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The group \(\operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}}^f)\) acts (on the left) on the space of automorphic forms for \(\operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}})\) by the formula \((g\cdot f)(x,y)=f(xg,y)\).
This function is well-defined, i.e., it sends a \(U\)-invariant form to a \(U\)-invariant form which is independent of the choice of \(g_i\).
An automorphic form is a function \(\phi :G(\mathbb {A}_N)\to \mathbb {C}\) satisfying the following conditions:
\(\phi \) is locally constant on \(G(\mathbb {A}_N^f)\) and \(C^\infty \) on \(G(N_\infty )\). In other words, for every \(g_\infty \), \(\phi (-,g_\infty )\) is locally constant, and for every \(g_f\), \(\phi (g_f,-)\) is smooth.
\(\phi \) is left-invariant under \(G(N)\);
\(\phi \) is right-\(U_\infty \)-finite (that is, the space spanned by \(x\mapsto \phi (xu)\) as \(u\) varies over \(U_\infty \) is finite-dimensional);
\(\phi \) is right \(K_f\)-finite, where \(K_f\) is one (or equivalently all) compact open subgroups of \(G(\mathbb {A}_N^f)\);
\(\phi \) is \(\mathcal{z}\)-finite, where \(\mathcal{z}\) is the centre of the universal enveloping algebra of the Lie algebra of \(G(N_\infty )\), acting via differential operators. Equivalently \(\phi \) is annihiliated by a finite index ideal of this centre, so morally \(\phi \) satisfies lots of differential equations of a certain type;
For all \(g_f\), the function \(g_\infty \mapsto \phi (g_f g\infty )\) is slowly-increasing in the sense above.
The group \(G(\mathbb {A}_N)\) acts on itself on the right, and this induces a left action of its subgroup \(G(\mathbb {A}_N^f)\times U_\infty \) on the spaces of automorphic forms and cusp forms. The Lie algebra \(\mathfrak {g}\) of \(G(N_\infty )\) also acts, via differential operators. Furthermore the actions of \(\mathfrak {g}\) and \(U_\infty \) are compatible in the sense that the differential of the \(U_\infty \) action is the action of its Lie algebra considered as a subalgebra of \(\mathfrak {g}\). We say that the spaces are \((G(\mathbb {A}_N^f)\times U_\infty ,\mathfrak {g})\)-modules.
An automorphic representation is an irreducible \((G(\mathbb {A}_N^f)\times U_\infty ,\mathfrak {g})\)-module isomorphic to an irreducible subquotient of the space of automorphic forms.
An irreducible admissible \((G(\mathbb {A}_N^f)\times U_\infty ,\mathfrak {g})\)-module is a restricted tensor product of irreducible representations \(\pi _v\) of \(G(N_v)\) as \(v\) runs through the finite places of \(N\), tensored with a tensor product of irreducible \((\mathfrak {g}_v,U_{\infty ,v})\)-modules as \(v\) runs through the infinite places of \(N\). The representations \(\pi _v\) are unramified for all but finitely many \(v\).
A smooth function \(f:\operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}}^f)\times \operatorname{GL}_n(\mathbb {R})\to \mathbb {C}\) is an \(O_n(\mathbb {R})\)-automorphic form on \(\operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}})\) if it satisfies the following five conditions.
(periodicity) For all \(g\in \operatorname{GL}_n(\mathbb {Q})\), we have \(f(gx,gy)=f(x,y)\).
(has a finite level) There exists a compact open subgroup \(U\subseteq \operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}}^f)\) such that \(f(xu,y)=f(x,y)\) for all \(u\in U\), \(x\in \operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}}^f)\) and \(y\in \operatorname{GL}_n(\mathbb {R})\).
(weight \(\rho \)) There exists a continuous finite-dimensional irreducible complex representation \(\rho \) of \(O_n(\mathbb {R})\) such that for every \((x,y)\in \operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}})\), the set of functions \(k\mapsto f(x,yk)\) span a finite-dimensional complex vector space isomorphic as \(O_n(\mathbb {R})\)-representation to a direct sum of copies of \(\rho \).
(has an infinite level) There is an ideal \(I\) of the centre \(Z_n\) described in the previous section, which has finite complex codimension, and which annihiliates the function \(y \mapsto f(x,y)\) for all \(x\in \operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}}^f)\). Note that this is a very fancy way of saying “the function satisfies some natural differential equations”. In the case of modular forms, the differential equations are the Cauchy-Riemann equations, which is why modular forms are holomorphic.
