The p-adic logarithm converts multiplicative local arithmetic into additive arithmetic. On units sufficiently close to one, multiplication becomes addition under log, exponentiation becomes scalar multiplication, and local unit groups begin to look like p-adic vector spaces.
This is not only a local analytic convenience. p-adic logarithms appear in unit computations, regulators, class-field algorithms, Diophantine reduction, Iwasawa theory and explicit p-adic methods.
This is Guide 43 in the Bukit Timah Tutor Computational Number Theory series. It extends the local-field framework of Guide 39 and the ordinary logarithmic regulator of Guide 19.
1. The formal logarithm
Start from the familiar series
log(1+x)=x−x²/2+x³/3−x⁴/4+...
Over the real numbers this converges for |x|<1.
Over Q_p the correct convergence condition is p-adic:
|x|_p<1
which means
v_p(x)>0.
Thus the natural domain is the group of 1-units.
2. 1-units
Inside Z_p^× define
U_1=1+pZ_p
for odd p.
For u=1+x in U_1, the logarithm series converges p-adically.
The resulting map
log_p:U_1→pZ_p
is a continuous homomorphism:
log_p(uv)=log_p(u)+log_p(v).
3. Worked example: log_5(6)
Since
6=1+5,
we have
log_5(6)=5−5²/2+5³/3−5⁴/4+...
Modulo25, every term after the first is divisible by25, so
log_5(6) ≡5 (mod25).
4. One more digit
Modulo125, the terms starting with 5³/3 vanish modulo125.
Hence
log_5(6) ≡5−25/2 (mod125).
Since 2^−1≡63 mod125,
−25/2 ≡−25·63 ≡50 (mod125).
Therefore
log_5(6) ≡55 (mod125).
This is exact p-adic modular arithmetic, not a decimal approximation.
5. The p-adic exponential
The inverse series is
exp(x)=1+x+x²/2!+x³/3!+...
Its p-adic convergence domain is smaller than all of pZ_p in general, but for odd p it converges on sufficiently p-divisible elements and gives a local inverse to log.
Thus a neighbourhood of 1 in the multiplicative group is analytically isomorphic to a neighbourhood of 0 in the additive group.
6. Why logarithms linearise exponent equations
If
u_1^a1 ... u_r^ar = v
with all terms in the logarithm domain, then
a1 log_p(u_1)+...+ar log_p(u_r)=log_p(v).
A multiplicative relation has become a linear equation over a p-adic field.
This is the local analogue of the real logarithmic embeddings used in unit and regulator computations.
7. Not every unit is a 1-unit
A general p-adic unit u may not satisfy
u≡1 (mod p).
For odd p there is a decomposition
u=ω(u)⟨u⟩
where ω(u) is the Teichmüller representative, a root of unity lifting u mod p, and
⟨u⟩∈1+pZ_p.
8. Teichmüller part
The Teichmüller representative satisfies
ω(u)^(p−1)=1
and
ω(u)≡u (mod p).
Because the p-adic logarithm is a homomorphism into a torsion-free additive group, roots of unity in its natural torsion subgroup map to zero.
Therefore define
log_p(u)=log_p(⟨u⟩).
9. Example: a non-1-unit in Q_5
Take u=2.
Its Teichmüller part is the unique fourth root of unity in Z_5 congruent to2 mod5.
Write
2=ω(2)⟨2⟩.
Then
log_5(2)=log_5(⟨2⟩).
Software usually handles this decomposition automatically, but the mathematical convention should be known because different branches can alter logarithm values outside the unit group.
10. Local unit filtration
For a finite extension L/Q_p with ring of integers O_L and maximal ideal P, define
U_0=O_L^×, U_n=1+P^n for n≥1.
This filtration captures progressively deeper congruence information.
The quotients U_n/U_(n+1) are closely related to the additive residue field.
11. Logarithm on deep units
For n sufficiently large, the p-adic logarithm gives an isomorphism
U_n ≅ P^n
between multiplicative and additive groups.
The threshold depends on p and the ramification of L.
This turns nonlinear multiplicative unit problems into linear algebra in the local field.
12. Valuation of log(1+x)
For x sufficiently p-adically small and away from exceptional low-p behaviour, the first term controls the valuation:
v_p(log(1+x))=v_p(x).
The higher terms are more p-divisible.
This makes the logarithm locally an isometry up to the chosen normalisation.
13. p=2 requires more care
The 2-adic logarithm converges on 1+2Z_2, but the clean isomorphism with the additive group is usually taken on a deeper subgroup such as 1+4Z_2.
Extra torsion and convergence subtleties make the p=2 case less uniform.
Implementations should never copy odd-prime assumptions blindly into Q_2.
14. Global units inside local units
Let K be a number field and choose a prime P above p.
Every global unit ε in O_K^× embeds into K_P^×.
After removing any local torsion/Teichmüller part, log_P(ε) is a p-adic number.
Collecting these logarithms across p-adic embeddings gives p-adic analogues of the real logarithmic unit lattice.
15. Ordinary regulator reminder
Dirichlet’s regulator is the determinant of a matrix of real logarithms
log |σ_i(ε_j)|
after deleting one dependent row.
Its nonzero value measures the covolume of the global unit lattice in the real logarithmic hyperplane.
16. p-adic regulator
A p-adic regulator is built analogously from p-adic logarithms of independent global units under chosen p-adic embeddings.
