redlib, redlib.py, redlib.js/.ts are Magma, Python and JavaScript packages (current version 3.1, 2024) with a collection of routines for working with

The Magma package works with plane or hyperelliptic curves, occassionally converting between the two types, and is based on

Tim Dokchitser, Classification of reduction types of curves, in preparation (Reduction types).

The Python and JavaScript libraries only implements the combinatorial aspects of reduction types, based on the third paper.

→ Download redlib as a zip archive ←

Included files:

manual.pdf	-	Package documentation
redlib.spec	-	Magma library specification, loaded with AttachSpec("redlib.spec");
redtype.py	-	Python library
redtype.js	-	JavaScript library for web browsers
redtype.ts	-	Same JavaScript library with TypeScript exports for standalone use
ex-redlib.m	-	Magma example of using the library
ex-redlib.py	-	Python example of using the library
ex-redlib.js	-	JavaScript example of using the library
*.m	-	all other .m files are individual packages for the Magma library
redlib.sty	-	LaTeX style file for auto-typesetting of reduction types
ex-redlib.tex	-	example file showing a reduction type (uses pbox, tikz and redlib.sty)
ex-redlib.pdf	-	compiled to pdf
data/*.txt	-	data files for shapes of reduction types in small genus

Classification of reduction types

As an example, here are all possible reduction types in genus 1,2 and 3.

Genus 1

For elliptic curves, there are 10 possible (families of) reduction types, and they are referred to by their standard Kodaira type names. Two of them (In and In*) are infinite families, with n>0 arbitrary.

1_g1

I₁

I^⁎₀

I^⁎₁

IV^⁎

III

III^⁎

II^⁎

Enter reduction type:

Genus: Label: Tamagawa number:

General genus 1 curves (possibly without no rational point to make them into an elliptic curve) have reduction types [d]K where K is one of the Kodaira types above, and d = 1 any multiple. For example, [3]I2*, [5]g1, [7]II etc.

Genus 2

Reduction types of genus 2 curves come in 104 families, and they have been classified by Namikawa and Ueno. Here is how to construct all 104=55+10+8+6+16+9 types by labels:

1. K1-K2 (55 types)

There are 55 types of the form K1-K2 where K1, K2 are any of the 10 Kodaira types. These are degenerations of two elliptic curves meeting at a point. For example,

1_g1₃1_g1

IV–IV^⁎

III–II

1_g1–1_g1

1_g1–I₁

1_g1–I^⁎₀

1_g1–I^⁎₁

1_g1–IV

1_g1–IV^⁎

1_g1–III

1_g1–III^⁎

1_g1–II

1_g1–II^⁎

I₁–I₁

I₁–I^⁎₀

I₁–I^⁎₁

I₁–IV

I₁–IV^⁎

I₁–III

I₁–III^⁎

I₁–II

I₁–II^⁎

I^⁎₀–I^⁎₀

I^⁎₀–I^⁎₁

I^⁎₀–IV

I^⁎₀–IV^⁎

I^⁎₀–III

I^⁎₀–III^⁎

I^⁎₀–II

I^⁎₀–II^⁎

I^⁎₁–I^⁎₁

I^⁎₁–IV

I^⁎₁–IV^⁎

I^⁎₁–III

I^⁎₁–III^⁎

I^⁎₁–II

I^⁎₁–II^⁎

IV–IV

IV–IV^⁎

IV–III

IV–III^⁎

IV–II

IV–II^⁎

IV^⁎–IV^⁎

IV^⁎–III

IV^⁎–III^⁎

IV^⁎–II

IV^⁎–II^⁎

III–III

III–III^⁎

III–II

III–II^⁎

III^⁎–III^⁎

III^⁎–II

III^⁎–II^⁎

II–II

II–II^⁎

II^⁎–II^⁎

Enter reduction type:

Genus: Label: Tamagawa number:

2. [2]K_D (10 types)

There are 10 types of the form [2]K_D where K is one of the 10 Kodaira types. There is a unique way to attach a D-tail (like in In*) in a minimal way to [2]K in every case. For example, for K=IV*, 1_g1, I_n:

[2]_g1,D

[2]_6,4D

[2]IV^⁎_D

[2]_g1,D

[2]_1,D

[2]I^⁎_0,D

[2]I^⁎_1,D

[2]IV_D

[2]IV^⁎_D

[2]III_D

[2]III^⁎_D

[2]II_D

[2]II^⁎_D

Enter reduction type:

Genus: Label: Tamagawa number:

3. K_n (8 types)

There are 8 types K_n obtained by adding a loop of gcd 1 to every Kodaira type except II, II*. For II, II* all the outgoing open chains have different initial multiplicities, so this is not possible, but it is possible for all the others, again in a unique minimal way. For example, 1_g1,n, IV_n, IV*_n, etc.

