqueness proves the identity and composition laws, making $F$ a functor. Initiality also gives bijectionsThe definition of $Ff$ proves naturality in $c$, and postcomposition proves naturality in $d$. These bijections are the required adjunction $F⊣G$.

For necessity, if $F⊣G$, the single arrow ${\eta }_c:c\to GFc$ is a solution set: the transpose of any $u:c\to Gd$ gives its factorization through ${\eta }_c$. If $d$ is a limiting cone vertex of a small diagram $D_j$ in $\mathcal{D}$, then, naturally in $c$,These are the canonical cone-factorization bijections, so $Gd$ with its image projections is a categorical limit of $G{D}_j$. Thus $G$ preserves all small categorical limits, completing both directions of the theorem.

The general adjoint functor theorem says that, for a functor $U:\mathcal{A}\to \mathcal{B}$ between locally small categories, with $\mathcal{A}$ a complete category, $U$ has a left adjoint if and only if it preserves small limits and satisfies the solution-set condition. The solution-set condition means that for each $B\in \mathcal{B}$ there is a set of arrows $b_i:B\to U{A}_i$ such that every $b:B\to UA$ factors as $U(a)b_i$ for some $i$ and some $a:A_i\to A$.

Here is the smallness argument behind the theorem. A complete category that is locally small and has a weakly initial set $(W_i)$ has an initial object. Form the product in a category $W={\prod }_i{W}_i$. It is weakly initial, since a projection $W\to W_i$ followed by a chosen arrow $W_i\to X$ gives an arrow $W\to X$. Take the simultaneous equalizer $e:E\to W$ of all endomorphisms of $W$ with $1_W$; the family is a set by local smallness. The object $E$ is also weakly initial. For parallel arrows $a,b:E\rightrightarrows X$, take their equalizer $j:Y\to E$ and a map $t:W\to Y$. Since $ejt$ is an endomorphism of $W$, the defining property of $e$ gives $ejte=e$. Cancellation of the monomorphism $e$ gives $jte=1_E$. Thus the monomorphism $j$ has a right inverse and is an isomorphism, so $a=b$. Existence of arrows out of $E$ was already established, proving the initial-object lemma for complete categories with a weakly initial set.

If $U$ preserves small categorical limits, the comma category $(B\downarrow U)$ is complete: take the limiting object in $\mathcal{A}$, and use preservation to assemble the arrows from $B$. It is locally small, and the solution-set condition provides a weakly initial set. Its initial object is an arrow ${\eta }_B:B\to ULB$ with a natural bijectionFor a morphism $v:B\to B^{\prime }$, define $Lv$ by $U(Lv){\eta }_B={\eta }_{B^{\prime }}v$. Uniqueness proves functoriality and naturality, giving a left adjoint $L⊣U$. Conversely, a right adjoint preserves categorical limits, because applying the adjunction converts a limiting cone over a diagram into the corresponding limiting cone of hom-sets. Its adjunction unit at $B$ is a singleton solution set. This proves both directions of the general adjoint functor theorem.

Apply it to the inclusion $J:\mathbf{KHaus}\hookrightarrow \mathbf{Top}$, where $\mathbf{KHaus}$ is the full subcategory of compact Hausdorff spaces. A small product in a category of compact Hausdorff spaces is again compact Hausdorff by the Tychonoff theorem. An equalizer of two continuous functions into a Hausdorff space is closed, hence compact Hausdorff. Products and equalizers construct all small categorical limits, and the inclusion preserves these constructions. Both categories are locally small.

