11Gxx Arithmetic algebraic geometry
This subtopic studies arithmetic algebraic geometry, connecting algebraic geometry with number-theoretic questions via Diophantine geometry and arithmetic of varieties.
Specific topics
11G05 Elliptic curves over global fields
Overview
11G05 studies elliptic curves over global fields within arithmetic algebraic geometry. It focuses on central definitions, foundational theorem patterns, and techniques that connect structure, computation, and asymptotic behavior.
Related Wikipedia Page
Elliptic curves over global fields (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and examples for elliptic curves over global fields
- Core proof tools used in modern number-theoretic research
- Interactions with algebraic, analytic, and computational methods
Typical Uses
Used to frame precise research questions, compare equivalent formulations, and select effective proof or computation strategies in advanced number theory.
Applications
- Development of core theory in arithmetic and analysis
- Algorithmic and computational investigations
- Cross-disciplinary links to algebraic geometry, dynamics, and cryptography
References
Recommended Textbooks
11G07 Elliptic curves over local fields
Overview
11G07 studies elliptic curves over local fields within arithmetic algebraic geometry. It focuses on central definitions, foundational theorem patterns, and techniques that connect structure, computation, and asymptotic behavior.
Related Wikipedia Page
Elliptic curves over local fields (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and examples for elliptic curves over local fields
- Core proof tools used in modern number-theoretic research
- Interactions with algebraic, analytic, and computational methods
Typical Uses
Used to frame precise research questions, compare equivalent formulations, and select effective proof or computation strategies in advanced number theory.
Applications
- Development of core theory in arithmetic and analysis
- Algorithmic and computational investigations
- Cross-disciplinary links to algebraic geometry, dynamics, and cryptography
References
Recommended Textbooks
11G10 Abelian varieties of dimension $> 1$
Overview
11G10 studies abelian varieties of dimension > 1 within arithmetic algebraic geometry. It focuses on central definitions, foundational theorem patterns, and techniques that connect structure, computation, and asymptotic behavior.
Related Wikipedia Page
Abelian varieties of dimension > 1 (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and examples for abelian varieties of dimension > 1
- Core proof tools used in modern number-theoretic research
- Interactions with algebraic, analytic, and computational methods
Typical Uses
Used to frame precise research questions, compare equivalent formulations, and select effective proof or computation strategies in advanced number theory.
Applications
- Development of core theory in arithmetic and analysis
- Algorithmic and computational investigations
- Cross-disciplinary links to algebraic geometry, dynamics, and cryptography
References
Recommended Textbooks
11G15 Complex multiplication and moduli of abelian varieties
Overview
11G15 studies complex multiplication and moduli of abelian varieties within arithmetic algebraic geometry. It focuses on central definitions, foundational theorem patterns, and techniques that connect structure, computation, and asymptotic behavior.
Related Wikipedia Page
Complex multiplication and moduli of abelian varieties (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and examples for complex multiplication and moduli of abelian varieties
- Core proof tools used in modern number-theoretic research
- Interactions with algebraic, analytic, and computational methods
Typical Uses
Used to frame precise research questions, compare equivalent formulations, and select effective proof or computation strategies in advanced number theory.
Applications
- Development of core theory in arithmetic and analysis
- Algorithmic and computational investigations
- Cross-disciplinary links to algebraic geometry, dynamics, and cryptography
References
Recommended Textbooks
11G20 Curves over finite and local fields
Overview
11G20 studies curves over finite and local fields within arithmetic algebraic geometry. It focuses on central definitions, foundational theorem patterns, and techniques that connect structure, computation, and asymptotic behavior.
Related Wikipedia Page
Curves over finite and local fields (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and examples for curves over finite and local fields
- Core proof tools used in modern number-theoretic research
- Interactions with algebraic, analytic, and computational methods
Typical Uses
Used to frame precise research questions, compare equivalent formulations, and select effective proof or computation strategies in advanced number theory.
Applications
- Development of core theory in arithmetic and analysis
- Algorithmic and computational investigations
- Cross-disciplinary links to algebraic geometry, dynamics, and cryptography
References
Recommended Textbooks
11G25 Varieties over finite and local fields
Overview
11G25 studies varieties over finite and local fields within arithmetic algebraic geometry. It focuses on central definitions, foundational theorem patterns, and techniques that connect structure, computation, and asymptotic behavior.
Related Wikipedia Page
Varieties over finite and local fields (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and examples for varieties over finite and local fields
- Core proof tools used in modern number-theoretic research
- Interactions with algebraic, analytic, and computational methods
Typical Uses
Used to frame precise research questions, compare equivalent formulations, and select effective proof or computation strategies in advanced number theory.
Applications
- Development of core theory in arithmetic and analysis
- Algorithmic and computational investigations
- Cross-disciplinary links to algebraic geometry, dynamics, and cryptography
References
Recommended Textbooks
11G35 Varieties over global fields
Overview
11G35 studies varieties over global fields within arithmetic algebraic geometry. It focuses on central definitions, foundational theorem patterns, and techniques that connect structure, computation, and asymptotic behavior.
Related Wikipedia Page
Varieties over global fields (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and examples for varieties over global fields
- Core proof tools used in modern number-theoretic research
- Interactions with algebraic, analytic, and computational methods
Typical Uses
Used to frame precise research questions, compare equivalent formulations, and select effective proof or computation strategies in advanced number theory.
Applications
- Development of core theory in arithmetic and analysis
- Algorithmic and computational investigations
- Cross-disciplinary links to algebraic geometry, dynamics, and cryptography
References
Recommended Textbooks
11G40 $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
Overview
11G40 studies l-functions of varieties over global fields; birch-swinnerton-dyer conjecture within arithmetic algebraic geometry. It focuses on central definitions, foundational theorem patterns, and techniques that connect structure, computation, and asymptotic behavior.
Related Wikipedia Page
L-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and examples for l-functions of varieties over global fields; birch-swinnerton-dyer conjecture
- Core proof tools used in modern number-theoretic research
- Interactions with algebraic, analytic, and computational methods
Typical Uses
Used to frame precise research questions, compare equivalent formulations, and select effective proof or computation strategies in advanced number theory.
Applications
- Development of core theory in arithmetic and analysis
- Algorithmic and computational investigations
- Cross-disciplinary links to algebraic geometry, dynamics, and cryptography
References
Recommended Textbooks