Women in numbers Europe III : research directions in number theory

Preface
Acknowledgements
Contents
From p-modular to p-adic Langlands Correspondences for U(1,1)(Qp2/Qp): Deformations in the Non-supercuspidal Case
	1 Introduction
		1.1 General Notation
	2 Non-supercuspidal Representations of G over Fp
		2.1 Principal Series Representations and Characters of G
		2.2 Special Series Representations
		2.3 Classification of Non-supercuspidal Representations of G
	3 A Non-supercuspidal Semisimple Langlands Correspondence
		3.1 Galois Representations and Dual Groups Associated with G
		3.2 From Langlands Parameters to C-Parameters
		3.3 A Langlands Correspondence for Non-supercuspidal Representations
	4 Deforming Non-supercuspidal Representations of G
		4.1 Deforming Parabolically Induced Representations
		4.2 Deformations of Special Series Representations
	5 Deforming Langlands Parameters
		5.1 Definition of C-Parameters in Characteristic Zero
		5.2 Inertial Types and Generic C-Parameters
			5.2.1 Genericity for Classical Inertial Types
			5.2.2 Genericity for C-Valued Inertial Types
			5.2.3 Genericity for C-Parameters and Their Inertial Types
		5.3 Deforming Galois Parameters
			5.3.1 Universal Framed Deformations
			5.3.2 Intermission: Frobenius-Twist Self-Dual Inertial Types
			5.3.3 Potentially Crystalline Deformations with Prescribed Hodge Type and Inertial Type
		5.4 From C-Parameters to Kisin Modules
			5.4.1 From C-Parameters to Genuine p-adic Galois Representations
			5.4.2 Kisin Modules with Prescribed Descent Data and Height
			5.4.3 Frobenius-Twist Self-Dual Kisin Modules and Associated Galois Representations
		5.5 Some Explicit Deformation Rings for C-Parameters
			5.5.1 Shape of a Kisin Module over k
			5.5.2 A Deformation Problem for Kisin Modules
			5.5.3 Some Consequences on Deformations of C-Parameters
	References
Explicit Connections Between Supersingular Isogeny Graphs and Bruhat–Tits Trees
	1 Introduction
		1.1 Contributions
	2 Background
		2.1 Elliptic Curves over Finite Fields
			2.1.1 Isogenies and Endomorphisms
			2.1.2 Supersingular -Isogeny Graphs
		2.2 Quaternion Algebras over Q
			2.2.1 Arithmetic of Quaternion Algebras
			2.2.2 -Ideal Graph of a Quaternion Algebra
			2.2.3 Norm Forms of Maximal Orders
		2.3 The Bruhat–Tits Tree for PGL2(Q)
	3 The Graph of the Bad Reduction of Shimura Curves
		3.1 Shimura Curves from Indefinite Quaternion Algebras
		3.2 The -Adic Upper Half-Plane
		3.3 -Adic Shimura Curves
		3.4 Computing the Graph of the Special Fibre of a Shimura Curve
	4 Different Views on Supersingular Isogeny Graphs
		4.1 Supersingular Elliptic Curves and Endomorphism Rings: Deuring's Correspondence
		4.2 The Bruhat–Tits Tree, an Unfolding of the Supersingular Isogeny Graph
			4.2.1 The Tate Module
			4.2.2 Translating Vertices of Bruhat–Tits Trees into Sublattices of the Tate Module
			4.2.3 Translating Sublattices of the Tate Module into Subgroups of Elliptic Curves
			4.2.4 Non-backtracking Walks in G as Level-Increasing Paths from the Root of Tl
		4.3 Bruhat–Tits Tree Quotients and Supersingular Isogeny Graphs: Ribet's Correspondence
		4.4 The Bruhat–Tits Tree and Quaternion Orders
	5 Towards Cryptographic Applications
		5.1 A Truncated Bruhat–Tits Tree from SIKE Parameters
		5.2 Isogenies from Paths in the Bruhat–Tits Tree
		5.3 Explicit Computations with the Bruhat–Tits Tree
		5.4 Computing and Exploiting Norm Equations
	6 Conclusion
	References
Semi-Regular Sequences and Other Random Systems of Equations
	1 Introduction
	2 Notation and Preliminaries
		2.1 Commutative Algebra Review
		2.2 Homogeneous Semi-Regular Sequences
		2.3 The Macaulay Matrix and the Solving Degree of a System of Equations
	3 Solving Degree, Degree of Regularity, and Castelnuovo-Mumford Regularity
	4 Solving Degree of Cryptographic Semi-regular Systems
		4.1 Homogeneous Cryptographic Semi-regular Sequences
		4.2 Inhomogeneous Cryptographic Semi-regular Sequences
	5 A Consequence of the Eisenbud-Green-Harris Conjecture
		5.1 Limits to the Applicability of Theorem 5.4 and Relation with the Degree of Regularity
	6 Values of r(n+k,n) for 2 ≤k,n ≤100
	References
Reduction Types of Genus-3 Curves in a Special Stratum of their Moduli Space
	1 Introduction
		1.1 Notation
	2 Stable Reduction and Admissible Covers
		2.1 The Set-up
		2.2 Stable Reduction of Covers
		2.3 Combinatorial Description of the Stable Reduction
		2.4 Computing the Stable Reduction
	3 The Smooth Plane Quartic Case
		3.1 Invariants
		3.2 Main Results
	4 Proofs of Main Results
		4.1 Main Result with Non-degenerate Conic
		4.2 Main Result with Degenerate Conic
	5 Hyperelliptic Case
		5.1 Invariants
		5.2 The Main Theorem and Its Proof
	Appendix: Admissible Covers
	References
The Complexity of MinRank
	1 Introduction
	2 Main Results
	References
Fields of Definition of Elliptic Fibrations on Covers of Certain Extremal Rational Elliptic Surfaces
	1 Introduction
		1.1 Relation to the Literature
	2 Preliminaries and Setting
	3 Rational Curves on K3 Surfaces
	4 Extremal Rational Elliptic Surfaces
		4.1 Minimal Models for Extremal RES Over k
	5 Double Covers of Extremal Rational Elliptic Surfaces
		5.1 Arithmetic Models of Extremal Rational Elliptic Surfaces
	6 The Surfaces R9 and X9
		6.1 Negative Curves on R9
		6.2 The K3 Surface X9
		6.3 Classification of All the Possible Fibrations of the K3 Surface X9
			6.3.1 Torsion of the Mordell–Weil Group for the Elliptic Fibrations Associated to X9
		6.4 Determining the Type of Each Fibration of X9
	7 The Surfaces R4, R3, R2 and the Surfaces X4, X3, X2
		7.1 The Rational Elliptic Surfaces R4, R3, and R2
		7.2 The K3 Surfaces X4, X3, X2
		7.3 Classification of All the Possible Fibrations on the K3 Surfaces X4, X3, and X2
		7.4 Determining the Type of Each Fibration of X4, X3, and X2
	References
Integers Represented by Ternary Quadratic Forms
	1 Introduction
	2 The Brauer-Manin Obstruction for Integral Points
	3 Local Solutions to Qa,b,c=n
	4 Generator of the Brauer Group of Xa,b,c
	5 Computation of the Local Evaluation Maps at Odd Primes
		5.1 Case p=∞
		5.2 Case p a Prime, 2