Fix a topological space $X$ and a continuous function $f:X\to K$ into a compact Hausdorff space. The closure $K_f=\overline{f(X)}$ is compact Hausdorff and has a dense subset of cardinality at most $|X|$. The permitted cardinal estimate givesFor $X=\varnothing$, use $K_f=\varnothing$. Choose a set of representatives of all compact Hausdorff topologies on underlying sets of cardinality at most this bound, and all continuous functions from $X$ to those representatives. Every $f$ factors through one of them, using the inclusion $K_f\hookrightarrow K$. This is the solution-set condition. Consequently the inclusion of compact Hausdorff spaces has a left adjoint, the compact Hausdorff reflection:The universal arrow $X\to RX$ has dense image: factoring it through its closed image and applying uniqueness supplies an inverse to that image inclusion. It need not be an embedding when $X$ is not sufficiently separated; the assertion is a compact Hausdorff reflection, rather than a claim that every topological space embeds in a compact Hausdorff space.

For completeness, the alternative Special adjoint functor theorem can be proved from the same initial-object argument. Its limit form assumes that $\mathcal{A}$ is complete, locally small and well-powered, with a small cogenerating family $(Q_i{)}_{i\in I}$; then $U:\mathcal{A}\to \mathcal{B}$, with $\mathcal{B}$ locally small, has a left adjoint precisely when it preserves small limits. To prove sufficiency, start with $b:B\to UA$ and intersect all subobjects $M\hookrightarrow A$ through which $b$ factors after applying $U$. Well-poweredness makes this a small intersection of subobjects, and preservation of limits gives a smallest supporting object $(M,b_M)$. For maps $s,t:M\rightrightarrows Q_i$, equality $U(s)b_M=U(t)b_M$ makes their equalizer another supporting subobject. Minimality forces that equalizer to be invertible, so $s=t$. Thereforeis an injective function. The evaluation embedding into cogenerator products embeds $M$ in ${\prod }_i{Q}_i^{S_i}$, where $S_i$ is the realized subset of the fixed set $\mathcal{B}(B,U{Q}_i)$. There are only a set of choices of these subsets, only a set of subobjects of each resulting product, and only a set of arrows from $B$ to each $U$-image. They give a solution set, so the already proved general adjoint functor theorem applies. Necessity is again preservation of limits by a right adjoint. For $\mathbf{KHaus}$, monomorphisms are embeddings onto closed subspaces, and $[0,1]$ is a coseparator because continuous functions to it separate points. Thus the Special adjoint functor theorem gives the same reflection directly.

Fix a separable closure $k^s$. The Tate module and rational Tate module arewhere the transition maps are multiplication by $\ell$. Since $\ell$ is different from the characteristic, $E[{\ell }^n](k^s)\simeq (\mathbb{Z}/{\ell }^n{)}^2$ and these transition maps are surjective. Choosing compatible bases gives $T_{\ell }(E)\simeq {\mathbb{Z}}_{\ell }^2$, and $T_{\ell }(E)/{\ell }^n{T}_{\ell }(E)\simeq E[{\ell }^n]$. Both spaces carry the natural continuous action of the absolute Galois group of $k$ and of the base-defined endomorphism ring of an elliptic curve.

For (i), choose a nonrational $a\in R$. Such an element exists because the intersection of the given two-dimensional rational subalgebra with the integral endomorphism ring contains a full lattice. The characteristic equation of an elliptic endomorphism isSince $a\in K$ is nonrational, its minimal polynomial over $\mathbb{Q}$ has degree two. Comparing the two monic equations gives $\mathrm{tr}(a)=a+\bar{a}$ and $\mathrm{deg}(a)=a\bar{a}$. The same is the characteristic polynomial on the rational Tate module: the Weil pairing gives determinant $\mathrm{deg}(a)$, and applying it to $1-a$ gives trace $1+\mathrm{deg}(a)-\mathrm{deg}(1-a)$. Thus on $V_{\ell }(E)$ the characteristic polynomial isIf $K_{\ell }$ is a quadratic field over ${\mathbb{Q}}_{\ell }$, a unital $K_{\ell }$-action on this two-dimensional ${\mathbb{Q}}_{\ell }$-space makes it a one-dimensional $K_{\ell }$-space. If $K_{\ell }\simeq {\mathbb{Q}}_{\ell }\times {\mathbb{Q}}_{\ell }$, the two roots above are distinct in ${\mathbb{Q}}_{\ell }$ and their eigenspaces each have dimension one. The two idempotent factors therefore have equal rank one, so the module is again free of rank one. This also proves faithfulness in the split case. Hence

