Hasse's theorem on elliptic curves

For an elliptic curve over a finite field with q elements, its number of rational points differs from q + 1 by at most 2√q.

Connect

Frobenius endomorphism: An endomorphism of an algebraic variety over a finite field induced by raising coordinates to the field's characteristic power. Its action on an elliptic curve encodes the point count that Hasse bounds.
Finite field: A field containing finitely many elements, whose size is a power of a prime. The theorem's parameter q is the number of elements in this field.
Helmut Hasse: A German mathematician who made major contributions to algebraic number theory and the theory of elliptic curves. He proved the point-count bound that bears his name.
Elliptic-curve point counting: The computational task of determining the number of points on an elliptic curve over a finite field. Hasse's interval narrows the possible answers and checks computed counts.
Sato–Tate conjecture: A conjecture describing the statistical distribution of Frobenius traces for elliptic curves in families. Hasse bounds each trace individually; Sato–Tate describes how traces are distributed across curves.
Trace of Frobenius: The trace of the Frobenius endomorphism acting on an elliptic curve's Tate module. The point count equals q + 1 minus this trace, whose absolute value Hasse bounds.
Rational point: A solution to algebraic equations whose coordinates belong to a specified field. The theorem counts points whose coordinates lie in the finite field.
André Weil: A French mathematician whose work shaped modern number theory and algebraic geometry. He generalized the estimate into conjectures about point counts on higher-dimensional varieties.
Schoof's algorithm: A polynomial-time algorithm for counting points on an elliptic curve over a finite field. It computes the trace of Frobenius, the quantity bounded by Hasse's theorem.
Deuring's theorem: A theorem classifying elliptic curves over finite fields by their endomorphism algebras and associated traces. It addresses which trace values occur, complementing Hasse's restriction on their size.