(growth condition) For every \(x\in \operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}}^f)\), the function \(y\mapsto f(x,y)\) on \(\operatorname{GL}_n(\mathbb {R})\) is slowly-increasing.
A function \(f:\operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}}^f)\times \operatorname{GL}_n(\mathbb {R})\to \mathbb {C}\) is smooth if it has the following three properties.
\(f\) is continuous.
For all \(x\in \operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}}^f)\), the function \(y\mapsto f(x,y)\) is smooth.
For all \(y\in \operatorname{GL}_n(\mathbb {R})\), the function \(x\mapsto f(x,y)\) is locally constant.
The weight of an automorphic form for \(\operatorname{GL}_n/\mathbb {Q}\) can be thought of as a finite-dimensional continuous complex representation \(\rho \) of a maximal compact subgroup of \(\operatorname{GL}_n(\mathbb {R})\), and it’s convenient to choose one (they’re all conjugate) so we choose \(O_n(\mathbb {R})\).
Let \(N\) be a number field. A compatible family of \(d\)-dimensional Galois representations over \(N\) is a finite set of finite places \(S\) of \(N\), a number field \(E\), a monic degree \(d\) polynomial \(F_{{\mathfrak {p}}}(X)\in E[X]\) for each finite place \({\mathfrak {p}}\) of \(K\) not in \(S\) and, for each prime number \(\ell \) and field embedding \(\phi : E\to \overline{\mathbb {Q}}_\ell \) (or essentially equivalently for each finite place of \(E\)), a continuous homomorphism \(\rho :\operatorname{Gal}(K^{\operatorname{sep}}/K)\to \operatorname{GL}_2(\overline{\mathbb {Q}}_\ell )\) unramified outside \(S\) and the primes of \(K\) above \(\ell \), such that \(\rho (\operatorname{Frob}_{\mathfrak {p}})\) has characteristic polynomial \(P_\pi (X)\) if \(\pi \) lies above a prime number \(p\not=\ell \) with \(p\not\in S\).
An affine algebraic group \(G\) of finite type over a field \(k\) is said to be connected if it is connected as a scheme, and reductive if \(G_{\overline{k}}\) has no nontrivial smooth connected unipotent normal \(k\)-subgroup.
An automorphic form is cuspidal (or “a cusp form”) if it furthermore satisfies \(\int _{U(N)\backslash U(\mathbb {A}_N)}\phi (ux)du=0\), where \(P\) runs through all the proper parabolic subgroups of \(G\) defined over \(N\) and \(U\) is the unipotent radical of \(P\), and the integral is with respect to the measure coming from Haar measure.
The cusp forms decompose as a (typically infinite) direct sum of irreducible \((G(\mathbb {A}_N^f)\times U_\infty ,\mathfrak {g})\)-modules.
A cuspidal automorphic representation is an irreducible \((G(\mathbb {A}_N^f)\times U_\infty ,\mathfrak {g})\)-module isomorphic to an irreducible summand of the space of cusp forms.
If \(E\) is an elliptic curve over a field \(k\), and \(n\) is a positive integer which is nonzero in \(k\), then the determinant of the 2-dimensional representation of \(\operatorname{Gal}(k^{\operatorname{sep}}/k)\) on \(E(k^{\operatorname{sep}})[n]\) is the mod \(n\) cyclotomic character.
Let \(n\) be a positive integer, let \(F\) be a separably closed field with \(n\) nonzero in \(F\), and let \(E\) be an elliptic curve over \(F\). Then the \(n\)-torsion \(E(F)[n]\) in the \(F\)-points of \(E\) is a finite group isomorphic to \((\mathbb {Z}/n\mathbb {Z})^2\).
If \(p\) is a prime and if \(E\) is an elliptic curve over a field \(K\) of characteristic not equal to \(p\), and if \(C\subseteq E(K^{\operatorname{sep}})[p]\) is a Galois-stable subgroup of order \(p\), then there’s an elliptic curve \(E':=\)“\(E/C\)” over \(K\) and an isogeny of elliptic curves \(E\to E'\) over \(K\) inducing a Galois-equivariant surjection \(E(K^{\operatorname{sep}})\to E'(K^{\operatorname{sep}})\) with kernel precisely \(C\).
