Hasse's Theorem on Elliptic Curves bounds the number of points on an elliptic curve defined over a finite field with q elements, showing that this count differs from q plus one by at most twice the square root of q. Named for Helmut Hasse, who proved it in the 1930s, it is a foundational estimate of arithmetic geometry, later generalized to curves of higher genus by the Weil conjectures and put to wide use in elliptic-curve cryptography.
Facts
StatementFor an elliptic curve E defined over a finite field with q elements, the number of points N on E satisfies the absolute value of N minus q plus 1 is at most twice the square root of q. 1 Classification
Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Proved By
Source Hasse's theorem on elliptic curves (Wikipedia)
Sources
1. Hasse's theorem on elliptic curves (Wikipedia)
Statement section
N - (q + 1) is bounded in absolute value by 2 times the square root of q
History section
It was proven by Hasse in 1933, with the proof published in a series of papers in 1936.
Proved By: Helmut Hasse, Lead paragraph
Hasse's theorem on elliptic curves, also referred to as the Hasse bound, provides an estimate of the number of points on an elliptic
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