Study on algorithms in computational algebraic geometryand their applications to finding special algebraic curves

Momonari Kudo

Institute of Mathematics for Industry, Kyushu University

7th December 2017 @Pusan National University

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Contents:

1. Introduction

2. General Introduction of Computational Algebraic Geometry

3. Computing the Frobenius on cohomology groups (Main result 1)

4. Application to finding superspecial curves (Main result 2)

5. Concluding remarks

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Motivation of this study In algebraic geometry, it is important to find varieties with special properties Special varieties are used in various areas (e.g., maximal curves in coding theory)

Such varieties are determined by invariants

Invariants can be computed from sheaf cohomology groups

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This talkA. Introduction of computational algebraic geometry for researchers in other areasB. Introduce our results related to the above topics 1 and 2

2. Finding varieties with special properties

Moduli of curves (future work) Arithmetic Geometry

Cryptography

Coding Theory

Related studiesMy researches in (Computational) Algebraic Geometry

1. Computing sheaf cohomologyand related invariants

Contents:

1. Introduction

2. General Introduction of Computational Algebraic Geometry

3. Computing the Frobenius on cohomology groups (Main result 1)

4. Application to finding superspecial curves (Main result 2)

5. Concluding remarks

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Background Computational Algebraic Geometry (CAG) Studies computational approaches in algebraic geometry

Algebraic geometry studies (algebraic) varieties

Aims to develop and implement algorithms for computing

1. Gröbner bases

2. Zeros of multivariate systems (Points of varieties)

3. Invariants of varieties, etc.

Implemented in mathematical software (e.g. MAGMA, SINGULAR, MAPLE) Applied to other areas; Number Theory, Cryptography, Coding Theory, etc.

Efficient implementations of algorithms in CAG are important

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Fig. 𝑦2 − 𝑥3 = 0

𝑥

𝑦

1 2

2

1

−1

−1

−2

O

Varieties Definition of (affine and classical) varieties For a set of polynomials 𝐹 ⊂ 𝐾[𝑥1, … , 𝑥𝑛], the variety 𝑉(𝐹) is defined by

𝑉 𝐹 : = 𝑎1, … , 𝑎𝑛 ∈ 𝐾𝑛 ∶ 𝑓 𝑎1, … , 𝑎𝑛 = 0 for all 𝑓 ∈ 𝐹

where 𝐾 is a field

Algebraic geometry aims to understand varieties Points, Dimension, Singularities, Invariants, etc.

Importance of the ideal corresponding to a variety There are close relationships between 𝑉(𝐹) and the ideal 𝐹 generated by 𝐹

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Varieties and Ideals Definition of ideals in a polynomial ring For a set of polynomials 𝐹 = {𝑓1, … , 𝑓𝑡} ⊂ 𝐾[𝑥1, … , 𝑥𝑛],

𝐹 ≔ ℎ1𝑓1 + ⋯ + ℎ𝑡𝑓𝑡 ∶ ℎ1, … , ℎ𝑡 ∈ 𝐾[𝑥1, … , 𝑥𝑛]

is called the ideal generated by 𝐹

𝐹 is called a basis of the ideal ⟨𝐹⟩

Note 𝑉 ⟨𝐹⟩ = 𝑉(𝐹)

What is a Gröbner basis? “Good” basis of an ideal 𝐼 ⊂ 𝐾[𝑥1, … , 𝑥𝑛]

Defined by a term order of monomials

Enables to compute various objects related to ideals and varieties

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Term orders and examples Definition of a term order of monomials in 𝐾 𝑥 = 𝐾[𝑥1, … , 𝑥𝑛]

A term order is a total order ≺ on the set of monomials 𝑥𝛼 = 𝑥1𝛼1 ⋯ 𝑥𝑛

𝛼𝑛 s.t.

