Additive characters, self-dual measures and Poisson summation on the adèles

Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI. Public domain (CC0).

Fourier analysis on the adèles works because one pairing simultaneously identifies every local field with its character group and the diagonal number field with its own annihilator. The normalization of that pairing determines the measures. At a ramified finite place the integral ring generally has mass less than one; the different supplies exactly the correction needed to make the global additive quotient have volume one.

We use the restricted-product duality theorem from Restricted products and profinite completions and the additive topology, approximation, and covolume from The adèle ring of a number field. The general closed-quotient duality and Poisson formula are the harmonic-analysis inputs stated precisely under Prerequisites and further directions. The number-field proofs and all constants in their application are given here. Our convention agrees with [Getz–Hahn draft, Appendix B] and [Poonen, §4.4]: the basic real character has negative exponent, and the Fourier kernel evaluates that character at the product of its arguments.

Characters and their local parameters

For \(x\in\mathbb Q_p\), choose its finite negative-power part \(\{x\}_p\in\mathbb Z[1/p]\cap[0,1)\), so that \(x-\{x\}_p\in\mathbb Z_p\). Define

For a finite completion \(F=K_v\) above \(p\), set

A character here means a continuous homomorphism from an additive group to the unit circle. Write \(\widehat F\) for its character group with the compact-open topology. The local inverse different is the trace-dual fractional ideal

Its inverse \(\mathfrak D_F\) is an integral ideal, and we write \(\mathfrak D_F=\varpi^d\mathcal O_F\), \(d\geq0\). The definition and integrality follow from Proposition 14.1 of The different and the discriminant; [Sutherland 12, Definition 12.2] is the scholarly reference. For \(q=\#(\mathcal O_F/\varpi\mathcal O_F)\), put \(N(\mathfrak D_F)=q^d\).

Proposition 5.1 (local self-duality). For every completion of a number field, the map

is a topological group isomorphism. At a finite place,

The largest fractional ideal on which \(\psi_F\) is trivial is \(\mathfrak D_F^{-1}\).

Proof. The negative-power part is additive modulo \(\mathbb Z\): its difference from the negative-power part of a sum belongs to both \(\mathbb Z_p\) and \(\mathbb Z[1/p]\), whose intersection is \(\mathbb Z\). Therefore \(\psi_{\mathbb Q_p}\) is a homomorphism. Its kernel is exactly \(\mathbb Z_p\), since the chosen negative-power part is in \([0,1)\) and its exponential is one only when it is zero. This open kernel also proves continuity.

We classify every character \(\chi\) of \(\mathbb Q_p\). Choose a small circle arc containing no nontrivial subgroup. Continuity gives \(m\geq0\) such that \(\chi(p^m\mathbb Z_p)\) lies in that arc. This image is a subgroup, so \(\chi\) is trivial on \(p^m\mathbb Z_p\). For \(n\geq0\), choose

The relation \(p\cdot p^{-n-1}=p^{-n}\) gives \(b_{n+1}\equiv b_n\pmod{p^{m+n}}\). These compatible residues specify \(b\in\mathbb Z_p\). Set \(a=p^{-m}b\). On \(p^{-n}\mathbb Z_p/p^m\mathbb Z_p\), a cyclic group generated by \(p^{-n}\), the character \(\psi_{\mathbb Q_p}(a\,\cdot)\) has the same value as \(\chi\) on that generator. Thus they agree on \(p^{-n}\mathbb Z_p\), and these subgroups cover \(\mathbb Q_p\). If \(a\ne0\), take \(x=a^{-1}/p\); then \(\psi_{\mathbb Q_p}(ax)\ne1\). This proves uniqueness.

For topology, a compact subset of \(\mathbb Q_p\) lies in some \(p^{-r}\mathbb Z_p\). Parameters differing by an element of \(p^r\mathbb Z_p\) therefore give identical character values on that compact set, proving continuity of (4). Conversely, for every integer \(s\), the image of \(p^s\mathbb Z_p\) is the annihilator of the compact subgroup \(p^{-s}\mathbb Z_p\). This annihilator is open in the character group: requiring the image of the compact subgroup to lie in a circle arc with no nontrivial subgroup requires it to be trivial. Hence (4) is open and is a topological isomorphism.