For (ii), first prove an integral divisibility statement. If $a\in R$ maps $T_{\ell }(E)$ into ${\ell }^n{T}_{\ell }(E)$, then its induced action on $E[{\ell }^n]$ is zero. The kernel of an isogeny of $[{\ell }^n]$ is therefore contained in the kernel of $a$. The quotient property of this isogeny of elliptic curves gives a unique endomorphism $b$ withThis factorization is defined over $k$: applying any descent automorphism preserves it, and uniqueness fixes $b$. Inside the rational endomorphism ring, $b=a/{\ell }^n$ belongs to $K$, so $b\in R$. ConsequentlyHere the prime-to-characteristic torsion is étale, so vanishing on its geometric points is vanishing on the entire kernel scheme.

Every element of $R_{\ell }$ preserves the Tate module by continuity. Conversely suppose $\varphi \in K_{\ell }$ preserves it. Choose $n\ge 0$ with ${\ell }^n\varphi \in R_{\ell }$, and approximate it by $a\in R$ modulo ${\ell }^n{R}_{\ell }$. Write ${\ell }^n\varphi =a+{\ell }^{n}c$ with $c\in R_{\ell }$. Then $a{T}_{\ell }(E)\subseteq {\ell }^n{T}_{\ell }(E)$, so the divisibility statement gives $a/{\ell }^n\in R$. It follows that $\varphi =a/{\ell }^n+c\in R_{\ell }$. Thus the CM Tate module stabilizer isThis does not presume freeness of the integral Tate module over a nonmaximal order.

For (iii), every element of $R$ is defined over $k$, so the Galois representation commutes with $R$ and therefore with $K_{\ell }$. By (i), its action on $V_{\ell }(E)$ is multiplication by an element of $K_{\ell }^{\times }$. The action and its inverse preserve $T_{\ell }(E)$; by (ii), this scalar and its inverse belong to $R_{\ell }$. Hence

The mapis an automorphism of the elliptic curve over $K$, fixes its identity point, and squares to $[-1]$. For the invariant differential on an elliptic curve $\omega =dx/(2y)$, it satisfies $[i{]}^{\ast }\omega =i\omega$. Thus $\mathbb{Z}[i]\subseteq {\mathrm{End}}_K(E)$, with the indicated embedding of $K$.

Under analytic complex uniformization of an elliptic curve $E(\mathbb{C})\simeq \mathbb{C}/L$, this automorphism acts as multiplication by $i$, so $L$ is stable under $\mathbb{Z}[i]$. It is a rank-one torsion-free module over the Gaussian integers. Since that ring is a principal ideal domain, $L=\Omega \mathbb{Z}[i]$. Every analytic endomorphism is multiplication by an $a\in \mathbb{C}$ with $aL\subseteq L$, which here says exactly $a\in \mathbb{Z}[i]$. Therefore the geometric endomorphism ring of an elliptic curve is $\mathbb{Z}[i]$, and all its elements are generated over $\mathbb{Z}$ by the displayed $K$-defined automorphism. Thus ${\mathrm{End}}_K(E)=\mathbb{Z}[i]$.

If an endomorphism is defined over $\mathbb{Q}$, its pullback on the rational differential $\omega$ has scalar in $\mathbb{Q}$. The differential action is injective in characteristic zero, because a nonzero isogeny of elliptic curves is separable and has nonzero differential. Its scalar must therefore lie in $\mathbb{Q}\cap \mathbb{Z}[i]=\mathbb{Z}$. Conversely integer multiplication is defined over $\mathbb{Q}$. Hence

Put $\lambda =1+i$. The valuations of the differences of the three nonidentity fourth roots of unity from $1$ areFor instance $2=-i{\lambda }^2$, and the other two differences are associates of $\lambda$. None is divisible by ${\lambda }^3$. The group $(\mathbb{Z}[i]/{\lambda }^3{)}^{\times }$ has $8-4=4$ elements, so the four units map bijectively to it. Every ideal prime to $\lambda$ consequently has a unique primary Gaussian integer generator.