1. 𝑥𝛼 ≺ 𝑥𝛽 ⟹ 𝑥𝛼+𝛾 ≺ 𝑥𝛽+𝛾 for all 𝛾 ∈ ℤ≥0𝑛

2. 1 ≺ 𝑥𝛼 for all 𝛼 ≠ (0, … , 0)

Examples of term orders Lexicographical order ≻𝑙𝑒𝑥 with 𝑥1 ≻ 𝑥2 ≻ ⋯ ≻ 𝑥𝑛

• 𝑥𝛼 ≻𝑙𝑒𝑥 𝑥𝛽 ⟺ ∃𝑖 s.t. 𝛼1 = 𝛽1, … , 𝛼𝑖−1 = 𝛽𝑖−1, 𝛼𝑖 > 𝛽𝑖

Graded reverse lexicographic order ≻𝑔𝑟𝑒𝑣𝑙𝑒𝑥 with 𝑥1 ≻ 𝑥2 ≻ ⋯ ≻ 𝑥𝑛

• 𝑥𝛼 ≻𝑔𝑟𝑒𝑣𝑙𝑒𝑥 𝑥𝛽 ⟺ 𝛼 > 𝛽 , or 𝛼 = 𝛽 かつ ∃𝑖 s.t. 𝛼𝑛 = 𝛽𝑛, … , 𝛼𝑖+1 = 𝛽𝑖+1, 𝛼𝑖 < 𝛽𝑖

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Leading monomials Leading monomials, terms, coefficients, multi-degree w.r.t. a term order ≻

For a polynomial 𝑓 = 𝛼∈ ℤ≥0𝑛 𝑐𝛼 𝑥𝛼 ∈ 𝐾 𝑥 ∖ {0} , we define

• mltdeg≻ 𝑓 ≔ max {𝛼 ∈ ℤ≥0𝑛: 𝑐𝛼 ≠ 0} Multi-degree

• LM≻ 𝑓 ≔ 𝑥mltdeg(𝑓) Leading Monomial

• LC≻ 𝑓 ≔ 𝑐mltdeg(𝑓) Leading Coefficients

• LT≻ 𝑓 ≔ LC 𝑓 ⋅ LM 𝑓 Leading Term

Examples

For ≻: =≻𝑙𝑒𝑥 with 𝑥 ≻ 𝑦 and 𝑓 = 𝑥2𝑦2 − 4𝑥5𝑦2 + 3𝑦3 − 𝑥 + 5 ∈ ℚ[𝑥, 𝑦],

• we have multdeg≻ 𝑓 = (5,2), LM≻ 𝑓 = 𝑥5𝑦2, LC≻ 𝑓 = −4, LT≻ 𝑓 = −4𝑥5𝑦2

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Gröbner bases Definition of a Gröbner basis for an ideal 𝐼 ⊂ 𝐾 𝑥 A finite subset 𝐺 ⊂ 𝐼 ∖ {0} is called a Gröbner basis (G.B.) for 𝐼 w.r.t. ≻ iff

LT≻ 𝑓 ∶ 𝑓 ∈ 𝐼 ∖ {0} = LT≻ 𝑔 ∶ 𝑔 ∈ 𝐺i.e. ∀𝑓 ∈ 𝐼 ∖ {0}, ∃𝑔 ∈ 𝐺 such that LT≻(𝑓) is divisible by LT≻(𝑔)

Example Case of 𝑛 = 3 and ℚ[𝑥, 𝑦, 𝑧]: we computationally obtain a G.B. w.r.t. ≻𝑙𝑒𝑥

• 𝐺 = {𝑥 + 2𝑧3 − 3𝑧, 𝑦2 − 𝑧2 − 1, 𝑧4 −3

2𝑧2 +

1

2}

How computed?Multivariate division Buchberger’s criterion

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Computing Gröbner bases (1/2) Division of a multivariate polynomial w.r.t. ≻ and 𝐺 = {𝑔1, … , 𝑔𝑠} ⊂ 𝐾 𝑥

For any 𝑓 ∈ 𝐾 𝑥 ∖ {0}, there exists an algorithm for computing ℎ1, … , ℎ𝑠 and 𝑟 s.t.

• 𝑓 = ℎ1𝑔1 + ⋯ + ℎ𝑠𝑔𝑠 + 𝑟

• No term of 𝑟 is divisible by LT≻(𝑔𝑖) for any 1 ≤ 𝑖 ≤ 𝑠

𝑟 is called the normal form of 𝑓 w.r.t. 𝐺 and ≻, and denoted by NF≻,𝐺(𝑓)

