Diophantine equations
Equations over integers and rational numbers, from elliptic curves to the limits of decision algorithms.
7 problemsArithmetic geometry
Arithmetic questions studied through geometric objects.
Also under Algebra and geometry
Collections
Browse collectionsFields and rational points
Field symmetries and rational points on curves and higher-dimensional varieties.
7 problemsA few problems to explore
10 totalBirch and Swinnerton-Dyer conjecture
An elliptic curve has both rational solutions and an analytic function. The conjecture predicts that these two ways of studying it encode the same rank.
Bombieri–Lang conjecture
Algebraic spaces of general type are predicted to have rational points confined to a smaller algebraic subset, rather than spread densely everywhere.
Artin’s primitive root conjecture
For a fixed eligible integer, how often do its powers generate every nonzero residue modulo a prime?
Leopoldt’s conjecture
Do the independent units of a number field remain independent when measured using p-adic arithmetic?
Uniform boundedness of rational torsion
Can rational torsion be bounded by abelian-variety dimension and number-field degree?
Tate conjecture over finite fields
Do all Galois-invariant cohomology classes of the relevant degree come from algebraic cycles over a finite field?