A logarithm rank lemma

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

The next lesson shows that an additive \(\ell\)-adic character of the absolute Galois group of a number field is determined by its values at Frobenius elements of infinitely many primes of degree one. Through class field theory, this comes down to one statement about \(\ell\)-adic logarithms of algebraic numbers: logarithms of enough points of a torus whose coordinates are multiplicatively independent span the whole space. This lesson proves that statement with an interpolation determinant: a determinant of values of exponential functions that is an algebraic integer, very small \(\ell\)-adically, and not too large at the archimedean places, which the product formula forbids. The zero estimate that makes the determinant nonzero is the component bound of Counting irreducible components. The statement is a special case of the \(\ell\)-adic rank theorem of Waldschmidt and Masser, treated in [Dasgupta 2023, Section 5]; the determinant method is Laurent's [Laurent 1991].

1. \(\ell\)-adic logarithms and exponentials

Let \(\ell\) be a prime and \(K\) a finite extension of \(\mathbb Q_\ell\), with ring of integers \(\mathcal O_K\), maximal ideal \(\mathfrak m_K\), and absolute value normalized by \(|\ell|=\ell^{-1}\). The logarithm series \(\log(1+x)\) converges on the principal units \(U^1=1+\mathfrak m_K\), and the exponential series converges for \(|z|<\ell^{-1/(\ell-1)}\); on a sufficiently deep subgroup \(U^{r_0}=1+\mathfrak m_K^{r_0}\) and the corresponding ideal \(\mathfrak m_K^{r_0}\) they are inverse isomorphisms of topological groups (Proposition 2.1 of The multiplicative group of a local field). Every unit is a root of unity of order prime to \(\ell\) times a principal unit, and we define \[ \log\colon\mathcal O_K^\times\longrightarrow K \] as the logarithm of the principal-unit factor; this is the restriction of the homomorphism \(\operatorname{Log}_K\) of Proposition 5.1 of that lesson. It is a continuous homomorphism, and its kernel is the group of roots of unity in \(K\): if \(\log u=0\), then \(u^{N}\in U^{r_0}\) for a suitable \(N\ge1\), since \(\mathcal O_K^\times/U^{r_0}\) is finite, and \(\log u^N=0\) forces \(u^N=1\) by injectivity on \(U^{r_0}\). Logarithms of points of \((\mathcal O_K^\times)^n\) are taken coordinatewise.

Lemma 1.1. For integers \(k\ge0\), \(|k!|^{-1}\le\ell^{k/(\ell-1)}\).

Proof. Legendre's formula gives \(v_\ell(k!)=\sum_{i\ge1}\lfloor k/\ell^i\rfloor\), and \(\sum_{i\ge1}\lfloor k/\ell^i\rfloor\le\sum_{i\ge1}k/\ell^i=k/(\ell-1)\). \(\square\)

2. The statement

Numbers \(c_1,\ldots,c_n\) in a field are multiplicatively independent modulo roots of unity if \(\prod_jc_j^{\gamma_j}\) is a root of unity only for \(\gamma=0\in\mathbb Z^n\). For points \(u_1,\ldots,u_m\in(E^\times)^n\) and \(a\in\mathbb Z^m\) write \(u^a=\prod_iu_i^{a_i}\), coordinatewise.

Theorem 2.1 (logarithm rank; [OpenAI-UC, Lemma 3.1]). Let \(n,m\ge1\), let \(E\) be a number field contained in a finite extension \(K/\mathbb Q_\ell\), and let \[ \begin{gathered} u_i=(u_{i1},\ldots,u_{in})\in(E^\times)^n,\\ 1\le i\le m, \end{gathered} \] have coordinates that are algebraic integers and units of \(K\). Suppose that for every \(a\in\mathbb Z^m\setminus\{0\}\) the coordinates of \(u^a\) are multiplicatively independent modulo roots of unity. If \(m>n(n-1)\), then \[ \operatorname{span}_K\{\log u_1,\ldots,\log u_m\}=K^n . \]

The hypothesis concerns every nonzero \(a\), not only the points \(u_i\) themselves; for \(n=1\) it says that \(u_{11},\ldots,u_{m1}\) are multiplicatively independent modulo roots of unity (Exercise 7.2). For \(n=2\) and \(m=3\) the theorem is an \(\ell\)-adic form of the six exponentials theorem.