Remark: 𝑓 ∈ 𝑔1, … , 𝑔𝑠 ⟺ NF≻,𝐺 𝑓 = 𝑟 ∈ 𝑔1, … , 𝑔𝑠

𝑆-polynomials w.r.t. a term order ≻

For 𝑓, 𝑔 ∈ 𝐾 𝑥 ∖ {0}, their 𝑆-polynomial is defined by

𝑆≻ 𝑓, 𝑔 ≔LCM LM≻ 𝑓 , LM≻ 𝑔

LT≻ 𝑓𝑓 −

LCM LM≻ 𝑓 , LM≻ 𝑔

LT≻ 𝑔𝑔

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Computing Gröbner bases (2/2) Buchberger’s criterion [Buc]

Theorem. 𝐺 ⊂ 𝐾 𝑥 ∖ {0} is a Gröbner basis for 𝐼 = 𝐺 w.r.t. ≻

⟺ NF≻,𝐺(𝑆≻ 𝑔, 𝑔′ ) = 0 for ∀𝑔, 𝑔′ ∈ 𝐺 with 𝑔 ≠ 𝑔′

Buchberger’s algorithm [Buc]

Input: a finite subset 𝐺 ⊂ 𝐾 𝑥 Output: a G.B. of 𝐺 w.r.t. ≻

Step 1. Test whether 𝐺 is G.B. or not by Buchberger’s criterion

Step 2. If 𝐺 is G.B., output 𝐺. Otherwise go to Step 2.

Step 3. Choose a pair {𝑔, 𝑔′} of elements in 𝐺 with 𝑟 ≔ NF≻,𝐺 𝑆≻ 𝑔, 𝑔′ ≠ 0

Step 4. Update 𝐺 by replacing 𝐺 by 𝐺 ∪ {𝑟}, and go back to Step 1

Note: This algorithm is guaranteed to be terminated by “Hilbert’s Basis Theorem”

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[Buc] B. Buchberger, “An Algorithm for Finding the Basis Elements of the Residue Class Ring of a Zero DimensionalPolynomial Ideal”, Ph. D. dissertation, University of Innsbruck, 1965.

An application: Elimination Elimination Theorem

Theorem. ≻𝑙𝑒𝑥: the lex order with 𝑥1 ≻ ⋯ ≻ 𝑥𝑛

𝐺: Gröbner basis for an ideal 𝐼 ⊂ 𝐾[𝑥1, … , 𝑥𝑛]

Then, for 1 ≤ ℓ ≤ 𝑛,

𝐺 ∩ 𝐾[𝑥ℓ, … , 𝑥𝑛] is a G.B. for 𝐼 ∩ 𝐾[𝑥ℓ, … , 𝑥𝑛]

where the order is the induced order from ≻𝑙𝑒𝑥 with 𝑥ℓ ≻ ⋯ ≻ 𝑥𝑛

Solving multivariate systems by using Elimination Theorem Once we compute a G.B. for 𝐼 w.r.t. ≻𝑙𝑒𝑥, we can eliminate variables from the system

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Applications of Gröbner bases Computing various objects related to ideals and varieties Points 𝑉(𝐹), Dimension dim 𝑉(𝐹), Singularities, Invariants, Free resolutions, etc.

Computing Sheaf cohomology (our main study 1)

Finding end enumerating varieties with special varieties (our main study 2)

Cryptanalysis

Automatic Theorem Proof

Integer Programing

Algebraic Statistics, etc.

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Contents:

1. Introduction

2. General Introduction of Computational Algebraic Geometry

3. Computing the Frobenius on cohomology groups (Main result 1)

4. Application to finding superspecial curves (Main result 2)

5. Concluding remarks

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Brief introduction to sheaf cohomology Cohomology group 𝐻𝑞 𝑋, ℱ• Defined as an abelian group for 𝑋 and ℱ 𝑋 : projective scheme over 𝐾, and ℱ : sheaf on 𝑋

If ℱ is coherent, 𝐻𝑞(𝑋, ℱ) is a finite-dimensional 𝐾-vector space

Importance of 𝐻𝑞(𝑋, ℱ)• In case of ℱ = 𝒪𝑋 (structure sheaf of 𝑋) A number of invariants of 𝑋 are obtained by computing dim𝐾 𝐻𝑞(𝑋, 𝒪𝑋)

• e.g. genus, Hilbert function, Euler characteristic

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Compute dim𝐾 𝐻𝑞(𝑋, ℱ) Obtain invariants of 𝑋

• Geometric genus: dim𝐾 𝐻𝑛(𝑋, 𝒪𝑋) with 𝑛 = dim𝑋• Arithmetic genus: 𝑞=0

∞ −1 𝑞 dim𝐾 𝐻𝑞(𝑋, 𝒪𝑋)