Let \(E\) be an elliptic curve over the field of fractions \(K\) of a valuation ring \(R\) with maximal ideal \(\mathfrak {m}\). We say \(E\) has good reduction over \(R\) if \(E\) has a model with coefficients in \(R\) and the reduction mod \(\mathfrak {m}\) is still non-singular. If \(E\) is an elliptic curve over a number field \(N\) and \(P\) is a maximal ideal of its integer ring \(\mathcal{O}_N\), then one says that \(E\) has good reduction at \(P\) if \(E\) has good reduction over the \(\mathcal{O}_{N,P}\), the localisation of \(\mathcal{O}_N\) at \(P\).
Let \(E\) be an elliptic curve over the field of fractions \(K\) of a valuation ring \(R\) with maximal ideal \(\mathfrak {m}\). We say \(E\) has multiplicative reduction over \(R\) if \(E\) has a model with coefficients in \(R\) and which reduces mod \(R/\mathfrak {m}\) to a plane cubic with one singularity, which is an ordinary double point. We say that the reduction is split if the two tangent lines at the ordinary double point are both defined over \(R/\mathfrak {m}\), and non-split otherwise.
If \(R\) is a commutative ring, then a finite flat group scheme over \(R\) is the spectrum of a commutative Hopf algebra \(H/R\) which is finite and flat as an \(R\)-module.
Given a Frey package \((a,b,c,p)\), the corresponding Frey curve (considered by Frey and, before him, Hellegouarch) is the elliptic curve \(E\) defined by the equation \(Y^2=X(X-a^p)(X+b^p).\)
If Fermat’s Last Theorem is false for \(p \ge 5\) and prime, then there exists a Frey package.
With notation as above, the characters \(\alpha \) and \(\beta \) are unramified at \(p\) for all primes \(p\not=\ell \).
If \((a,b,c,\ell )\) is a Frey package, then the semisimplification of the restriction of the \(\ell \)-torsion \(\rho \) in the associated Frey curve to \(\operatorname{Gal}(\overline{\mathbb {Q}}_2/\mathbb {Q}_2)\) is unramified.
If \(p\not=\ell \) is a prime not dividing \(abc\) then \(\rho \) is unramified at \(p\).
If \(E\) is the Frey curve \(Y^2=X(X-a^\ell )(X+b^\ell )\) associated to a Frey package \((a,b,c,\ell )\), and if \(p\) is a prime not dividing \(abc\) (and in particular if \(p{\gt}2\)), then \(E\) has good reduction at \(p\).
Let \(\rho \) be the Galois representation on the \(\ell \)-torsion of the Frey curve coming from a Frey package \((a,b,c,\ell )\). Then \(\rho \) is hardly ramified.
The \(\ell \)-torsion in the Frey curve associated to a Frey package \((a,b,c,\ell )\) is irreducible.
Let \(\rho \) be the \(\ell \)-torsion in the Frey curve associated to a Frey package \((a,b,c,\ell )\). Then the restriction of \(\rho \) to \(\operatorname{Gal}(\overline{\mathbb {Q}}_\ell /\mathbb {Q}_\ell )\) comes from a finite flat group scheme.
If \(E\) is the Frey curve \(Y^2=X(X-a^\ell )(X+b^\ell )\) associated to a Frey package \((a,b,c,\ell )\), and if \(p\) is an odd prime which divides \(abc\), then \(E\) has multiplicative reduction at \(p\).
If \(E\) is the Frey curve \(Y^2=X(X-a^\ell )(X+b^\ell )\) associated to a Frey package \((a,b,c,\ell )\) then \(E\) has multiplicative reduction at 2.
If \(\rho \) is reducible, then either \(\rho \) has a trivial 1-dimensional submodule or a trivial 1-dimensional quotient (here “trivial” means that the Galois group \(\operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\) acts trivially).
If \(\rho \) has a trivial 1-dimensional submodule then the Frey curve has a non-trivial point of order \(\ell \).
If \((a,b,c,\ell )\) is a Frey package, if \(2{\lt}p\mid abc\) is a prime with \(p\not=\ell \), then the \(\ell \)-torsion in the Frey curve is unramified at \(p\).
If \((a,b,c,\ell )\) is a Frey package, then the \(\ell \)-torsion in the Frey curve is unramified at all primes \(p\not=2,\ell \).
