Arithmetical Geometry, Clock Arithmetic, and Zeta Functions
Counting Solutions to Polynomial Equations
A collaboration with Raemeon A. Cowan
and Lauren M. White, undergraduates at
California State University, Northridge,
who have now graduated
Daniel J. Katz
California State University, Northridge
You can try a different modulus, say modulo 7
Arithmetical Geometry
5 + 4 = 2 (mod 7)
Finding whole number solutions to polynomial equations
(5 + 4 = 9, but 9 is too big, so use 9 − 7 = 2)
Example: Pythagorean Theorem
z
(2 + 5 = 7, and we actually prefer to use 7 − 7 = 0 rather than 7)
x
Whole Number (Arithmetical) Solutions
3 × 5 = 1 (mod 7)
means we only accept solutions where
x, y, and z are all whole
so z =
If x = 1 and y = 2,
then z 2 = 5,
so z =
If x = 2 and y = 3,
then z 2 = 13,
so z =
√
√
√
y
2 = 1.414 . . . not a whole number solution
arise from (are “lifted from”) each solution modulo 3
5 = 2.236 . . . not a whole number solution
Igusa’s Zeta Function
Most solutions (smooth solutions) lift predictably
13 = 3.605 . . . not a whole number solution
The Igusa zeta function is an object that counts solutions to equations in clock arithmetic
but some solutions (non-smooth solutions) lift
If x = 3 and y = 4, then z 2 = 25, so z = 5
a whole number solution
If x = 5 and y = 12, then z 2 = 169, so z = 13
a whole number solution
Named after Jun-ichi Igusa (1924–2013)
For example: 2x2 + 2y 2 = z 2 modulo 3
Most x and y you try will make z irrational—not a whole number
If x = 2, y = 0, z = 2, then 2x2 + 2y 2 = 8 = 2 (mod 3), but z 2 = 4 = 1 (mod 3): not a solution
But there are more whole number solutions!
If x = 2, y = 1, z = 1, then 2x2 + 2y 2 = 10 = 1 (mod 3), and z 2 = 1 (mod 3): this is a solution
(x = 65, y = 72, z = 97)
Hensel lifting, named after Kurt Hensel (1861–1941),
tracks how many solutions modulo 32, 33, 34, . . .
(3 × 5 = 15, but 15 is too big, so try 15 − 7 = 8, but 8 is too big, so use 8 − 7 = 1)
If x = 1 and y = 1,
turned out to be unproblematic
2 + 5 = 0 (mod 7)
Polynomial Equation: x2 + y 2 = z 2
then z 2 = 2,
Coordinate Transforms and generating functions
(x = 799, y = 960, z = 1249)
(x = 12709, y = 13500, z = 18541)
These big examples were taken from tablet Plimpton 322 (Larsa, Sumer, ca. 1800 BC)
unpredictably: some have too many lifts and
others have no lifts at all
The following figure illustrates Hensel lifting:
In total, there are nine solutions modulo 3:
(x, y, z) = (0, 0, 0), (1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2), (2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2)
• smooth solutions modulo 3 lift predictably: each gives rise to three new solutions modulo
32 = 9, and each of these gives rise to three new solutions modulo 33 = 27, and so forth
What about another modulus, say modulo 9?
• non-smooth solutions modulo 3 lift unpredictably: some gives rise to many solutions
modulo 32 = 9, and some to none at all
Any solution modulo 9 will always work modulo 3
• non-solutions never lift at all (and so cause no difficulty)
(you can always subtract nine by subtracting three thrice)
mod 33 = 27
Hensel
lift
mod 32 = 9
Hensel
lift
But only some solutions modulo 3 work modulo 9:
(x, y, z) = (2, 1, 1) is a solution to 2x2 + 2y 2 = z 2 modulo 3, and also modulo 9
(2x2 + 2y 2 = 10 = 1 (mod 9) and z 2 = 1 (mod 9))
(x, y, z) = (2, 1, 2) is a solution to 2x2 + 2y 2 = z 2 modulo 3, but not modulo 9
(2x2 + 2y 2 = 10 = 1 (mod 9) and z 2 = 4 (mod 9))
reduce
mod 32
reduce
mod 3
mod 3
smooth
An Igusa zeta function records number of solutions in infinitely many modular systems
For example, one Igusa zeta function records number of solutions modulo 3, modulo 32 = 9,
modulo 33 = 27, modulo 34 = 81, and so on ad infinitum
The oldest documented mathematics!
Clock Arithmetic
Project Objectives
An easier system of arithmetic that helps with the original problem
Add numbers as you do with a clock
• To develop new techniques to make it easier to calculate Igusa zeta functions
• To calculate zeta functions that Igusa already calculated, and then to check if our results
agree with his
• To employ these new techniques to calculate the Igusa zeta functions for important new
classes of equations
not
non-smooth
solutions
A Breakthrough
• We found a way to mathematically stitch together infinitely many generating functions into
a new object that we call a p-adic generating function
• This new p-adic generating function largely circumvents the counting difficulties inherent
in Hensel lifting!
Results
• We discovered a new way of calculating Igusa zeta functions
10
+
5
=
3
(10 + 5 = 15, but 15 is too big, so use 15 − 12 = 3)
Arithmetic modulo 12: keep subtracting 12 (“reduce modulo 12”) to keep the number in range
Methods
• This makes it easy to recapitulate existing calculations by Igusa
Our approach combines three ingredients:
• It allowed us to prove new theoretical results about general properties of the zeta functions
1. Use coordinate transforms to convert our equations into simpler forms
• We wrote up our findings in a research paper, which is publicly available online
2. Use generating functions to count solutions in the smallest cases (for example, modulo 3)
• We submitted our paper to a peer-reviewed journal
3. Use Hensel lifting to count solutions in larger cases (for example, modulo 32 = 9, 33 = 27,
34 = 81, and so forth)
• The undergraduate participants (Raemeon Cowan and Lauren White) had their first experience with mathematical research
• It enabled us to calculate hitherto unknown zeta functions