For a finite extension \(F/\mathbb Q_p\), choose a basis \(e_1,\ldots,e_n\). Its coordinate map is a topological linear isomorphism \(\mathbb Q_p^n\simeq F\). A character of this finite product is a product of characters on its axes, and therefore has the form

The trace pairing is nondegenerate because the extension is separable. Consequently there is a unique \(a\in F\) with \(\operatorname{Tr}(ae_i)=c_i\) for every \(i\). Equation (6) is then \(\psi_F(ax)\). This parameter change is an invertible \(\mathbb Q_p\)-linear map. Finite-product character identification is topological: restriction to each axis is continuous for the compact-open topology, and continuity of the product construction follows by projecting compact sets onto those finitely many axes. The already proved \(\mathbb Q_p\) case therefore proves the topological assertion for \(F\).

For \(\mathbb R\), a continuous character has the form \(e^{2\pi i c x}\). To verify the classification, lift its values near zero to a continuous real argument of absolute value less than \(1/4\). On a sufficiently small interval that argument is locally additive, since the difference of arguments is an integer of absolute value less than one. Continuous local additivity gives a linear argument there, by the rational approximation proof of the continuous Cauchy equation; subdividing any real number into pieces in that interval proves the formula globally. Our negative sign in (1) merely replaces the parameter \(c\) by \(-c\). Uniform convergence on compact intervals follows from convergence of parameters. In the other direction, if \(|a|\geq\varepsilon\), its character takes the value \(-1\) at \(x=1/(2a)\) in the fixed compact interval \([-1/(2\varepsilon),1/(2\varepsilon)]\). A compact-open neighborhood excluding that value forces \(|a|<\varepsilon\). Thus the parameter identification is topological.

For \(\mathbb C\simeq\mathbb R^2\), write \(a=\alpha+i\beta\), \(x=u+iv\). The pairing in (4) is

The coefficient map \((\alpha,\beta)\mapsto(-2\alpha,2\beta)\) is an invertible real linear map, so the finite-product real argument proves the complex case.

Finally, (2) and the exact kernel \(\mathbb Z_p\) prove (5) directly from (3). The ideal \(\mathfrak D_F^{-1}\) is contained in \(\ker\psi_F\). If a fractional ideal \(I\) is contained in that kernel, then \(a\mathcal O_F\subset I\) for each \(a\in I\); hence \(a\) satisfies (3). Thus \(I\subset\mathfrak D_F^{-1}\), proving maximality. \(\square\)

For an extension of degree greater than one, the entire kernel of \(\psi_F\) can be larger than its conductor ideal: every trace-zero element lies in the kernel. Equation (3) describes the elements that annihilate the whole integral ring, which is the condition Fourier transforms of its indicator require.

Measures fixed by inversion

Use the Fourier transform

A Haar measure is self-dual when \(\widehat{\widehat f}(x)=f(-x)\) for Schwartz–Bruhat functions. At a finite place these are locally constant compactly supported functions; at real or complex places they are the usual Schwartz functions on the underlying real vector space.

Proposition 5.2 (self-dual measures and basic transforms). The self-dual measures for (1)–(2) are \(dx\) on \(\mathbb R\), \(2\,du\,dv\) on \(\mathbb C\), and at a finite place the measure satisfying

In particular,

More generally, for a finite place and \(H=b\mathcal O_F\),

Proof. Integrating a character over a compact group gives its volume if the character is trivial, and zero otherwise. In the second case translate by an element on which the character is not one; invariance makes the integral equal to that nonunit scalar times itself, so it must vanish. Apply this to \(H\) and use translation by \(a\), obtaining (11).

Write \(c=\operatorname{vol}(\mathcal O_F)\). Haar scaling gives \(\operatorname{vol}(\mathfrak D_F^{-1})=q^d c\). Applying the transform twice to \(\mathbf1_{\mathcal O_F}\) gives \(q^d c^2\mathbf1_{\mathcal O_F}\), so self-duality requires \(c=q^{-d/2}\). This normalization is sufficient as well. For \(H=b\mathcal O_F\), (5) gives

The annihilator of \(H^\perp=b^{-1}\mathfrak D_F^{-1}\) is \(H\), by (5). Applying (11) twice therefore gives \(\mathbf1_{-a+H}(x)=\mathbf1_{a+H}(-x)\). Every locally constant compactly supported function is a finite linear combination of such coset indicators: compactness gives one sufficiently small common open subgroup on which it is invariant, and its support meets only finitely many cosets. Thus inversion holds for every such function. Formula (11) also shows that the transform remains locally constant and compactly supported.

