11Fxx Automorphic forms
This subtopic studies automorphic forms, including modular forms and related analytic objects that encode deep arithmetic and representation-theoretic information.
Specific topics
11F03 Modular and automorphic functions
Overview
11F03 studies modular and automorphic functions within automorphic forms. It focuses on central definitions, foundational theorem patterns, and techniques that connect structure, computation, and asymptotic behavior.
Related Wikipedia Page
Modular and automorphic functions (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and examples for modular and automorphic functions
- 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
11F06 Structure of modular groups and generalizations
Overview
11F06 studies structure of modular groups and generalizations within automorphic forms. It focuses on central definitions, foundational theorem patterns, and techniques that connect structure, computation, and asymptotic behavior.
Related Wikipedia Page
Structure of modular groups and generalizations (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and examples for structure of modular groups and generalizations
- 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
11F11 Holomorphic modular forms of integral weight
Overview
11F11 studies holomorphic modular forms of integral weight within automorphic forms. It focuses on central definitions, foundational theorem patterns, and techniques that connect structure, computation, and asymptotic behavior.
Related Wikipedia Page
Holomorphic modular forms of integral weight (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and examples for holomorphic modular forms of integral weight
- 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
11F12 Automorphic forms, one variable
Overview
11F12 studies automorphic forms, one variable within automorphic forms. It focuses on central definitions, foundational theorem patterns, and techniques that connect structure, computation, and asymptotic behavior.
Related Wikipedia Page
Automorphic forms, one variable (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and examples for automorphic forms, one variable
- 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
11F20 Dedekind eta function, Dedekind sums
Overview
11F20 studies dedekind eta function, dedekind sums within automorphic forms. It focuses on central definitions, foundational theorem patterns, and techniques that connect structure, computation, and asymptotic behavior.
Related Wikipedia Page
Dedekind eta function, Dedekind sums (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and examples for dedekind eta function, dedekind sums
- 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
11F30 Fourier coefficients of automorphic forms
Overview
11F30 studies fourier coefficients of automorphic forms within automorphic forms. It focuses on central definitions, foundational theorem patterns, and techniques that connect structure, computation, and asymptotic behavior.
Related Wikipedia Page
Fourier coefficients of automorphic forms (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and examples for fourier coefficients of automorphic forms
- 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
11F37 Forms of half-integer weight; nonholomorphic modular forms
Overview
11F37 studies forms of half-integer weight; nonholomorphic modular forms within automorphic forms. It focuses on central definitions, foundational theorem patterns, and techniques that connect structure, computation, and asymptotic behavior.
Related Wikipedia Page
Forms of half-integer weight; nonholomorphic modular forms (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and examples for forms of half-integer weight; nonholomorphic modular forms
- 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
11F66 Langlands $L$-functions; one variable Dirichlet series
Overview
11F66 studies langlands l-functions; one variable dirichlet series within automorphic forms. It focuses on central definitions, foundational theorem patterns, and techniques that connect structure, computation, and asymptotic behavior.
Related Wikipedia Page
Langlands L-functions; one variable Dirichlet series (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and examples for langlands l-functions; one variable dirichlet series
- 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