3. Zariski density of cyclic subgroups

Let \(\overline E\) be an algebraic closure of \(E\). We use the Zariski topology of the torus \(\mathbb G_m^n(\overline E)\) as in Section 1 of Counting irreducible components.

Lemma 3.1. Let \(c\in(\overline E^\times)^n\) have coordinates multiplicatively independent modulo roots of unity. Then \(\{c^t:t\ge0\}\) is Zariski dense in \(\mathbb G_m^n(\overline E)\). Consequently, if \(V\subseteq\mathbb G_m^n(\overline E)\) is closed, nonempty and satisfies \(cV\subseteq V\), then \(V=\mathbb G_m^n(\overline E)\).

Proof. Let \(Q=\sum_{\alpha\in\mathcal A}q_\alpha x^\alpha\) be a polynomial, with \(\mathcal A\subset\mathbb Z_{\ge0}^n\) finite, vanishing at all \(c^t\). The numbers \(\lambda_\alpha=c^\alpha\) are pairwise distinct: \(c^\alpha=c^\beta\) means \(c^{\alpha-\beta}=1\), so \(\alpha=\beta\). The equations \(\sum_\alpha q_\alpha\lambda_\alpha^t=0\) for \(0\le t<|\mathcal A|\) have an invertible Vandermonde matrix, so \(Q=0\). A closed set containing all \(c^t\) is cut out by polynomials vanishing at all \(c^t\), hence by \(0\). For the second claim, take \(x\in V\). Then \(c^tx\in V\) for all \(t\ge0\), and multiplication by \(x\) is a homeomorphism of the torus, so \(V\) contains the closure \(x\cdot\mathbb G_m^n=\mathbb G_m^n\). \(\square\)

4. The zero estimate

Fix \(n,m,E,K\) and \(u_1,\ldots,u_m\) as in Theorem 2.1. For integers \(D,S\ge1\) let \(\mathcal M_{D,S}\) be the matrix with rows indexed by \(\alpha\in\{0,\ldots,D\}^n\), columns indexed by \(a\in\{0,\ldots,nS\}^m\), and entries \[ (u^a)^\alpha=\prod_{j=1}^n\Bigl(\prod_{i=1}^mu_{ij}^{a_i}\Bigr)^{\alpha_j}\in E . \]

Proposition 4.1. If \((S+1)^m>(nD)^n\), then \(\mathcal M_{D,S}\) has rank \((D+1)^n\), the number of its rows.

Proof. Otherwise the rows are linearly dependent over \(E\): there is a nonzero polynomial \(P=\sum_\alpha p_\alpha x^\alpha\) of degree at most \(D\) in each variable with \(P(u^a)=0\) for all \(a\in\{0,\ldots,nS\}^m\). For \(0\le j\le n\) put \[ \begin{aligned} Z_j=\{&x\in\mathbb G_m^n(\overline E):P(u^ax)=0\\ &\text{for all }a\in\{0,\ldots,jS\}^m\}. \end{aligned} \] Each \(x\mapsto P(u^ax)\) is a polynomial of total degree at most \(nD\), so \(Z_j\) is closed, and \(Z_0\supseteq Z_1\supseteq\cdots\supseteq Z_n\ni1\). Since \(P\ne0\) and the torus is irreducible of dimension \(n\), \(\dim Z_0\le n-1\). The \(n+1\) dimensions \(\dim Z_0\ge\cdots\ge\dim Z_n\) lie in \(\{0,\ldots,n-1\}\), so \(\dim Z_j=\dim Z_{j+1}=:d\) for some \(j<n\).