The full real and complex inversion statements follow from the whole-space proof below, with its dimension equal to one or two. That proof uses only the Gaussian transform we establish next, so it does not assume inversion in deriving that transform. Its real normalization is Lebesgue measure; its complex coefficient matrix is \(\operatorname{diag}(2,-2)\), and the measure factor is \(2\). Their determinant relation \(2^2/4=1\) gives complex self-duality. A positive scalar multiple \(c\) of either chosen measure multiplies the double transform by \(c^2\), proving uniqueness.

For the real Gaussian, its integral is one: the square of the integral is the polar-coordinate integral of \(e^{-\pi(u^2+v^2)}\), which equals one. Differentiating its Fourier transform under the integral and integrating by parts gives \(F'(y)=-2\pi yF(y)\). The boundary term vanishes by Gaussian decay and \(F(0)=1\), so \(F(y)=e^{-\pi y^2}\). Scaling gives the transform of \(e^{-\pi a x^2}\) as \(a^{-1/2}e^{-\pi y^2/a}\) for \(a>0\). For the complex Gaussian, (7) separates the integral into two real Gaussian transforms with \(a=2\) and frequencies \(2\operatorname{Re}w\), \(-2\operatorname{Im}w\). Their constants multiply to \(1/2\), canceled by the factor \(2\) in the measure. Their exponents give \(-2\pi|w|^2\), proving (10). \(\square\)

For \(F=\mathbb Q_2(i)\), the integral ring is \(\mathbb Z_2[i]\), by the integral completion identity in Proposition 2.1 and the calculation of \(\mathcal O_{\mathbb Q(i)}\) in lesson 2. Write \(x=r+si\). The trace conditions against \(1\) and \(i\) are \(2r,2s\in\mathbb Z_2\). Hence

The ideal identity uses \((1+i)^2=2i\). This also follows from the derivative \(2i\) of the monic polynomial \(X^2+1\), in the monogenic different formula [Sutherland 12, Proposition 12.24].

One character on the whole adèle ring

For \(x\in\mathbb A_K\), define

At every finite place \(\psi_{K_v}\) is trivial on \(\mathcal O_v\), since traces of integral elements are integral. All but finitely many factors in (14) are therefore one. On any open restricted-product stage the integral tail is trivial and only finitely many local characters remain, so this is a continuous character.

Proposition 5.3 (the diagonal is annihilated). The character (14) is trivial on \(K\). It satisfies

Proof. For \(r\in\mathbb Q\), only finitely many \(\{r\}_p\) are nonzero. The rational number

belongs to every \(\mathbb Z_p\): its difference at a selected \(p\) is integral there, and the other negative-power parts have denominators prime to \(p\). A rational number integral at every prime is an integer. Thus

which proves the rational case. At a rational finite prime, the trace on \(K\otimes\mathbb Q_p\simeq\prod_{v\mid p}K_v\) is the sum of the local traces. At infinity it is the sum of the real components and of \(z+\overline z\) over complex components. Therefore (1)–(2) imply (15). For diagonal \(a\in K\), that trace is the diagonal rational number \(\operatorname{Tr}_{K/\mathbb Q}(a)\). Apply (16). \(\square\)

The field is exactly its annihilator

Theorem 5.4 (global self-duality and covolume). The pairing

induces a topological isomorphism \(\mathbb A_K\simeq\widehat{\mathbb A_K}\). Under this identification,

The restricted product of the local measures of Proposition 5.2 is self-dual, and, with counting measure on \(K\),

Proof. Proposition 5.1 identifies the local dual of \(K_v\) with \(K_v\). At a finite place the annihilator of the distinguished subgroup \(\mathcal O_v\) is \(\mathfrak D_v^{-1}\). Theorem 1.3 identifies the global dual with the restricted product for those local annihilators, including its compact-open topology. Only finitely many local differents are nontrivial: the local/global identity is Proposition 4.1 of Ramification groups and the different of a local extension, and the ramification criterion is Theorem 14.4 of The different and the discriminant. The scholarly locators are [Sutherland 12, Proposition 12.4 and Theorem 12.19]. Changing distinguished compact open subgroups at finitely many places changes neither restricted-product set nor topology. Thus that dual is \(\mathbb A_K\), with the pairing (17).

