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.
Arithmetic geometry
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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?
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?
Serre’s uniformity conjecture
Do non-CM elliptic curves over the rationals always have full mod-p Galois image for primes p>37?
Lehmer’s nonvanishing conjecture
Is every coefficient of Ramanujan’s discriminant modular form nonzero?