Let \(V\) be an irreducible component of \(Z_{j+1}\) of dimension \(d\), and let \(b\in\{0,\ldots,S\}^m\). For \(x\in V\) and \(a\in\{0,\ldots,jS\}^m\), we have \(a+b\in\{0,\ldots,(j+1)S\}^m\), so \(P(u^au^bx)=P(u^{a+b}x)=0\). Thus \(u^bV\subseteq Z_j\). Translation by \(u^b\) is a homeomorphism of the torus, so \(u^bV\) is irreducible, closed and of dimension \(d\). It lies in a component \(C\) of \(Z_j\), of dimension at most \(d\); a proper closed subset of \(C\) would have smaller dimension, so \(u^bV=C\) is a component of \(Z_j\).

These \((S+1)^m\) translates are distinct. If \(u^bV=u^{b'}V\) with \(b\ne b'\), then \(c=u^{b-b'}\) satisfies \(cV=V\), and the coordinates of \(c\) are multiplicatively independent modulo roots of unity by hypothesis. Lemma 3.1 would give \(V=\mathbb G_m^n\), contrary to \(V\subseteq Z_0\). So \(Z_j\) has at least \((S+1)^m>(nD)^n\) irreducible components, while Theorem 5.2 of the counting lesson, applied over \(\overline E\) with \(\delta=nD\), allows at most \((nD)^n\). \(\square\)

5. The interpolation determinant

Proof of Theorem 2.1. Suppose that the logarithms span a subspace of dimension \(r<n\).

The case \(r=0\). Then \(\log u_{1j}=0\) for all \(j\), so every coordinate of \(u_1=u^{e_1}\) is a root of unity, contrary to the hypothesis with \(a=e_1\). Hence \(1\le r<n\).

Normalization. Replacing every \(u_i\) by \(u_i^N\) for a common integer \(N\ge1\) preserves the hypotheses, since \((u^N)^a=u^{Na}\) and \(Na\ne0\), and does not change the span of the logarithms. Choose \(N\) so that every coordinate lies in \(U^{r_0}\), where exponential and logarithm are inverse. The \(\mathcal O_K\)-module generated by \(\log u_1,\ldots,\log u_m\) is finitely generated and torsion-free over the principal ideal domain \(\mathcal O_K\), hence free, of rank \(r\) since it spans an \(r\)-dimensional space; choose a basis \(b_1,\ldots,b_r\in K^n\). A further common power \(\ell^s\) multiplies the logarithms and the basis by \(\ell^s\), so we may assume that \[ \begin{gathered} \rho=\max_{t,j}|b_{t,j}|\quad\text{satisfies}\\ \eta=\rho\,\ell^{1/(\ell-1)}<1 . \end{gathered} \] Then \(\log u_i=\sum_{t=1}^rz_{i,t}b_t\) with \(z_i=(z_{i,1},\ldots,z_{i,r})\in\mathcal O_K^r\). From now on the points, \(E\), \(K\), the basis and \(\eta\) are fixed.

Exponential functions. For \(\alpha\in\mathbb Z_{\ge0}^n\) and \(z\in\mathcal O_K^r\) put \[ \begin{gathered} f_\alpha(z)=\exp\Bigl(\sum_{t=1}^r\beta_{\alpha,t}z_t\Bigr),\\ \beta_{\alpha,t}=\sum_{j=1}^n\alpha_jb_{t,j}. \end{gathered} \] The argument has absolute value at most \(\rho<\ell^{-1/(\ell-1)}\), so the series converges. Expanding, \(f_\alpha(z)=\sum_{\nu\in\mathbb Z_{\ge0}^r}c_{\alpha,\nu}z^\nu\) with \(c_{\alpha,\nu}=\beta_\alpha^\nu/\nu!\), and Lemma 1.1 gives \[ \begin{gathered} |c_{\alpha,\nu}|\le\eta^{|\nu|},\\ |\nu|=\nu_1+\cdots+\nu_r, \end{gathered}\tag{5.1} \] uniformly in \(\alpha\), since integers have absolute value at most \(1\). For \(a\in\mathbb Z_{\ge0}^m\) put \(z(a)=\sum_ia_iz_i\in\mathcal O_K^r\). Then \(\sum_t\beta_{\alpha,t}z(a)_t=\sum_j\alpha_j\log\bigl((u^a)_j\bigr)\), which lies in \(\mathfrak m_K^{r_0}\), and since \(\exp\) is a homomorphism inverse to \(\log\) there, \[ f_\alpha(z(a))=(u^a)^\alpha . \]