The Grössencharacter of the square-lattice elliptic curve is explicitlyFor integral ideals prime to $2$, the generator is an odd Gaussian integer; extend multiplicatively to fractional ideals prime to $2$. Products of primary generators are primary, so this is a Hecke character with ideal infinity type $(1,0)$: on a principal ideal with generator $a\equiv 1(\mathrm{mod}{\lambda }^3)$, its value is $a$. Its conductor divides ${\lambda }^3$, and cannot divide ${\lambda }^2$, since $-1\equiv 1(\mathrm{mod}{\lambda }^2)$ would force ${\psi }_E((-1))=-1$, whereas the unit ideal has value $1$. Thus the displayed conductor is exact.

We next verify directly that this character gives the Frobenius values of this particular curve, fixing the unit ambiguity. The pointsatisfies, by the chord-and-tangent addition law,For the first addition, $[i]P=(-i,1+i)$ and the chord has slope $-1$. Since $P$ is killed by ${\lambda }^3$ but not by ${\lambda }^2$, it generates the eight-element $\mathbb{Z}[i]$-module $E[{\lambda }^3]\simeq \mathbb{Z}[i]/{\lambda }^3$. Thus all this torsion is rational over $K$.

The model has discriminant $64$, so it has good reduction at every $\gamma \ne (\lambda )$. First suppose $\gamma$ lies over a split rational prime $p$. Then its residue field is ${\mathbb{F}}_p$, $p\ge 5$, and $\pi ={\psi }_E(\gamma )$ has norm $p$. Reduction of $[\pi ]$ still has degree $p$ and has zero differential, since its differential scalar $\pi$ belongs to $\gamma$. It is therefore purely inseparable of degree $p$. Factoring out the Frobenius isogeny givesfor an automorphism $u$ of the reduction. In characteristic at least five, the automorphisms of this model fixing $O$ are the four Gaussian units: comparing coefficients in $(x,y)\mapsto (v^{2}x,v^{3}y)$ gives $v^4=1$. Prime-to-$p$ torsion reduces injectively. Both $[\pi ]$ and ${\mathrm{Frob}}_p$ fix the reduction of $E[{\lambda }^3]$, the former by the primary congruence and the latter by its rationality over the residue field. Thus $u$ fixes this entire module. The unit congruences proved above force $u=1$.

For an inert rational prime $p\equiv 3(\mathrm{mod}4)$, we have $\gamma =(p)$, $N\gamma =p^2$, and its primary generator is $\pi =-p$, because $4\mid p+1$ implies ${\lambda }^3\mid (-p-1)$. To determine Frobenius without excluding $p=3$, writeReplacing $x$ by $-x$ changes each nonzero quadratic-character value to its negative, because $-1$ is a nonsquare. Hence $a_p=0$. The Frobenius characteristic equation ${\mathrm{Frob}}_p^2-[a_p]{\mathrm{Frob}}_p+[p]=0$ gives ${\mathrm{Frob}}_{p^2}=[-p]$. This proves the asserted Frobenius value at inert primes as well, and identifies the displayed Hecke character with that of $E$.

Finally, the formal parameter $t=-x/y$ is defined over the local ring at $\gamma$. Since the reduced endomorphism is the $q$-power Frobenius isogeny, with $q=N\gamma$, its pullback sends $t$ to $(-x/y{)}^q=t^q$. Passing to the completion at the identity gives the equality of formal series

The construction must first specify the field of definition of the complex multiplication. Let $E/F$ be a CM elliptic curve and let $K$ be its geometric CM field, embedded by its action on an invariant differential. In characteristic zero this action is injective. Consequently every geometric endomorphism is defined over $F^{\prime }=FK$: a Galois automorphism fixing $FK$ fixes its differential scalar and therefore fixes the endomorphism itself. Initially assume $F$ already contains $K$; we return to the other case below.