Case of ℱ = 𝒪𝑋

Known algorithms computing cohomology Two main strategies for computing the dimensions of sheaf cohomology groups:

1. Polynomial ring-based method [Eis]

Based on Serre’s local duality

Compute free resolutions of modules over a polynomial ring

2. Exterior algebra-based method [EFS]

Based on Bernstein-Gel’fand-Gel’fand correspondence

Compute free resolutions of modules over an exterior algebra

Both algorithms are recently analyzed in [Kud1]

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[Eis] D. Eisenbud, “Computing Cohomology”, pp. 219-226, a chapter in Computational Methods in Commutative Algebra and Algebraic Geometry, edited by W. Vasconcelos, AMS, 2, Springer (1998)

[EFS] D. Eisenbud, G. Fløystad and F.-O. Schreyer, “Sheaf Cohomology and Free Resolutions over Exterior Algebra”,Trans. Amer. Math. Soc., 355, no. 11, pp. 4397-4426 (2003)

[Kud1] M. Kudo, “Analysis of an algorithm to compute the cohomology groups of coherent sheaves andits applications”, Japan Journal of Industrial and Applied Mathematics, 35, pp. 131-169 (2017)

Introduction to Frobenius on cohomology Frobenius action 𝐹∗: 𝐻𝑞 𝑋, 𝒪𝑋 ⟶ 𝐻𝑞(𝑋, 𝒪𝑋)• Induced from the absolute Frobenius 𝐹: 𝑋 → 𝑋 on a sheme 𝑋 over a finite field in char. 𝑝

• 𝐹 : morphism with the identity map on 𝑋, and 𝑎 ↦ 𝑎𝑝 on sections

Importance of 𝐹∗

• In case of dim 𝑋 = 1

A representation matrix for 𝐹∗ is said to be the Hasse-Witt matrix

Hasse-Witt matrix of 𝑋 is zero ⟺ 𝑋 is a maximal curve (i.e., a curve with many rational points)

• Maximal curves are often used in coding theory

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Investigate 𝐹∗ Clarifying properties of 𝑋

The Hasse-Witt matrix of 𝑋 is determined

• Determine whether 𝑋 is maximal or not• Determine the (non-)existence of étale covering for 𝑋

Case of dim 𝑋 = 1

cf. For a hyperelliptic curve, 𝐻0(𝐶, Ω𝐶1) is usually used instead of 𝐻1(𝐶, 𝒪𝐶)

Conventional method for computing 𝐹∗

Given specific equations for 𝑋, conventional methods (e.g. [PW]) takeStep 1. Choose an open covering 𝒰 = {𝑈𝑖}𝑖 of 𝑋

Step 2. Compute a basis of 𝐻𝑞 𝑋, 𝒪𝑋 ≅ 𝐻𝑞(𝒰, 𝒪𝑋) compatible with 𝐹∗

Require to clarify the structure of the 𝐾-vector space 𝐻𝑞(𝒰, 𝒪𝑋) by • Investigating the sections 𝒪X 𝑈 = Γ(𝑈, 𝒪𝑋 𝑈) for all 𝑈 not algorithmically

• Using some specific properties of 𝑋

Step 3. Compute the image of each basis element by 𝐹∗

• Represent each image element as a linear combination of the origin basis

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Difficult to compute symbolically and algorithmically Objects computed in these steps depends on

• (specific) properties of 𝑋• one’s choice of an open covering of 𝑋

[PW] R. Pries and C. Weir, The Ekedahl-Oort type of Jacobians of Hermitian curves, arXiv: 1302.6261 [math.NT] (2013)

Our results Give a general-purpose algorithm [Kud2]• Given 𝑝, 𝑞 and 𝑋, compute 𝐹∗ algorithmically and symbolically

• Specifically, compute a representation matrix of 𝐹∗ via a suitable basis

Our techniques for algorithmic and symbolic computation: a. Reduce the computation of basis to that over cohomology groups on 𝐏𝑛

a-1. Constructing a decomposition of 𝐹: 𝑋 → 𝑋

a-2. The isomorphism between the origin cohomology groups and cohomology groups on 𝐏𝑛

b. Compute algorithmically a basis of 𝐻𝑞(𝑋, 𝒪𝑋), compatible with 𝐹∗ and its image by 𝐹∗

b-1. Construct a decomposition of 𝐹∗: 𝐻𝑞 𝑋, 𝒪𝑋 ⟶ 𝐻𝑞(𝑋, 𝒪𝑋)

b-2. Compute the basis and its image by using algorithms in CAG

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[Kud2] M. Kudo, Computing representation matrices for the action of Frobenius to cohomology groups,arXiv: 1704.08110 [math.AG] (2017)