The determinant. Let \(D\ge1\), \(A=(D+1)^n\), and \(S=\lceil(nD)^{n/m}\rceil\), so that \((S+1)^m>(nD)^n\) and \(S\le(nD)^{n/m}+1\). By Proposition 4.1 choose distinct columns \(a^{(1)},\ldots,a^{(A)}\) of \(\mathcal M_{D,S}\) with \[ \Delta_D=\det\bigl((u^{a^{(k)}})^\alpha\bigr)_{\alpha,k}\ne0 . \] It is an algebraic integer of \(E\), since its entries are.

\(\ell\)-adic smallness. Write the rows as \(f_\alpha(z(a^{(k)}))\). Truncate each series at \(|\nu|\le T\). By multilinearity, the truncated determinant is a finite sum over families \((\nu_\alpha)_\alpha\) of terms \(\prod_\alpha c_{\alpha,\nu_\alpha}\cdot\det\bigl(z(a^{(k)})^{\nu_\alpha}\bigr)_{\alpha,k}\). A family with \(\nu_\alpha=\nu_{\alpha'}\) for some \(\alpha\ne\alpha'\) gives a determinant with two equal rows. For a family of \(A\) distinct multi-indices, the second factor has absolute value at most \(1\), and by (5.1) the first is at most \(\eta^{\sum_\alpha|\nu_\alpha|}\). At most \(\binom{T'+r}r\le(T'+1)^r\) multi-indices have \(|\nu|\le T'\). With \(T'+1=\lfloor(A/2)^{1/r}\rfloor\), at least \(A/2\) of the \(\nu_\alpha\) have \(|\nu_\alpha|\ge\lfloor(A/2)^{1/r}\rfloor\), so for \(A\) large \[ \begin{aligned} \sum_\alpha|\nu_\alpha|&\ge\tfrac A2\bigl((A/2)^{1/r}-1\bigr)\\ &\ge c_rA^{1+1/r},\qquad c_r=2^{-2-1/r}. \end{aligned} \] By the ultrametric inequality the truncated determinant has absolute value at most \(\eta^{c_rA^{1+1/r}}\). As \(T\to\infty\) the truncated series converge to the entries, and the determinant is continuous, so \[ \begin{gathered} \log|\Delta_D|\le-c\,A^{1+1/r},\\ c=c_r\log(1/\eta)>0, \end{gathered}\tag{5.2} \] for all large \(D\). Here \(\log|\cdot|\) is the real logarithm of the absolute value of \(K\).

Archimedean size. Let \(H\ge1\) bound \(|\tau(u_{ij})|\) for all embeddings \(\tau\colon E\to\mathbb C\) and all \(i,j\). An entry \((u^a)^\alpha\) is a product of at most \(\sum_ia_i\sum_j\alpha_j\le m\,nS\cdot nD\) coordinates, so \(|\tau((u^a)^\alpha)|\le H^{mn^2DS}\). Expanding the determinant into \(A!\le A^A\) products, \[ \begin{aligned} \log|\tau(\Delta_D)|&\le C'ADS\\ &\quad+A\log A,\\ C'&=mn^2\log H, \end{aligned}\tag{5.3} \] with \(C'\) independent of \(D\).

The product formula. Let \(\lambda\) be the place of \(E\) induced by \(E\subset K\). By the product formula (Theorem 5.2 of Places of number fields in extensions and the product formula), written with the absolute values extending the usual ones on \(\mathbb Q\) and with each complex place counted through its two embeddings, \[ \begin{aligned} 0={}&\sum_{\tau\colon E\to\mathbb C}\log|\tau(\Delta_D)|\\ &+\sum_{v\ \mathrm{finite}}[E_v:\mathbb Q_{p_v}]\log|\Delta_D|_v, \end{aligned} \] where \(p_v\) is the rational prime below \(v\). Every finite term is at most \(0\) because \(\Delta_D\) is an algebraic integer, and the term at \(\lambda\) is \([E_\lambda:\mathbb Q_\ell]\log|\Delta_D|\). With (5.2) and (5.3), \[ \begin{gathered} c\,[E_\lambda:\mathbb Q_\ell]\,A^{1+1/r}\\ \le[E:\mathbb Q]\bigl(C'ADS+A\log A\bigr). \end{gathered} \] As \(D\to\infty\), the left side has order \(D^{n+n/r}\) and the right side is \(O(D^{n+1+n/m}+D^n\log D)\). Since \(r\le n-1\) and \(m>n(n-1)\), \[ \frac nr\ge\frac n{n-1}=1+\frac1{n-1}>1+\frac nm , \] so the inequality fails for large \(D\). This contradiction proves \(r=n\). \(\square\)