Given an automorphic representation \(\pi \) for an inner form of \(\operatorname{GL}_2\) over a totally real field and with reflex field \(E\), such that \(\pi \) is weight 2 discrete series at every infinite place, there exists a compatible family of 2-dimensional Galois representations associated to \(\pi \), with \(S\) being the places at which \(\pi \) is ramified, and \(F_{{\mathfrak {p}}}(X)\) being the monic polynomial with roots the two Satake parameters for \(\pi \) at \({\mathfrak {p}}\).
If \(N\) is a finite extension of \(\mathbb {Q}\) then there are two “canonical” isomorphisms of topological groups between the profinite abelian groups \(\pi _0(\mathbb {A}_N^\times /N^\times )\) and \(\operatorname{Gal}(\overline{N}/N)^{\operatorname{ab}}\); one sends local uniformisers to arithmetic Frobenii and the other to geometric Frobenii; each of the global isomorphisms is compatible with the local isomorphisms above.
If \(E\) is an elliptic curve over a number field \(N\) and \(E\) has good reduction at a maximal ideal \(P\) of \(\mathcal{O}_N\) containing the prime number \(p\), then the Galois representation on the \(p\)-torsion of \(E\) comes from a finite flat group scheme over the localisation \(\mathcal{O}_{N,P}\).
If \(E\) is an elliptic curve over a number field \(N\) and \(E\) has good reduction at a maximal ideal \(P\) of \(\mathcal{O}_N\), and if furthermore \(n\not\in P\), then the Galois representation on the \(n\)-torsion of \(E\) is unramified.
Let \(\ell \geq 5\) be a prime and let \(V\) be a 2-dimensional vector space over \(\mathbb {Z}/\ell \mathbb {Z}\). A representation \(\rho : \operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\to \operatorname{GL}(V)\) is said to be hardly ramified if it satisfies the following four axioms:
\(\det (\rho )\) is the mod \(\ell \) cyclotomic character;
\(\rho \) is unramified outside \(2\ell \);
The semisimplification of the restriction of \(\rho \) to \(\operatorname{Gal}(\overline{\mathbb {Q}}_2/\mathbb {Q}_2)\) is unramified;
The restriction of \(\rho \) to \(\operatorname{Gal}(\overline{\mathbb {Q}}_\ell /\mathbb {Q}_\ell )\) comes from a finite flat group scheme.
The Hurwitz quaternions are the set \(\mathcal{O}:= \mathbb {Z}\oplus \mathbb {Z}\omega \oplus \mathbb {Z}i\oplus \mathbb {Z}i\omega \) (as an abstract abelian group or as a subgroup of the usual quaternions). Here \(\omega =\frac{-1+(i+j+k)}{2}\) and note that \((i+j+k)^2=-3\). We have \(\overline{\omega }=\omega ^2=-(\omega +1)\). A general quaternion \(a+bi+cj+dk\) is a Hurwitz quaternion if either \(a,b,c,d\in \mathbb {Z}\) or \(a,b,c,d\in \mathbb {Z}+\frac{1}{2}\).
There’s a conjugation map (which we’ll call "star") from the Hurwitz quaternions to themselves, sending integers to themselves and purely imaginary elements like \(2\omega +1\) to minus themselves. It satisfies \((x^*)^*=x\), \((xy)^*=y^*x^*\) and \((x+y)^*=x^*+y^*\). In particular, the Hurwitz quaternions are a "star ring" in the sense of mathlib.
If \(N\) is a positive natural then the obvious map \(\mathcal{O}\to \widehat{\mathcal{O}}/N\widehat{\mathcal{O}}\) is surjective.
Thus the centre \(\mathbb {Z}_n\) of this universal enveloping algebra also acts on the smooth complex functions.
This extends to is a natural complex Lie algebra action of the complexification of the real Lie algebra, on the smooth complex functions on \(\operatorname{GL}_n(\mathbb {R})\).
There is a natural action of the real Lie algebra of \(\operatorname{GL}_n(\mathbb {R})\) on the complex vector space of smooth complex-valued functions on \(\operatorname{GL}_n(\mathbb {R})\).
By functoriality, we get an action of the universal enveloping algebra of this complexified Lie algebra on the smooth complex functions.
If \(D\) is a central simple algebra over \(K\) and \(L/K\) is a field extension, then \(L\otimes _KD\) is a central simple algebra over \(L\).