I_g1,5

IV₁

IV^⁎_-1

I_g1,1

I_1,1

I^⁎_0,0

I^⁎_1,0

IV₀

IV^⁎_-1

III₀

III^⁎_-1

Enter reduction type:

Genus: Label: Tamagawa number:

4. K_D (6 types)

There are 6 types K_D obtained by adding a D-tail to a Kodaira type whose principal component has even multiplicity, namely I0*, I1*, III, III*, II, II*. For example,

I^⁎_0,D

III_D

II_D

I^⁎_0,D

I^⁎_1,D

III_D

III^⁎_D

II_D

II^⁎_D

Enter reduction type:

Genus: Label: Tamagawa number:

5. Cores with χ=-2 (16 types)

There are 16 types from cores with χ = -2, consisting of one principal component of genus 0 and multiplicity m, and open chains with initial multiplicities d₁,...,d_k ∈ ℤ/mℤ and ∑ d_i = 0.

2^1,1,1,1,1,1

6^2,4,3,3

10^3,2,5

2^1,1,1,1,1,1

3^1,1,2,2

4^1,3,2,2

5^1,1,3

5^1,2,2

5^2,4,4

5^3,3,4

6^1,1,4

6^5,5,2

6^2,4,3,3

8^1,3,4

8^5,7,4

10^1,4,5

10^3,2,5

10^7,8,5

10^9,6,5

Enter reduction type:

Genus: Label: Tamagawa number:

6. Remaining ones from cores 1, D, T and 4^1,3 (9 types)

There are 9 remaining types derived from cores 1, D=2^1,1, T=3^1,2, 4^1,3 that have χ=2: 1g2 (good reduction), 1---1, [2]_D,D,D (core(s) 1), Dg1, D_{2-2}, D-=D (core(s) D), T=T, [2]T_{6}D (core(s) T), and 4^1,3_D (core 4^1,3).

1_g2

D_g1

D_{2-2}

[2]_D,D,D

4^1,3_D

[2]T_{6}D

T=T

D– =D

1– – –1

1_g2

D_g1

D_{2-2}

[2]_D,D,D

4^1,3_D

[2]T_{6}D

T=T

D– =D

1– – –1

Enter reduction type:

Genus: Label: Tamagawa number:

Genus 3

Reduction types of genus 3 curves come in 1901 families, and this is how we can classify them. Recall, for example that there are 55 genus 2 families of type K1-K2, two Kodaira types with a chain between them of gcd 1. Equivalently, such a type is glued together from two principal types with χ=-1 and one loose chain of gcd 1. There are 10 choices for these principal types,

1¹₍₁₀₎: 1_g1– I₁– I^⁎₀– I^⁎₁– IV– IV^⁎– III– III^⁎– II– II^⁎–. and we write 1¹₍₁₀₎ for this collection. The superscript 1 stands for a sequence of gcds of loose chains, first '1' for -χ, and the subscript (10) for the number of such types. Now we say that the 55 types of the form K1-K2 have the following shape: Shape is a graph whose components are collections of all principal types with given χ and gcd's of loose links (like 1¹₍₁₀₎ above), and edges merge loose links into edges.

Similarly, the 8 principal types with χ=-2 and two loose chains of gcd 1 form the set

2^1,1₍₈₎: 1_g1– – I₁– – I^⁎₀– – I^⁎₁– – IV– – IV^⁎– – III– – III^⁎– – II– – II^⁎– –. and in genus 3 one of the possible shape configuration is representing all possible degenerations of 3 elliptic curves forming a chain 1g1-1g1-1g1. The two outside leaves are in 1¹₍₁₀₎ and the inside one in 2^1,1₍₈₎. For example, here are three such types:

1_g1–1_g1–1_g1

I₁₂IV–II

IV–I^⁎₀–II

The total number of types of this shape is Binomial(10,2)*8 = 55*8 = 420.