We prove the annihilator assertion first over \(\mathbb Q\). Suppose \(y\in\mathbb A_{\mathbb Q}\) satisfies \(\psi_{\mathbb Q}(ry)=1\) for every \(r\in\mathbb Q\). Subtract a diagonal rational number to put \(y\) in the fundamental domain \([0,1)\times\widehat{\mathbb Z}\) from Theorem 2.2. Subtraction preserves the hypothesis by Proposition 5.3. Testing \(r=1\) gives \(e^{-2\pi iy_\infty}=1\), so \(y_\infty=0\). For a prime \(p\), test \(r=p^{-n}\), \(n\geq1\). At every other finite prime this multiplier is integral, so all those local factors are one. The remaining equation is \(\psi_{\mathbb Q_p}(p^{-n}y_p)=1\), which says \(y_p\in p^n\mathbb Z_p\). For every \(n\) this forces \(y_p=0\). Hence the reduced \(y\) is zero, proving \(\mathbb Q^\perp=\mathbb Q\).

For general \(K\), choose a rational basis \(e_1,\ldots,e_n\) and its trace-dual basis \(e_1^*,\ldots,e_n^*\), with \(\operatorname{Tr}(e_i e_j^*)=\delta_{ij}\). The adelic tensor identification of Proposition 2.1 writes

If \(y\in K^\perp\), test (17) against \(r e_j^*\in K\) for every \(r\in\mathbb Q\). By (15), the result is \(\psi_{\mathbb Q}(r b_j)=1\). The rational case gives \(b_j\in\mathbb Q\), so (20) lies in \(K\). Conversely \(K\subset K^\perp\) by Proposition 5.3. This proves (18)'s first equality. Since \(K\) is closed, the dual of the quotient \(\mathbb A_K/K\) identifies topologically with this annihilator by Theorem 2.1 of Subgroups, quotients and annihilators. It is the discrete subgroup \(K\), proving the second assertion.

Let \(d_K\) be the discriminant of \(K\). The norm-discriminant identity and the local factorization of the different give

The norm identity is Theorem 14.3 of The different and the discriminant, and completion of the trace dual is Proposition 4.1 of Ramification groups and the different of a local extension. The scholarly locators are [Sutherland 12, Propositions 12.4 and 12.16 and Theorem 12.17]. The product of our local measures exists by Proposition 1.2, since its integral-ring masses are one outside a finite set. It differs from the measure of Proposition 2.4 by the scalar \(|d_K|^{-1/2}\): the infinite measures are unchanged and (9), (21) supply the finite scalar. Since Proposition 2.4 gave additive covolume \(\sqrt{|d_K|}\), the new covolume is one.

The remaining Fourier-inversion assertion, for the full adelic test space rather than only products over individual infinite places, is proved in the next section. Together with this covolume calculation it completes the theorem. \(\square\)

Fourier inversion on the whole archimedean space

Put

where \(K_\infty\simeq\mathbb R^{r_1}\times\mathbb C^{r_2}\), the first space is the usual Schwartz space on the entire underlying real vector space of dimension \(d=r_1+2r_2\), and \(C_c^\infty\) here means locally constant compactly supported functions. The tensor product in (22) is algebraic: its elements are finite sums \(\phi(x_\infty)h(x_f)\). This separates the whole infinite block from the finite block; it imposes no separation among the coordinates or places within \(K_\infty\). A function \(\phi\) belongs to \(\mathcal S(\mathbb R^d)\) when it is smooth and every \(x^\alpha\partial^\beta\phi(x)\) is bounded, for all multi-indices \(\alpha,\beta\). The finite part is a finite linear combination of products of local coset indicators. Indeed, compact support lies in one restricted-product stage. Choose at each point of the support a product-form compact open subgroup on whose coset the function is constant, and pass to a finite subcover. Intersect those finitely many subgroups and, if necessary, shrink the finitely many exceptional local factors to fractional-ideal balls. This gives a common compact open subgroup of product form. The support is a finite union of its cosets; translating within this subgroup preserves that union and each constant value, so the function is invariant on the whole group, including its zero set. The coset indicators separate into local products with an integral tail. This is the same argument used for finite adèles in lesson 1.

The distinction between a whole archimedean block and separated infinite-place tests is useful when comparing definitions. Getz–Hahn, Appendix B, §B.3, takes Schwartz functions on the whole real vector space at infinity and tensors that block with finite-place tests. Poonen, §5.4, describes the test space using tensor products over individual places. The inversion and periodization proved below apply to the full space (22), including functions whose infinite coordinates do not separate.