If \(G\) is an affine algebraic group of finite type over \(K=\mathbb {R}\) or \(\mathbb {C}\) then \(G(K)\) is naturally a real or complex Lie group.
If \(K\) is a finite extension of \(\mathbb {Q}_p\) then there are two “canonical” isomorphisms of topological abelian groups, between \(K^\times \) and the abelianisation of the Weil group of \(K\).
If \(h^i(M)\) denotes the order of \(H^i(G_K,M)\) then \(h^0(M)-h^1(M)+h^2(M)=0\).
If \(\mu =\bigcup _{n\geq 1}\mu _n\) and \(M':=\operatorname{Hom}(M,\mu )\) is the dual of \(M\) then for \(0\leq i\leq 2\) the cup product pairing \(H^i(G_K,M)\times H^{2-i}(G_K,M')\to H^2(G_K,\mu )=\mathbb {Q}/\mathbb {Z}\) is perfect.
(i) There is a “canonical” isomorphism \(H^2(K,\mu _\infty )=\mathbb {Q}/\mathbb {Z}\); (ii) The pairing above is perfect.
\(H^2(G_K,\mu _n)\) is “canonically” isomorphic to \(\mathbb {Z}/n\mathbb {Z}\).
If \(X\) is as in the previous definition and \(X\to \mathbb {A}^n_K\) is a closed immersion, then the induced map from \(X(K)\) with its manifold structure to \(K^n\) is an embedding of manifolds.
Let \(K\) be a field equipped with an isomorphism to the reals, complexes, or a finite extension of the \(p\)-adic numbers. Let \(X\) be a smooth affine algebraic variety over \(K\). Then the points \(X(K)\) naturally inherit the structure of a manifold over \(K\).
If \(n\geq 1\) then the \(n\times n\) matrices \(M_n(K)\) are a central simple algebra over \(K\).
The maximal unramified extension \(K^{un}\) in a given algebraic closure of \(K\) is Galois over \(K\) with Galois group “canonically” isomorphic to \(\widehat{\mathbb {Z}}\) in two ways; one of these two isomorphisms identifies \(1\in \widehat{\mathbb {Z}}\) with an arithmetic Frobenius (the endomorphism inducing \(x\mapsto x^q\) on the residue field of \(K^{un}\), where \(q\) is the size of the residue field of \(K\)). The other identifies 1 with geometric Frobenius (defined to be the inverse of arithmetic Frobenius).
Let \(E\) be an elliptic curve over \(\mathbb {Q}\). Then the torsion subgroup of \(E\) has size at most 16.
If \(\overline{\rho }\) is modular of level \(\Gamma _1(S)\) and \(\rho :G_F\to \operatorname{GL}_2(\mathcal{O})\) is an \(S\)-good lift of \(\overline{\rho }\) to \(\mathcal{O}\), the integers of a finite extension of \(\mathbb {Q}_\ell \), then \(\rho \) is also modular of level \(\Gamma _1(S)\).
Let \(K^{\operatorname{avoid}}/K\) be a Galois extension of number fields. Suppose also that \(S\) is a finite set of places of \(K\). For \(v\in S\) let \(L_v/K_v\) be a finite Galois extension. Suppose also that \(T /K\) is a smooth, geometrically connected curve and that for each \(v\in S\) we are given a nonempty, \(\operatorname{Gal}(L_v/K_v)\)-invariant, open subset \(\Omega _v\subseteq (L_v)\). Then there is a finite Galois extension \(L/K\) and a point \(P ∈ T (L)\) such that
\(L/K\) is Galois and linearly disjoint from \(K^{\operatorname{avoid}}\) over \(K\);
if \(v\in S\) and \(w\) is a prime of \(L\) above \(v\) then \(L_w /K_v\) is isomorphic to \(L_v/K_v\);
and \(P \in \Omega _v\subseteq T (L_v) \cong (L_w)\) via one such \(K_v\)-algebra morphism (this makes sense as \(\Omega _v\) is \(\operatorname{Gal}(L_v/K v)\)-invariant).