Decomposition 𝑋 → 𝑋𝑝 → 𝑋 of the absolute Frobenius 𝐹

• For 𝑋 = 𝑉(𝑓1, … , 𝑓𝑡), define 𝑋𝑝 ≔ 𝑉(𝑓1𝑝, … , 𝑓𝑡

𝑝)

• 𝑋 → 𝑋𝑝 and 𝑋𝑝 → 𝑋 correspond to 𝑎 ↦ 𝑎 and 𝑎 ↦ 𝑎𝑝 on sections

• This decomposition induces the following commutative diagram:

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Our technique a

𝐻𝑞(𝑋(𝑝), 𝒪𝑋(𝑝))

𝐻𝑞(𝑋, 𝒪𝑋)

𝐻𝑞(𝑋, 𝒪𝑋)

𝐻𝑞(𝑋𝑝, 𝒪𝑋𝑝)

⟶⟶

𝐹∗

Spec(𝐾) Spec(𝐾)

𝑋(𝑝) 𝑋

𝑋

𝐹𝐹𝑋

𝐾 : a perfect field, 𝐹𝑋 is said to be the relative Frobenius

𝑋𝑝

Our techniques a and b Computing 𝐹∗ is reduced to computing a map of cohomology groups on 𝐏𝑛

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𝐻𝑞(𝑋, 𝒪𝑋)

𝐻𝑞(𝑋, 𝒪𝑋)

𝐻𝑞(𝑋𝑝, 𝒪𝑋𝑝)

𝐻𝑞+1(𝐏𝑛, 𝐼)

𝐻𝑞+1(𝐏𝑛, 𝐼)

𝐻𝑞+1(𝐏𝑛, 𝐼𝑝)

⟶

⟶

⟶

⟶

⟶

⟶

≃

≃

≃≃

≃

≃⟶

⟶⟶⟶⟶

⟶

𝐼 = ⟨𝑓1, … , 𝑓𝑡⟩ and 𝐼𝑝 = ⟨𝑓1𝑝, … , 𝑓𝑡

𝑝⟩

𝜑𝑖, 𝜑𝑖(𝑝)

and 𝜓𝑖 are certain homomorphisms of free modules over 𝑆 = 𝐾[𝑥0, … 𝑥𝑛]

Obtained by computing “free resolutions” of 𝑆/𝐼 and 𝑆/𝐼𝑝 via Gröbner bases alg.

𝐹∗

power 𝑝

𝐻𝑛−𝑞(𝜓𝑖∼)

Ker(𝐻𝑛 𝜑𝑖∼ )/Im(𝐻𝑛 𝜑𝑖+1

∼ )

Ker(𝐻𝑛 𝜑𝑖∼ )/Im(𝐻𝑛 𝜑𝑖+1

∼ )

Ker(𝐻𝑛 𝜑𝑖𝑝 ,∼

)/Im(𝐻𝑛 𝜑𝑖+1𝑝 ,∼

)

Computation over 𝐏𝑛 (with Gröbner basis algorithmsand linear algebra computation)

Case of complete intersections A representation matrix of 𝐹∗ for a complete intersection

Proposition ([Kud2], Prop. 4.2.1)

Assume (𝑓1, . . , 𝑓𝑡) is 𝑆-regular with 𝑞 = dim 𝑋 = 𝑛 − 𝑡

i.e. 𝑋 = 𝑉(𝑓1, … , 𝑓𝑡) is a complete intersection

Write 𝑘0, … , 𝑘𝑛 ∈ ℤ<0𝑛+1: 𝑖=0

𝑛 𝑘𝑖 = − 𝑗=1𝑡 deg 𝑓𝑗 = 𝑘0

𝑗, … , 𝑘𝑛

𝑗

𝑗

Then 𝐹∗ is represented by a matrix, each (𝑖, 𝑗)-entry of which is the coefficient of

𝑥0

−𝑘0(𝑖)