For example, for \(K=\mathbb Q(\sqrt2)\), the function

is Schwartz: each derivative is a polynomial times this function, and Gaussian decay bounds every further polynomial weight. It is not a finite sum of products \(u_j(x)v_j(y)\). Indeed, for any distinct \(y_1,\ldots,y_m\), its slices in the \(x\)-variable are linearly independent. Dividing a putative slice relation by \(e^{-\pi x^2}\), and differentiating at zero through orders \(0,\ldots,m-1\), gives the invertible Vandermonde matrix \(((iy_j)^k)_{k,j}\); the factors \(e^{-\pi y_j^2}\) are nonzero. A sum of \(r\) separated products could have slice span of dimension at most \(r\), contradicting this for \(m=r+1\). Thus whole-space inversion is needed even for elementary adelic tests.

Lemma 5.4A (whole-space Fourier inversion). Identify \(K_\infty\) with \(\mathbb R^d\), using the real coordinate at each real place and the real and imaginary coordinates at each complex place. Put

The product character has kernel \(e^{-2\pi i x^{\mathsf T}By}\), and the product infinite measure is \(c\,dx\), where \(dx\) is ordinary \(d\)-dimensional Lebesgue measure. For every \(\phi\in\mathcal S(\mathbb R^d)\), the transform

is Schwartz and satisfies \(T^2\phi(x)=\phi(-x)\).

Proof. First prove ordinary Euclidean Fourier inversion, writing

Schwartz decay makes every polynomial-weighted derivative of \(\phi\) integrable. Differentiation under the integral expresses \(\partial_\xi^\beta\mathcal F\phi\) as the transform of \((-2\pi i x)^\beta\phi\). Multiplication by a monomial \(\xi^\alpha\) can then be transferred, by repeated integration by parts, to derivatives of this polynomial-weighted function. Boundary terms vanish by rapid decay. The resulting integral is bounded by a constant times its integrand's \(L^1\)-norm, uniformly in \(\xi\). Thus every Schwartz seminorm of \(\mathcal F\phi\) is finite; in particular \(\mathcal F\phi\) is Schwartz and integrable.

For \(\varepsilon>0\), the Gaussian calculation in Proposition 5.2, scaling and finite-dimensional Fubini give

Both \(\phi\) and this frequency Gaussian are integrable. Absolute Fubini therefore gives

The kernels \(k_\varepsilon\) are nonnegative and have integral one. Their mass outside any fixed ball of radius \(\delta>0\) tends to zero, by substituting \(u=\sqrt\varepsilon z\) and taking a Gaussian tail. Continuity of \(\phi\) at \(x\) controls its difference from \(\phi(x)\) inside that ball; boundedness of \(\phi\) controls the vanishing tail. Hence the right side tends to \(\phi(x)\). On the left, integrability of \(\mathcal F\phi\) permits dominated convergence. We obtain ordinary inversion and therefore

For the number-field pairing, symmetry of \(B\) gives \(T\phi=c(\mathcal F\phi)\circ B\). Composition with an invertible linear map preserves all Schwartz seminorms, so \(T\phi\) is Schwartz. The change of variables \(u=By\), for any Schwartz function \(h\), gives

Since \(B^{-\mathsf T}B=I_d\) and \(|\det B|=4^{r_2}=c^2\), ordinary inversion yields

This proves inversion for arbitrary whole-space Schwartz functions, including the nonseparated example above. \(\square\)

Applying Poisson to every adelic test function

With the product measure, define

Take first \(f=\phi\otimes h\) in (22), with \(\phi\) arbitrary on the whole infinite block. Decompose \(h\) into the finite coset products described above, choosing a common exceptional finite set containing all ramified places. The remaining local factors are \(\mathbf1_{\mathcal O_v}\), each with mass one and unchanged transform by (9)–(10). On the exceptional factors Proposition 5.2 proves inversion and preservation of the local test space; every ramified mass \(q_v^{-d_v/2}\) is retained. The infinite transform is \(T\phi\) from Lemma 5.4A. All factors are integrable, so absolute Fubini on a restricted-product stage factors (23) into this whole-block transform and the finite transforms. Applying it twice reflects both blocks. Finite linearity now proves that (23) preserves the entire space (22) and satisfies \(\widehat{\widehat f}(x)=f(-x)\) there. This completes Theorem 5.4's self-dual-measure assertion in its full stated scope.