Choose a complex embedding and an analytic complex uniformization of an elliptic curvewhere $\mathfrak{a}$ is a lattice in $K$, stable under the CM order $R$. It need not be the maximal order or a principal ideal. Its torsion is described by the finite adelic quotients of this lattice.

The input from the CM reciprocity theorem is precise: if $s$ is an idele of $F$, $n=N_{F/K}s$, and $\sigma ={\mathrm{Art}}_F(s)$, then $\sigma$ takes the CM torus to the torus with lattice $n_f^{-1}L$; on torsion the comparison is induced by multiplication by $n_f^{-1}$. This reciprocity statement comes from the ideal action on CM lattices: an ideal $\mathfrak{b}$ gives the isogeny to ${\mathfrak{b}}^{-1}L$, composition multiplies ideals, and adding torsion level structure removes automorphism ambiguity. The resulting action on algebraic CM points is the ray-class Artin action. This is the deep CM input, not an assertion that an arbitrary abelian Galois representation automatically comes from an algebraic Hecke character.

Because $\sigma$ fixes $F$, the algebraic curve $E^{\sigma }$ is $E$. Comparing the reciprocity uniformization with the original one therefore gives multiplication by a scalarThe scalar belongs to $K$: a complex scalar carrying one lattice contained in $K$ to another is the quotient of two nonzero lattice elements. Its torsion action isThe uniqueness of the torsion-compatible comparison is important: two possible scalars would induce the same action on all torsion, hence their difference would annihilate torsion of arbitrarily large order and be zero. Multiplying the torsion-action formulas for $s_f$ and $t_f$ now proves $a(s_f{t}_f)=a(s_f)a(t_f)$.

For $b\in F^{\times }$, Artin reciprocity sends its principal idele to the identity. The archimedean factors are complex and connected, so its finite part also has trivial Artin image. The preceding formula therefore givesIt follows thatis trivial on principal ideles and defines a character of the idèle class group of $F$. This establishes the multiplicativity and principal-idele condition, rather than merely prescribing Frobenius values and hoping that they extend.

We must still prove continuity and a finite conductor. For a finite place $v$ and a prime $\ell$ different from its residue characteristic, an idele supported on the units at $v$ has trivial $\ell$-component of its norm. Thus its action on the Tate module is exactly multiplication by $a(u)$. At a place of good reduction, inertia acts trivially, so $a(u)=1$. At a bad place, a CM curve has potentially good reduction, and inertia on the prime-to-residue-characteristic Tate module has finite image. Hence $a$ is trivial on an open subgroup of the local units there. There are only finitely many bad places. Together these facts give an open finite-congruence kernel on the unit ideles and prove continuity. The archimedean norm factor is already continuous. The potentially-good-reduction and inertia facts are the reduction-theoretic inputs in Serre and Tate's treatment of CM abelian varieties.

Consequently ${\psi }_E$ is a Hecke character. In ideal notation, for $b\equiv 1(\mathrm{mod}\mathfrak{f})$,so its infinity type of a Hecke character is the type determined by the chosen CM embedding, with the inverse archimedean norm in the idele convention. Nonmaximal orders merely contribute additional finite-level conditions; the proof retains their lattice and does not replace it by the maximal-order lattice without justification.

At a good prime $v$, choose a uniformizer idele and a prime $\ell$ different from the residue characteristic. Its norm has trivial $\ell$-component, so arithmetic Frobenius acts on the Tate module by ${\psi }_E(v)$. Specialization of prime-to-residue-characteristic torsion identifies this with the reduced Frobenius isogeny. ThusThe corresponding Hasse-Weil L-function factors as $L(E/F,s)=L({\psi }_E,s)L({\bar{\psi }}_E,s)$, including the local factors determined by inertia. This explains why the Hecke character encodes both the torsion representations and the Euler factors.

