Fields and rational points

Problems in this collection

7 problems

Inverse Galois problem

Can every finite group occur as the Galois group of a finite extension of the rational numbers?

Birch 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.

Uniform boundedness of rational torsion

Can rational torsion be bounded by abelian-variety dimension and number-field degree?

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.

Tate conjecture over finite fields

Do all Galois-invariant cohomology classes of the relevant degree come from algebraic cycles over a finite field?

Dynamical uniform boundedness conjecture

Can finite arithmetic orbits be bounded by map degree and number-field degree alone?

Grothendieck’s section conjecture

Does the arithmetic fundamental group of a curve identify exactly its rational points?

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