Theorem 5.5 (Poisson summation and adelic Riemann–Roch). For \(f\in\mathcal S(\mathbb A_K)\), the following sums converge absolutely and satisfy

For any idèle \(a\in J_K\), writing \(|a|=\prod_v|a_v|_v\),

Proof. We verify the hypotheses of the general Poisson formula for the discrete cocompact subgroup \(K\subset\mathbb A_K\). Both \(f\) and \(\widehat f\) are integrable, because each is a finite sum of Schwartz functions times compactly supported bounded finite functions. First take \(f(x)=\phi(x_\infty)h(x_f)\) in (22). Its finite support is contained in a compact set \(B\subset\mathbb A_{K,f}\). Every compact set of finite adèles is contained in \(N^{-1}\widehat{\mathcal O}_K\) for some positive rational integer \(N\): finitely many charts cover the set, only finitely many local denominators occur there, and an integer clears all of them. Thus the \(k\in K\) contributing to the sum lie in \(N^{-1}\mathcal O_K\).

The archimedean image of \(N^{-1}\mathcal O_K\) is a full lattice in the real space of dimension \(n=[K:\mathbb Q]\). Its lattice-point count in a ball of radius \(R\) is \(O((1+R)^n)\): a lattice basis identifies it with \(\mathbb Z^n\), and an invertible linear map compares its norm with the coordinate maximum. For any \(A>n\), Schwartz decay gives \(|\phi(z)|\leq C_A(1+\|z\|)^{-A}\). Summing over dyadic annuli gives convergence, since their contributions are bounded by a geometric series with ratio \(2^{n-A}<1\). The finite factor \(h\) is bounded. Hence \(\sum_{k\in K}|f(k)|<\infty\).

These estimates also give uniform convergence of periodization over a compact set of additive representatives. Such a set exists by Theorem 2.2. Its finite projection is compact, so the differences between that projection and \(B\) are contained in one denominator lattice as above. Its infinite projection is bounded, and bounded translates of a Schwartz function have the same rapid-decay bound with a uniform constant. The dyadic estimates therefore apply uniformly to \(\sum_k f(y+k)\) on that representative set. Since \(\widehat f\) is again in (22), the same argument proves absolute convergence of its lattice sum and the corresponding uniform periodization. Finite sums of tensors retain every asserted convergence property.

Theorem 1.1 of The Poisson summation formula now applies, with annihilator \(K^\perp=K\) and covolume one by Theorem 5.4. The provider defines its Fourier transform with the conjugate character. Under our identification \(y\mapsto[x\mapsto\psi(xy)]\), its transform at \(y\) is our \(\widehat f(-y)\). The absolutely convergent sum over \(K\) is unchanged by \(y\mapsto-y\), so the theorem gives (24) with our convention. This is the precise point where the general harmonic-analysis theorem is used; the adelic hypotheses and both sums have been checked.

Multiplication by an idèle preserves (22): locally it is an invertible linear scaling; at almost all finite places it preserves the integral indicator. Haar scaling on open restricted-product stages gives \(d(ax)=|a|\,dx\). A change of variables in (23) therefore gives

Apply (24) to \(f(a\,\cdot)\) to obtain (25), including absolute convergence. \(\square\)

The Gaussian integers and the different's factor

Let \(K=\mathbb Q(i)\), so \(\mathcal O_K=\mathbb Z[i]\) and \(|d_K|=4\). The trace calculation in (13), now applied globally, gives \(\mathfrak D_K=(2)\). Thus

For \(t>0\), take the adelic function

This is obtained from the basic Gaussian \(e^{-2\pi|z|^2}\) by multiplying the infinite coordinate by \(\sqrt{t/2}\), leaving the finite coordinates fixed; hence (25) applies to it. The complex Gaussian calculation with measure \(2\,du\,dv\) gives

Its factor \(2/t\) combines with the finite factor \(1/2\) in (27). The left side of (24) is

On the right side, the finite support restricts the summation to \(\frac12\mathbb Z[i]\). Substitute \(\alpha/2\) for its points to get

Equivalently \(\theta_{\mathbb Z[i]}(1/t)=t\theta_{\mathbb Z[i]}(t)\). Absolute convergence permits separation of the sum over \(\alpha=m+ni\); hence \(\theta_{\mathbb Z[i]}(t)=\theta_{\mathbb Z}(t)^2\). Squaring the classical identity \(\theta_{\mathbb Z}(1/t)=\sqrt t\,\theta_{\mathbb Z}(t)\) gives exactly (29). The factor supplied by the different is essential in matching these normalizations.

Exercises

1. Easy. Prove directly that \(\psi_{\mathbb Q_p}\) is a continuous character whose kernel is exactly \(\mathbb Z_p\). Explain why a choice of a different negative-power representative does not change the exponential.