If $F$ does not contain $K$, apply the construction over $FK$. The two conjugate CM characters are exchanged by the nontrivial automorphism of $FK/F$, and the two-dimensional Galois representation over $F$ is their induced representation. In that case the natural description is $L(E/F,s)=L({\psi }_E/FK,s)$ by induction, rather than a pair of one-dimensional characters over $F$. Question 3 supplies a concrete example: $\mathbb{Q}$ does not define $[i]$, whereas $\mathbb{Q}(i)$ does. This completes the existence argument for an arbitrary CM curve over a number field, with its field of definition correctly accounted for.

An Artinian module is an $A$-module satisfying the descending chain condition on submodules: every chain $M_0\supseteq M_1\supseteq \cdots$ is eventually constant. An Artinian ring is a ring that is Artinian as a module over itself, so its descending chains of ideals stabilize.

We first prove the closure properties needed for finite generation. A submodule of an Artinian module is Artinian, because any descending chain in it is also a chain in the original module. A quotient module is Artinian, because inverse images turn a descending chain in the quotient into one upstairs. Conversely, consider a short exact sequencewith $N$ and $Q$ Artinian. For a descending chain $(M_i)$, the chains $M_i\cap N$ and $\pi (M_i)$ both stabilize. Choose $r$ beyond both stabilization indices. If $i\ge r$ and $x\in M_i$, equality of the images gives $y\in M_{i+1}$ with $\pi (y)=\pi (x)$. Thenso $x\in M_{i+1}$. Hence $M_i=M_{i+1}$. This proves the extension property of Artinian modules in a short exact sequence.

Apply this property inductively to $0\to A^{n-1}\to A^n\to A\to 0$. Every finite direct sum $A^n$ is Artinian when $A$ is an Artinian ring. If $m_1,\dots ,m_n$ generate $M$, the map $(a_i)\mapsto \sum a_i{m}_i$ is a surjection $A^n\twoheadrightarrow M$. The quotient property therefore provesThis argument also covers the zero module; it does not require a structure theorem for Artinian rings.

For a counterexample without finite generation, take any field $k$ and the moduleThe ring $k$ is Artinian, since its only ideals are $0$ and $k$. But $M_i⊋M_{i+1}$ for every $i$, since $e_i\notin M_{i+1}$. Thus an infinite-dimensional vector space over a field provides the requested non-Artinian module.

Finally, let $f:M\to M$ be an injective endomorphism with $M$ Artinian. Its image chain stabilizes, so $f^{r}M=f^{r+1}M$ for some $r$. Given $x\in M$, choose $y$ with $f^{r}x=f^{r+1}y=f^r(fy)$. Since $f^r$ is injective, $x=fy$. ThereforeThis is precisely the assertion that Artinian modules are co-Hopfian.

For a section represented by a column $u$, expand the covariant exterior derivative using the graded Leibniz rule:Therefore the Cartan curvature matrix equation isThe product includes endomorphism composition; it need not vanish for a matrix-valued one-form. For a scalar frame change $g$, $A^{\prime }=A+g^{-1}dg$ and $d(g^{-1}dg)=0$, confirming that the line-bundle two-forms $dA$ patch globally.

The connection difference as an endomorphism-valued one-form follows because the graded Leibniz rule terms cancel:Thus $a=D_1-D$ is a global endomorphism-valued one-form. To define the endomorphism bundle connection, requireThe right side is $C^{\infty }$-linear in $s$, so it is indeed an endomorphism-valued one-form, and replacing $\varphi$ by $f\varphi$ gives $df\otimes \varphi +f\tilde{D}\varphi$. In a frame with connection matrix $A$, its extension to an endomorphism-valued $k$-form $b$ isThe endomorphism-valued exterior product combines the wedge of differential forms with composition in $\mathrm{End}E$. In particular,Expanding $d(A+a)+(A+a)\wedge (A+a)$ proves the curvature difference formula:

The Chern connection is characterized by metric compatibility and $D^{0,1}={\bar{\partial }}_E$. Take Hermitian inner products linear in the first variable. In a holomorphic local frame write $⟨eu,ev⟩=v^{†}Hu$, with $H$ a positive Hermitian matrix. The second condition forces $A^{0,1}=0$, and metric compatibility isIts $(1,0)$ component determines $A$ uniquely:Conversely this formula and its conjugate transpose satisfy the metric equation. Under a holomorphic local frame change, $H^{\prime }=g^{†}Hg$ and $\partial g^{†}=0$, so the displayed formula transforms by $A^{\prime }=g^{-1}Ag+g^{-1}\partial g$. Hence the local matrices patch, proving existence and uniqueness.