Schematically, for units ε_j and embeddings τ_i into a p-adic field, form
M_(ij)=log_p(τ_i(ε_j)).
A suitable determinant or minor gives the p-adic regulator.
Normalisation conventions vary, so a computational result should state the exact definition used.
17. Leopoldt’s conjecture
Leopoldt’s conjecture predicts that the p-adic logarithms of global units have the expected rank.
Equivalently, the relevant p-adic regulator should not vanish after the standard dependencies are removed.
The conjecture is known in many cases but not in full generality for arbitrary number fields.
This is a reminder that a regulator computation can touch genuinely open arithmetic territory.
18. Split primes and multiple p-adic embeddings
If p splits into several prime ideals in K, each completion K_P supplies a different p-adic embedding direction.
For example, in Q(√5), the prime11 splits into two degree-one primes.
The two embeddings of the fundamental unit into Q_11 are obtained by sending
ω=(1+√5)/2
to the two 11-adic roots of
x²−x−1.
Modulo11 these begin at4 and8.
19. Hensel lifting supplies the embeddings
The roots4 and8 modulo11 are simple.
Hensel’s lemma lifts each uniquely to a root in Z_11.
These lifted roots define the two embeddings
K→Q_11.
Evaluating global units at those roots gives the p-adic numbers whose logarithms enter the regulator matrix.
20. p-adic logarithms in S-unit equations
Guide40 used real and p-adic logarithmic reduction to shrink exponent bounds.
The mechanism is now visible: an S-unit identity produces a p-adically small multiplicative expression; log converts it into a small linear form in p-adic logarithms.
Explicit lower bounds and lattice reduction then constrain the integer exponents.
21. p-adic logarithm precision
When computing
log_p(1+x)
modulo p^N, stop when every omitted term has valuation at least N.
Because denominators n can contain powers of p, the valuation of x^n/n is
n v_p(x)−v_p(n).
Term truncation must account for v_p(n); degree alone is not enough.
22. Precision loss in determinants
A p-adic regulator is a determinant.
If rows or columns are nearly p-adically dependent, the determinant may have high valuation and lose many digits of relative precision.
Therefore input logarithms need more precision than the desired final regulator in ill-conditioned cases.
The determinant valuation itself is arithmetic information, not merely numerical noise.
23. Branch choices for nonunits
Extending log from units to all of Q_p^× requires a convention for log_p(p).
A common Iwasawa branch sets
log_p(p)=0.
More general branches may assign another value.
Software interfaces can expose branch parameters; reproducible computations must record them.
24. Software behaviour
SageMath’s p-adic logarithm documentation explicitly notes that the power series converges on 1-units and extends to all units by separating the Teichmüller root-of-unity part.
Its p-adic rings track finite precision, so outputs include terms such as
O(p^N).
This precision marker is part of the value.
25. Verification by exponentiation
For y=log_p(u) in a domain where exp and log are inverse, verify
exp_p(y)=u
to the available precision.
For a multiplicative relation, compare
log_p(uv)
with
log_p(u)+log_p(v).
Independent precision paths are useful for catching truncation mistakes.
26. Verification receipt for a regulator
Retain:
number field and maximal order prime p and primes P above p fundamental global units chosen p-adic embeddings branch convention input precision logarithm matrix determinant/minor definition output precision and valuation.
This makes the normalisation and numerical reliability auditable.
27. Common mistakes
1. Applying the power series to an arbitrary unit without removing its torsion part. 2. Treating p-adic convergence as real convergence. 3. Ignoring the special p=2 behaviour. 4. Dropping O(p^N) precision annotations.
5. Forgetting denominator valuations in series truncation. 6. Calling any determinant of p-adic logs “the regulator” without stating the normalisation. 7. Assuming nonvanishing in all number fields without acknowledging Leopoldt. 8. Mixing branches for log_p(p).
28. Practice set
1. State the p-adic log series. 2. Where does it converge naturally? 3. Compute log_5(6) mod25. 4. Compute it mod125.
5. What is a 1-unit? 6. What is Teichmüller decomposition? 7. Why does the torsion part have logarithm zero? 8. Define U_n.
9. What does a p-adic regulator measure? 10. What does Leopoldt predict? 11. Why can determinant precision drop? 12. What must be recorded about logarithm branches?
29. Answers
1. log(1+x)=Σ_(n≥1)(−1)^(n+1)x^n/n. 2. On p-adically small x, especially 1+pZ_p for odd p. 3. 5 mod25. 4. 55 mod125.
5. A unit congruent to1 modulo the maximal ideal. 6. u=ω(u)⟨u⟩ with root-of-unity part and 1-unit part. 7. A homomorphism from finite torsion to a torsion-free additive group kills torsion. 8. U_0=O_L^× and U_n=1+P^n.
9. p-adic linear independence/covolume data of global units under p-adic logs. 10. The expected rank of p-adic logarithms of global units. 11. Near p-adic dependence makes the determinant highly divisible by p. 12. The convention, especially the value assigned to log_p(p) when extending beyond units.
Sources and further study
SageMath’s p-adics reference documents the p-adic logarithm, its convergence on 1-units and its extension to all units through Teichmüller decomposition. Standard local-number-theory texts develop unit filtrations, local logarithms and p-adic regulator theory.
Continue through Batch 11
Return to Guide 42. Continue to Guide 44: PSLQ, Algebraic Number Recognition and Minimal-Polynomial Reconstruction.