𝑝+𝑘0(𝑗)

⋯ 𝑥𝑛−𝑘𝑛

𝑖𝑝+𝑘𝑛

(𝑗)

in 𝑓1 ⋯ 𝑓𝑡𝑝−1

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Contents:

1. Introduction

2. General Introduction of Computational Algebraic Geometry

3. Computing the Frobenius on cohomology groups (Main result 1)

4. Application to finding superspecial curves (Main result 2)

5. Concluding remarks

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Superspecial curves (1/2) The notion of superspecial curves

Def. A non-singular curve 𝐶 of genus 𝑔 over 𝐾 = 𝔽𝑞 is superspecial (s.sp.)

⟺ 𝐽(𝐶) is isomorphic to 𝐸𝑔 over 𝐾

• 𝐽(𝐶) : Jacobi variety of 𝐶

• 𝐸 : a supersinguler elliptic curve

Equivalently the Frobenius on 𝐻1(𝐶, 𝒪𝐶) is zero

Importance of superspecial curves A tool for investigating the moduli of curves, or the moduli of abelian varieties

Superspecial curves can be useful in other areas such as coding theory

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Superspecial curves (2/2) Any superspecial curve descends to a maximal or minimal curves over 𝔽𝑝2

Minimal curves : #𝐶 𝔽𝑝2 = 1 − 2𝑔𝑝 + 𝑝2 (Hasse-Weil lower bound)

Maximal curves : #𝐶 𝔽𝑝2 = 1 + 2𝑔𝑝 + 𝑝2 (Hasse-Weil upper bound)

However, finding superspecial/maximal curves is very difficult in algebraic geometry Fixed 𝑔, 𝑞 , it is known that s.sp./max. curves of genus 𝑔 over 𝔽𝑞 are very rare

Ekedahl asked in his paper:

Problem ([Eke], p. 173): Does there exist a superspecial curve of genus 4 or 5 for any 𝑝 ≠ 2,3 ?

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Curves with many rational points permitlarger threshold of error correction in algebraic geometric codes!

[Eke] T. Ekedahl, “On supersingular curves and abelian varieties”, Math. Scand. 60 (1987), pp. 151-178.

Related works for enumeration

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Table 1. Main references to enumerations of 𝐾-isom. classes of s.sp. curves over 𝔽𝑞

𝒈＼ 𝒒 𝑝 ≤ 3 52𝑒−1 52𝑒 72𝑒−1 72𝑒 112𝑒−1 112𝑒 𝑝 ≥ 13

1 𝐾 = 𝔽𝑞 : Deuring, 𝐾 = 𝔽𝑞 : Xue-Yang-Yu

2 𝐾 = 𝔽𝑞 with 𝑞 = 𝑝2𝑒 ：Hashimoto-Ibukiyama, 𝐾 = 𝔽𝑞 with 𝑞 = 𝑝2𝑒−1 : Ibukiyama-Katsura

3 𝐾 = 𝔽𝑞 with 𝑞 = 𝑝2𝑒 : Hashimoto, 𝐾 = 𝔽𝑞 with 𝑞 = 𝑝2𝑒−1 : Ibukiyama

4 Non-Existenceby Ekedahl

𝐾 = 𝔽𝑞 : Fuhrmann et al.

𝐾 = 𝔽𝑞 and 𝔽𝑞:

Thm. A [KH1] and C [KH2]

Thm. B [KH1]

Thm. D[KH2]

Exists Nogeneralresult

Different from the cases of

𝑔 ≤ 3

[KH1] M. Kudo and S. Harashita, “Superspecial curves of genus 4 in small characteristic”,Finite Fields and Their Applications, 45, 131-169 (2017)

[KH2] M. Kudo and S. Harashita, “Enumerating superspecial curves of genus 4 over prime fields”,In: Proceedings of The Tenth International Workshop on Coding and Cryptography (WCC2017), Full paper: arXiv:1702.05313 [math.AG] (2017)

Our resultsMain Theorems ([KH1] and [KH2])

The number of 𝐾-isom. classes of nonhyperelliptic s.sp. curves of 𝑔 = 4 over 𝔽𝑞 is