Another possible shape in genus 3 is

with total number of types Binomial(10,3) = 10*11*12/5 = 220.

All in all, in genus 3 there are 35 possible shapes, and here they are with a breakdown of possible number of reduction type families in each shape:

Here are the simplest reduction types of each shape (with components of largest possible genus):

1_g1–1_g1–1_g1 (440 types)

1_g2–1_g1 (390 types)

1_g1–1–1_g1&1_g1–c₂ (220 types)

[2]_g1–D–1_g1 (180 types)

[2]_g1–[2]_g1 (171 types)

1_g3 (150 types)

1_g1– –1–1_g1 (80 types)

1_g1–1– –1–1_g1 (55 types)

1_g1–D=D–1_g1 (55 types)

1_g1– –1_g1 (36 types)

1_g1^1,2T=T (20 types)

[3]_g1–T (15 types)

1_g1–1–1– –1–c₂ (10 types)

1_g1–D– =D (10 types)

1_g1–1– – –1 (10 types)

1_g1–1–D=D–c₂ (10 types)

1_g1–1– –1–c₁ (8 types)

1_g1–D=D–c₁ (8 types)

I^⁎₀– =D (6 types)

1_g1– – –1 (5 types)

T^2,2=T (3 types)

D= =D (3 types)

4^1,3=4^1,3 (3 types)

D=D–D=c₁ (2 types)

D– – =D

[2]T=[2]T

1– – – –1

[2]– – –[2]

D– –1–D=c₁

1– –1–1– –c₁

[2]D– =[2]D

1–1– –1–1– –c₁

D–1– –1–D=c₁

D–D=D–D=c₁

1–1–1–1–c₁–c₃&c₂–c₄

The buttons below cycle through all the reduction types of a given shape:

1_g1–1_g1–1_g1 (440 types)

1_g2–1_g1 (390)

1_g1–1–1_g1&1_g1–c₂ (220)

[2]_g1–D–1_g1 (180)

[2]_g1–[2]_g1 (171)

1_g3 (150)

1_g1– –1–1_g1 (80)

1_g1–1– –1–1_g1 (55)

1_g1–D=D–1_g1 (55)

1_g1– –1_g1 (36)

1_g1^1,2T=T (20)

[3]_g1–T (15)

1_g1–1–1– –1–c₂ (10)

1_g1–D– =D (10)

1_g1–1– – –1 (10)

1_g1–1–D=D–c₂ (10)

1_g1–1– –1–c₁ (8)

1_g1–D=D–c₁ (8)

I^⁎₀– =D (6)

1_g1– – –1 (5)

T^2,2=T (3)

D= =D (3)

4^1,3=4^1,3 (3)

D=D–D=c₁ (2)

D– – =D

[2]T=[2]T

1– – – –1

[2]– – –[2]

D– –1–D=c₁

1– –1–1– –c₁

[2]D– =[2]D

1–1– –1–1– –c₁

D–1– –1–D=c₁

D–D=D–D=c₁

1–1–1–1–c₁–c₃&c₂–c₄

Enter reduction type:

Genus: Label: Tamagawa number:

Large genus examples

Finally, here are some examples in larger genus:

Depths

Loops

D-tails

Enter reduction type:

Genus: Label: Tamagawa number:

redlib: Reduction types of curves in Magma, Python and JavaScript

by Tim Dokchitser

Classification of reduction types

Genus 1

Genus 2

1. K1-K2 (55 types)

2. [2]K_D (10 types)

3. K_n (8 types)

4. K_D (6 types)

5. Cores with χ=-2 (16 types)

6. Remaining ones from cores 1, D, T and 4^1,3 (9 types)

Genus 3

Large genus examples

redlib: Reduction types of curves in Magma, Python and JavaScript

by Tim Dokchitser

Classification of reduction types

Genus 1

Genus 2

1. K1-K2 (55 types)

2. [2]KD (10 types)

3. Kn (8 types)

4. KD (6 types)

5. Cores with χ=-2 (16 types)

6. Remaining ones from cores 1, D, T and 41,3 (9 types)

Genus 3

Large genus examples

2. [2]K_D (10 types)

3. K_n (8 types)

4. K_D (6 types)

6. Remaining ones from cores 1, D, T and 4^1,3 (9 types)