6. Remarks

The proof uses the hypothesis on \(u^a\) only through Lemma 3.1 in the zero estimate, and in the case \(r=0\). The exponent \(n\) in the component bound \((nD)^n\) produces the threshold \(m>n(n-1)\): the zero estimate needs \(S\) of order \(D^{n/m}\), and the comparison of \(D^{n+n/r}\) with \(D^{n+1+n/m}\) is decided by \(n/(n-1)\) against \(1+n/m\). Theorem 2.1 is a special case of the \(\ell\)-adic rank theorem of Waldschmidt and Masser for matrices of logarithms of algebraic numbers ([Dasgupta 2023, Section 5]); the threshold \(m>n(n-1)\) is exactly the condition that its rank bound \(mn/(m+n)\) exceed \(n-1\).

7. Exercises

7.1. Let \(n=2\) and \(u_i=(x_i,x_i)\) with \(x_i\in E\) units of \(K\) that are not roots of unity. Show that the logarithms span a line, and identify the hypothesis of Theorem 2.1 that fails.

7.2. Prove Theorem 2.1 for \(n=1\) directly.

7.3. Show that the minimum of \(\sum_{k=1}^A|\nu_k|\) over \(A\) distinct \(\nu_k\in\mathbb Z_{\ge0}\) is \(A(A-1)/2\), and compare with the bound \(c_1A^2\) of Section 5.

7.4. Let \(m=3\), \(n=2\). Show that \(m>n(n-1)\), and state the conclusion of Theorem 2.1 as a statement about the rank of a \(3\times2\) matrix of \(\ell\)-adic logarithms.

7.5. Let \(\ell\) be odd, \(k\ge1\), and \(u\in1+\ell^k\mathbb Z_\ell\) with \(u\notin1+\ell^{k+1}\mathbb Z_\ell\). Show that \(v_\ell(\log u)=k\), and deduce that replacing \(u\) by \(u^{\ell^s}\) multiplies \(|\log u|\) by \(\ell^{-s}\).

8. Solutions

7.1. \(\log u_i=(\log x_i,\log x_i)\) lies on the diagonal line. The coordinates of \(u^a\) are equal, so they are not multiplicatively independent: \(c_1c_2^{-1}=1\).

7.2. For \(n=1\) the condition \(m>0\) always holds. The span is \(K\) unless every \(\log u_{i1}=0\), that is, unless every \(u_{i1}\) is a root of unity; the hypothesis with \(a=e_1\) excludes this for \(u_{11}\).

7.3. The smallest possible values are \(0,1,\ldots,A-1\), with sum \(A(A-1)/2\). The bound of Section 5 with \(r=1\) is \(c_1A^2=A^2/8\), smaller by a constant factor, as it must be since it is derived from a cruder count.

7.4. \(3>2\). The \(3\times2\) matrix with rows \(\log u_1,\log u_2,\log u_3\) has rank \(2\): its rows span \(K^2\).

7.5. Write \(u=1+x\) with \(v_\ell(x)=k\). For \(j\ge2\), \(v_\ell(x^j/j)=jk-v_\ell(j)\), and \(v_\ell(j)\le\log_3j<j-1\le(j-1)k\), so \(v_\ell(x^j/j)>k\). The first term of \(\log(1+x)=x-x^2/2+\cdots\) therefore determines the valuation: \(v_\ell(\log u)=k\). Since \(\log\) is a homomorphism, \(\log u^{\ell^s}=\ell^s\log u\), whose absolute value is \(\ell^{-s}|\log u|\).

References