2. Medium. Compute the inverse different, the self-dual mass of the integral ring, and the transform of its indicator for \(\mathbb Q_3(\sqrt3)\) and \(\mathbb Q_2(\sqrt2)\). Include a verification of their integral rings.

3. Medium. Verify both Gaussian transforms in (10), including the factor \(2\) in the complex Haar measure.

4. Hard. Prove \(K^\perp=K\) in \(\mathbb A_K\), starting with the rational fundamental domain and then using a trace-dual rational basis. Identify exactly where discreteness, approximation and separability enter.

Solutions

Solution 1. Two representatives in \(\mathbb Z[1/p]\) whose difference from \(x\) is integral differ by an element of \(\mathbb Z[1/p]\cap\mathbb Z_p=\mathbb Z\); their exponentials therefore agree. The representatives of \(x\), \(y\), and \(x+y\) satisfy \(\{x\}_p+\{y\}_p-\{x+y\}_p\in\mathbb Z\). Exponentiating proves the homomorphism property. With the representative in \([0,1)\), its exponential is one precisely when the representative is zero, equivalently \(x\in\mathbb Z_p\). The open kernel proves local constancy and continuity. For example, \(\psi_{\mathbb Q_p}(p^{-1})=e^{2\pi i/p}\ne1\), verifying nontriviality.

Solution 2. Let \(p=3\) or \(2\), and set \(F=\mathbb Q_p(\pi)\), \(\pi^2=p\). The polynomial is irreducible because a square in \(\mathbb Q_p\) has even valuation, so \([F:\mathbb Q_p]=2\). If \(e\) is the ramification index, then \(2\operatorname{ord}_F(\pi)=e\). Since \(1\leq e\leq2\), it follows that \(e=2\) and \(\operatorname{ord}_F(\pi)=1\). The residue degree is one, so \(q=p\). For \(a,b\in\mathbb Q_p\), the two terms of \(a+b\pi\) have valuations \(2\operatorname{ord}_p(a)\) and \(2\operatorname{ord}_p(b)+1\). Whenever both are nonzero these valuations have opposite parity and cannot cancel; the sum has their minimum. Consequently \(a+b\pi\) is integral exactly when \(a,b\in\mathbb Z_p\). Hence \(\mathcal O_F=\mathbb Z_p[\pi]\).

For \(x=r+s\pi\), the trace pairing against its integral basis gives \(\operatorname{Tr}(x)=2r\), \(\operatorname{Tr}(x\pi)=2ps\). Thus the trace-dual integral module is

For \(p=3\), \(2\) is a unit, so \(\mathfrak D_F=(\pi)\), \(d=1\), and

For \(p=2\), \(\operatorname{ord}_F(2)=2\), so \(\operatorname{ord}_F(2\pi)=3\), \(d=3\), and

These follow from (9)–(10). The monogenic derivative formula also gives the different \((2\pi)\), agreeing with the direct trace computation. The usual local degree formula \([F:\mathbb Q_p]=ef\) and the valuation criterion for integrality are the local-field prerequisites; no unramified measure normalization was used at these ramified places.

Solution 3. Put \(g(x)=e^{-\pi x^2}\) and \(F(y)=\int g(x)e^{-2\pi ixy}\,dx\). Its integral at \(y=0\) is one by the polar-coordinate calculation of the square of the Gaussian integral. Since \(g'(x)=-2\pi xg(x)\), differentiation and integration by parts give

Every interchange is justified by Gaussian integrability, including its polynomial multiples. Solve this first-order equation with \(F(0)=1\) to get \(F(y)=e^{-\pi y^2}\). Substitution \(u=\sqrt2x\) then gives \(\int e^{-2\pi x^2}e^{-2\pi isx}\,dx=2^{-1/2}e^{-\pi s^2/2}\).

For \(z=u+iv\), \(w=\xi+i\eta\), the trace is \(zw+\overline{zw}=2(u\xi-v\eta)\). Hence the complex transform is

Ordinary complex multiplication, rather than a Hermitian pairing, accounts for the sign in the second factor; the even Gaussian gives the same value for either frequency sign.

Solution 4. Proposition 5.3 gives \(K\subset K^\perp\). Over \(\mathbb Q\), choose a diagonal rational \(r\) so that \(y-r\in[0,1)\times\widehat{\mathbb Z}\), using Theorem 2.2. If \(y\) annihilates \(\mathbb Q\), so does \(y-r\), by (16). Testing against \(1\) makes its real component the integer in \([0,1)\), namely zero. Testing against \(p^{-n}\) then makes its \(p\)-component divisible by \(p^n\) for every \(n\). All components vanish. Thus \(y=r\) is diagonal rational.