If \(E\) is an elliptic curve over a field \(K\) complete with respect to a nontrivial nonarchimedean (real-valued) norm and with perfect residue field, and if \(E\) has multiplicative reduction, then there’s an unramified character \(\chi \) of \(\operatorname{Gal}(K^{\operatorname{sep}}/K)\) whose square is 1, such that for all positive integers \(n\) with \(n\not=0\) in \(K\), the \(n\)-torsion \(E(K^{\operatorname{sep}})[n]\) is an extension of \(\chi \) by \(\epsilon \chi \), where \(\epsilon \) is the cyclotomic character. Furthermore, the element of \(K^\times /(K^\times )^\ell \) corresponding to this extension is given by the \(q\)-invariant of the curve.
The sum of \(\mathbb {Q}\) and \(\widehat{\mathbb {Z}}\) in \(\widehat{\mathbb {Q}}\) is \(\widehat{\mathbb {Q}}\). More precisely, every element of \(\widehat{\mathbb {Q}}\) can be written as \(q+z\) with \(q\in \mathbb {Q}\) and \(z\in \widehat{\mathbb {Z}}\), or more precisely as \(q\otimes _t 1+1\otimes _t z\).
The product of \(\mathbb {Q}^\times \) and \(\widehat{\mathbb {Z}}^\times \) in \(\widehat{\mathbb {Q}}^\times \) is all of \(\widehat{\mathbb {Q}}^\times \). More precisely, every element of \(\widehat{\mathbb {Q}}^\times \) can be written as \(qz\) with \(q\in \mathbb {Q}^\times \) and \(z\in \widehat{\mathbb {Z}}^\times \).
We need the definition of (the canonical model over \(F\) of) the Shimura curve attached to an inner form of \(\operatorname{GL}_2\) with precisely one split infinite place, and the same for the Shimura surface associated to an inner form split at two infinite places (and ramified elsewhere, so it’s compact).
Let \(S\) be a finite set of places of a number field \(K\) . For each \(v \in S\) let \(L_v/K_v\) be a finite Galois extension. Then there is a finite solvable Galois extension \(L/K\) such that if \(w\) is a place of \(L\) dividing \(v \in S\), then \(L_w/K_v\) is isomorphic to \(L_v/K_v\) as \(K_v\)-algebra. Moreover, if \(K^{\operatorname{avoid}} /K\) is any finite extension then we can choose \(L\) to be linearly disjoint from \(K^{\operatorname{avoid}}\).
A function \(f : G(N_\infty )\to \mathbb {C}\) is slowly-increasing if there exists some \(C{\gt}0\) and \(n\geq 1\) such that \(|f(x)\leq C||x||_\rho ^n\).
If \(E\) is an elliptic curve over a field complete with respect to a nontrivial nonarchimedean (real-valued) norm \(K\) and if \(E\) has split multiplicative reduction, then there is a Galois-equivariant injection \((K^{\operatorname{sep}})^\times /q^{\mathbb {Z}}\to E(K^{\operatorname{sep}})\), where \(q\in K^\times \) satisfies \(|q|=|j(E)|^{-1}\).
If \(X\) is as above and \(X\to \mathbb {A}^n_K\) is a closed immersion, then the induced map from \(X(R)\) with its topology as above to \(R^n\) is an embedding of topological spaces (that is, a homeomorphism onto its image).
If \(X\) is an affine scheme of finite type over \(K\), and if \(R\) is a \(K\)-algebra which is also a topological ring, then we define a topology on the \(R\)-points \(X(R)\) of \(K\) by embedding the \(K\)-algebra homomorphisms from \(A\) to \(R\) into the set-theoretic maps from \(A\) to \(R\) with its product topology, and giving it the subspace topology.
The profinite completion \(\widehat{\mathbb {Z}}\) of \(\mathbb {Z}\) is the set of all compatible collections \(c=(c_N)_N\) of elements of \(\mathbb {Z}/N\mathbb {Z}\) indexed by \(\mathbb {N}^+:=\{ 1,2,3,\ldots \} \). A collection is said to be compatible if for all positive integers \(D\mid N\), we have \(c_N\) mod \(D\) equals \(c_D\).
The infinite sum \(0!+1!+2!+3!+4!+5!+\cdots \) looks like it makes no sense at all; it is the sum of an infinite series of larger and larger positive numbers. However, the sum is finite modulo \(N\) for every positive integer \(N\), because all the terms from \(N!\) onwards are multiples of \(N\) and thus are zero in \(\mathbb {Z}/N\mathbb {Z}\). Thus it makes sense to define \(e_N\) to be the value of the finite sum modulo \(N\). Explicitly, \(e_N=0!+1!+\cdots +(N-1)!\) modulo \(N\).