Differentiating $H^{-1}$ givesTherefore the $(2,0)$ vector-bundle curvature is zero. There is no $(0,2)$ component, andThe last equality is ${\bar{\partial }}^{2}A=0$ in holomorphic local frames, so is intrinsic. This is Dolbeault closedness of Chern curvature. The induced endomorphism bundle connection is itself the Chern connection for the Hilbert-Schmidt metric on $\mathrm{End}E$: the tensor product of a metric-compatible connection and its dual preserves the induced metric, and its $(0,1)$ part is the induced holomorphic Dolbeault operator. These two properties and uniqueness suffice; no further matrix calculation is needed.

For two metrics, both Chern connections have the same $(0,1)$ part, so $a\in A^{1,0}(\mathrm{End}E)$. The $(1,1)$ part of the curvature difference formula is just ${\bar{\partial }}_{\mathrm{End}E}a$; the other two terms have type $(2,0)$, which must cancel because both vector-bundle curvatures have type $(1,1)$. ThusBy Dolbeault theorem, this proves that the curvature Dolbeault classis independent of the metric.

Endomorphism composition and wedge product of differential forms define the powers in $A^{k,k}(\mathrm{End}E)$, rather than in a tensor power of the endomorphism bundle. The Dolbeault operator is a derivation, so $\bar{\partial }({\Theta }^k)=0$. The noncommutative telescoping identity gives the stronger explicit independence statementAll powers of vector-bundle curvature have even total degree, so the derivation introduces no extra signs before $a$. Consequentlyis metric independent. For $k>n$, the form space and the resulting class are zero. This proves metric independence of curvature Dolbeault powers without assuming that endomorphism-valued forms commute.

A weakly initial set is a set of objects $(W_i{)}_{i\in I}$ such that every object receives at least one map from some $W_i$. An initial object is itself a singleton weakly initial set. We prove the converse without replacing uniqueness by weak initiality.

Let $\mathcal{E}$ be a complete category that is locally small. Form $W={\prod }_i{W}_i$. For every object $A$, choose $W_i\to A$ and compose with the projection $W\to W_i$, so $W$ is weakly initial. Completeness supplies a terminal object, so the given weakly initial set cannot be empty.

The endomorphisms of $W$ form a set. Take a simultaneous equalizer $e:I\to W$ of all of them with $1_W$; equivalently, equalize the two maps from $W$ into ${\prod }_{s\in \mathcal{E}(W,W)}W$ with components $s$ and $1_W$. Thus $se=e$ for every endomorphism $s$. The object $I$ is weakly initial, since $I\stackrel{e}{\to }W\to A$ exists for every $A$. Weak initiality of $W$ gives a map $u:W\to I$. Applying the equalizing identity to $eu$ yields $eue=e$, so monicity of $e$ gives $ue=1_I$.

For any endomorphism $v:I\to I$, the map $evu$ is an endomorphism of $W$. Therefore $evue=e$. Since $ue=1_I$, this becomes $ev=e$, giving $v=1_I$. Now let $f,g:I\rightrightarrows A$ and form their equalizer $j:E\to I$. Weak initiality of $I$ gives $h:I\to E$. The composite $jh$ is an endomorphism of $I$, hence the identity. Consequently $f=fjh=gjh=g$.

There is at least one map from $I$ to every object, and the argument proves that there is at most one. The object $I$ is an initial object. This establishes the initial-object lemma for complete categories with a weakly initial set using only small products in a category and equalizers.