MSC2020 · 11Rxx

Algebraic number theory: global fields

Algebraic number theory: global fields \{For complex multiplication, see 11G15\}

10 mapped entries · Browse and filter all →

Counts include all statuses and descendants. Classifications are editorial or inferred from community fields, not an AMS assessment of the problem.

Subcategories

11R04Algebraic numbers; rings of algebraic integers0 11R06PV-numbers and generalizations; other special algebraic numbers; Mahler measure4 11R09Polynomials (irreducibility, etc.)2 11R11Quadratic extensions1 11R16Cubic and quartic extensions0 11R18Cyclotomic extensions0 11R20Other abelian and metabelian extensions0 11R21Other number fields0 11R23Iwasawa theory0 11R27Units and factorization0 11R29Class numbers, class groups, discriminants4 11R32Galois theory0 11R33Integral representations related to algebraic numbers; Galois module structure of rings of integers0 11R34Galois cohomology0 11R37Class field theory0 11R39Langlands-Weil conjectures, nonabelian class field theory0 11R42Zeta functions and

L

-functions of number fields0 11R44Distribution of prime ideals0 11R45Density theorems0 11R47Other analytic theory0 11R52Quaternion and other division algebras: arithmetic, zeta functions0 11R54Other algebras and orders, and their zeta and

L

-functions0 11R56Adèle rings and groups0 11R58Arithmetic theory of algebraic function fields0 11R59Zeta functions and

L

-functions of function fields0 11R60Cyclotomic function fields (class groups, Bernoulli objects, etc.)0 11R65Class groups and Picard groups of orders0 11R70

K

-theory of global fields0 11R80Totally real fields0 11R99Algebraic number theory: global fields0

Mapped problems

OP-10085Number theoryStatus unassessed

Uniform bounds for eventual stability

A question from Bulletin of the American Mathematical Society (2019). Question 19.5; inspected PDF p. 55.

Robert Benedetto — Current trends and open problems in arithmetic dynamics· Journal question · editorial summary

OP-10084Number theoryStatus unassessed

Eventual stability of rational dynamics over number fields

A question from Bulletin of the American Mathematical Society (2019). Conjecture 19.4 and Definition 19.2; inspected PDF p. 55.

Robert Benedetto — Current trends and open problems in arithmetic dynamics· Journal question · editorial summary

OP-12103Number theoryOpen

Martinet's constant for totally real number fields ( $C_{87}$ )

Martinet's constant for totally real number fields: determine C 87 or improve its bounds; the repository records 60.8395 (lower) and 857.567 (upper).

Optimization Constants in Mathematics — Davis, Ivanisvili & Tao· Licensed source statement· 2026

OP-12102Number theoryOpen

Schur–Siegel–Smyth trace constant ( $C_{86}$ )

Schur–Siegel–Smyth trace constant: determine C 86 or improve its bounds; the repository records 1.80203 (lower) and 1.8216 (upper).

Optimization Constants in Mathematics — Davis, Ivanisvili & Tao· Licensed source statement· 2026

OP-12097Number theoryOpen

Essential minimum of the Zhang-Zagier height ( $C_{82}$ )

Essential minimum of the Zhang-Zagier height: determine C 82 or improve its bounds; the repository records 0.24874 (lower) and 0.2536331090204145 (upper).

Optimization Constants in Mathematics — Davis, Ivanisvili & Tao· Licensed source statement· 2026

OP-12051Number theoryOpen

Asymptotic Dobrowolski constant for Lehmer’s problem ( $C_{40b}$ )

Asymptotic Dobrowolski constant for Lehmer’s problem: determine C 40b or improve its bounds; the repository records 9/4 (lower) and infty (upper).

Optimization Constants in Mathematics — Davis, Ivanisvili & Tao· Licensed source statement· 2026

OP-12050Number theoryOpen

Lehmer’s Mahler measure constant ( $C_{40a}$ )

Lehmer’s Mahler measure constant: determine C 40a or improve its bounds; the repository records 1 (lower) and 1.176280... (upper).

Optimization Constants in Mathematics — Davis, Ivanisvili & Tao· Licensed source statement· 2026

OP-11986Number theoryOpen

A Prescribed Class-Group Structure for an Imaginary Quadratic Field

A Prescribed Class-Group Structure for an Imaginary Quadratic Field. Solid result problem from Epoch AI’s FrontierMath: Open Problems; answers are checked by a verifier program.

Epoch AI — FrontierMath: Open Problems· Licensed source statement· 2026

OP-11980Number theoryOpen

Class Group Algorithms

Class Group Algorithms. Solid result problem from Epoch AI’s FrontierMath: Open Problems; answers are checked by a verifier program.

Epoch AI — FrontierMath: Open Problems· Licensed source statement· 2026

OP-11946Number theoryOpen

Large $\ell$ -Rank in Class Groups of Imaginary Quadratic Fields

Large ell -Rank in Class Groups of Imaginary Quadratic Fields. Moderately interesting problem from Epoch AI’s FrontierMath: Open Problems; answers are checked by a verifier program.

Epoch AI — FrontierMath: Open Problems· Licensed source statement· 2026

Journals & articles in MSC 11

These are selected sources for the parent class. Article coverage is a first reviewed selection, and may not cover this exact subcategory.

Algebra & Number Theory ↗

Mathematical Sciences Publishers

Algebra, number theory and their connections

EMS Surveys in Mathematical Sciences ↗

European Mathematical Society / EMS Press

Mathematical surveys

Current trends and open problems in arithmetic dynamics
2019 · 3 questions extracted

View source evidence →

Subcategories

Mapped problems

Uniform bounds for eventual stability

Eventual stability of rational dynamics over number fields

Martinet's constant for totally real number fields (C87C_{87}C87​)

Schur–Siegel–Smyth trace constant (C86C_{86}C86​)

Essential minimum of the Zhang-Zagier height (C82C_{82}C82​)

Asymptotic Dobrowolski constant for Lehmer’s problem (C40bC_{40b}C40b​)

Lehmer’s Mahler measure constant (C40aC_{40a}C40a​)

A Prescribed Class-Group Structure for an Imaginary Quadratic Field

Class Group Algorithms

Large ℓ\ellℓ-Rank in Class Groups of Imaginary Quadratic Fields

Journals & articles in MSC 11

Martinet's constant for totally real number fields ( $C_{87}$ )

Schur–Siegel–Smyth trace constant ( $C_{86}$ )

Essential minimum of the Zhang-Zagier height ( $C_{82}$ )

Asymptotic Dobrowolski constant for Lehmer’s problem ( $C_{40b}$ )

Lehmer’s Mahler measure constant ( $C_{40a}$ )

Large $\ell$ -Rank in Class Groups of Imaginary Quadratic Fields