For general \(K\), separability makes the trace pairing invertible, providing the dual basis \(e_j^*\). The adelic tensor isomorphism writes \(y=\sum_i e_i b_i\). Testing against \(r e_j^*\), for every rational \(r\), shows that \(b_j\) annihilates \(\mathbb Q\), so every \(b_j\) is rational by the first step. Hence \(y\in K\). The rational principal-part approximation is used to obtain its fundamental-domain representative; the general-field tensor identification transfers that argument. Discreteness and closedness of \(K\) are needed for the later topological quotient-dual and Poisson applications, but this algebraic annihilator equality itself uses the fundamental-domain construction and trace nondegeneracy, not a new discreteness argument.

Prerequisites and further directions

• The restricted-product topological duality theorem: the dual is the restricted product of the local duals with respect to annihilators of the compact open subgroups. This is Theorem 1.3 of Restricted products and profinite completions.
• Additive adelic tensor identification, closed discreteness of \(K\), its compact quotient and the covolume \(\sqrt{|d_K|}\) for integral finite-ring masses one. These are Proposition 2.1, Theorem 2.2 and Proposition 2.4 of The adèle ring of a number field.
• Different theory is proved in The different and the discriminant, Propositions 14.1–14.2 and Theorems 14.3–14.4, and Ramification groups and the different of a local extension, Proposition 4.1. These give trace-dual integrality, completion, the norm identity, and the monogenic formula when the full integral ring is a power basis. Scholarly locators are [Sutherland 12, Definition 12.2, Propositions 12.4, 12.16 and 12.24 and Theorem 12.17]. The trace computations in the examples are given here independently.
• For a closed subgroup \(H\) of an LCA group \(G\), pullback identifies \(\widehat{G/H}\) with \(H^\perp\) topologically. This is Theorem 2.1 in Subgroups, quotients and annihilators. Its hypotheses and topological conclusion are the ones used in Theorem 5.4.
• The general LCA Poisson formula from The Poisson summation formula, Theorem 1.1: for a discrete cocompact subgroup \(\Lambda\subset G\), self-dual measures, and an integrable test function with continuous uniformly convergent periodization and an absolutely summable dual-lattice transform, \(\sum_{\lambda\in\Lambda}f(\lambda)=\operatorname{vol}(G/\Lambda)^{-1}\sum_{\eta\in\Lambda^\perp}\widehat f(\eta)\). Theorem 5.5 verifies all these conditions on the adèles. [Getz–Hahn draft, Theorem B.3.2] records the resulting adelic formula.

References

• [Getz–Hahn draft] J. R. Getz and H. Hahn, An Introduction to Automorphic Representations: With a View toward Trace Formulae, author-hosted draft dated April 22, 2022, §3.5 and Appendix B, §§B.1–B.3, PDF pages 91–92 and 501–505. The published book appeared in 2024; the locators here refer to the draft.
• [Sutherland 12] A. V. Sutherland, The different and the discriminant, MIT 18.785 Number Theory I lecture notes, Fall 2021, Lecture 12, Definition 12.2, Propositions 12.4, 12.16 and 12.24, and Theorems 12.17 and 12.19: the different, the discriminant, their localization and completion, and the ramification criterion.
• H. Jacquet and R. P. Langlands, Automorphic Forms on GL(2), IAS retyped edition, 1970, Chapter I, §1, printed pages 1–2: trace pairing and self-dual Haar measure. Its Weil-representation constructions are further directions.
• A. Connes and M. Marcolli, Noncommutative Geometry, Quantum Fields and Motives, author version, Chapter 2, §4.1, Definition 2.8 and Proposition 2.9, and §8.3, (2.297)–(2.300), PDF pages 335–337 and 366–367. The first normalization gives norm-one mass one on a multiplicative modulated group. Our additive self-dual measure is determined by (8); the two normalizations have different purposes.
• [Poonen] Bjorn Poonen, Tate’s Thesis, MIT 18.786, Spring 2015, §§4.4–4.5, Proposition 4.8, and §§5.4–5.5, Lemma 5.6 and Theorem 5.7: local character, measure and periodization comparisons. The proof here retains the whole archimedean Schwartz space of (22).