A) 21 (resp. 1) if 𝐾 = 𝔽25 (resp. 𝐾 = 𝔽25) for 𝑞 = 25

B) 0 in characteristic 𝑝 = 7

C) 7 (resp. 1) if 𝐾 = 𝔽5 (resp. 𝐾 = 𝔽5) for 𝑞 = 5

D) 30 (resp. 9) if 𝐾 = 𝔽11 (resp. 𝐾 = 𝔽11) for 𝑞 = 11

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𝐾 ＼ 𝑞 5 25 7 49 11 121

𝔽𝑞 7 (Thm. C) 21 (Thm. A)0 (Thm. B)

30 (Thm. D) In progress

𝔽𝑞1 (Thm. C) 1 (Thm. A) 9 (Thm. D) In progress

Table 2. The num. of 𝐾-isom. classes of (nonhyperelliptic) s.sp. curves over 𝔽𝑞

Remark We have explicit defining equations, but omit them in the statement.

Rough sketch of the proofs

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Theorems A - D

Computational Problem:Enumerate quadratic forms 𝑄 ∈ 𝔽𝑞[𝑥, 𝑦, 𝑧, 𝑤] and cubic forms 𝑃 ∈ 𝔽𝑞[𝑥, 𝑦, 𝑧, 𝑤],

such that 𝐶 = 𝑉(𝑄, 𝑃) is superspecial

Reduced

Solved over Computer Algebra Systems

Problem: Given 𝑞, enumerate all superspecial curves of genus 4 over 𝔽𝑞

(1) Equations for (non-hyperelliptic) curves 𝐶 of 𝑔 = 4

(2) Criterion for superspecialityBased on computing the Frobenius on cohomology

(3) Algorithm to enumerate s.sp. curves of 𝑔 = 4(3-1) Reduction of cubic forms(3-2) Heuristic hybrid method to solve multivariate systems

(4) Isomorphism testing

Enumerated superspecial curvesWe found the following s.sp. curves 𝑉(𝑄, 𝑃) over 𝔽𝑝

𝐶1 = 𝑉(𝑄1, 𝑃1) over 𝔽5 with 𝑄1 = 2𝑦𝑤 + 𝑧2 and 𝑃1 = 𝑥3 + 𝑦2𝑧 + 𝑧𝑤2

𝐶2 = 𝑉(𝑄2, 𝑃2) over 𝔽11 with 𝑄2 = 2𝑥𝑤 + 𝑦2 + 9𝑧2 and 𝑃2 = 𝑥2𝑧 + 5𝑦3 + 4𝑧𝑤2

𝐶3 = 𝑉(𝑄3, 𝑃3) over 𝔽11 with 𝑄3 = 2𝑦𝑤 + 𝑧2 and 𝑃3 = 𝑥3 + 𝑦3 + 𝑤3

The above curves provide max. curves over 𝔽𝑝2

𝐶1 is max. over 𝔽25 i.e. . #𝐶 𝔽𝑝2 = 𝑝2 + 1 + 2𝑔𝑝 = 66 for 𝑔, 𝑝 = (4,5)

𝐶2 and 𝐶3 are max. over 𝔽121 i.e. #𝐶 𝔽𝑝2 = 𝑝2 + 1 + 2𝑔𝑝 = 210 for 𝑔, 𝑝 = (4,11)

All curves enumerated in this work are summarized at

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http://www2.math.kyushu-u.ac.jp/~m-kudo/kudo-harashita-comp-PF.html

Contents:

1. Introduction

2. General Introduction of Computational Algebraic Geometry

3. Computing the Frobenius on cohomology groups (Main result 1)

4. Application to finding superspecial curves (Main result 2)

5. Concluding remarks

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Concluding remarkSummary of this talkWe first give a general intro. of CAG, in particular Gröbner bases

As our main results, we gave an algorithm for computing

The Frobenius on cohomology groups

Enumerating superspecial (s.sp.) curves

Using our algorithm, we determined the s.sp. curves over 𝔽𝑞 for 𝑞 = 5, 25, 7, 49, 11

Enumerated s.sp. curves provide max. curves

Future works

Enumeration of s.sp. or max. curves for larger 𝑝 (or 𝑞) or higher 𝑔

Investigate the structure of s.sp. curves

For 𝑞 = 𝑝 = 11, automorphisms groups have been computed in [KHS]

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[KHS] M. Kudo, S. Harashita, and H. Senda, “Automorphism groups of superspecial curves of genus 4 over 𝔽11”,arXiv:1709.03627 (2017).