Reduction modulo primes and Hasse's bound

Over Q_p, choose an integral Weierstrass equation. A discriminant that is a p-adic unit gives a smooth projective special fiber, hence good reduction. Conversely, good reduction means that such a smooth integral model exists; it does not require every integral equation for the curve to remain smooth. A locally minimal integral Weierstrass equation has discriminant of minimal valuation. For that model, a positive discriminant valuation characterizes bad reduction.

For example, at an odd prime p the equations y²=x³−x and y²=x³−p⁴x define isomorphic curves over Q_p after scaling x and y by p² and p³. The second equation has discriminant 64p¹² and singular displayed reduction, but the first has unit discriminant and proves good reduction. At genuinely bad reduction the minimal special fiber is singular; multiplicative reduction is associated with a node and additive reduction with a cusp in its Weierstrass cubic, with splitness requiring a further residue-field check.

Let the residue field have odd cardinality q and consider a smooth short equation. For each x, its cubic value contributes one y if it is zero, two if it is a nonzero square, and none if it is a nonsquare. The unique projective point O contributes one more. Equivalently, the count is q+1 plus the sum of the quadratic character of the cubic, taking the character of zero to be zero. Forgetting O shifts every inferred Frobenius trace by one.

Write a_q=q+1−#E(F_q). Hasse's theorem bounds its absolute value by two square roots of q. This is a theorem for any elliptic curve over a finite field, including characteristics where a general Weierstrass model is required. It is not a bound on the number of rational points over Q. One proof uses the degree form on endomorphisms and the q-power Frobenius map; an arbitrary cubic with a singularity does not satisfy the theorem's elliptic-curve hypothesis.

The group E(F_q) is finite abelian and has at most two cyclic factors. Its cardinality alone does not decide those factors, and a cyclic toy model of the same size need not represent its group law. Rational two-torsion can immediately exclude cyclicity: if all three distinct cubic roots lie in the field, the group contains a copy of two cyclic factors of order two. Determining group structure requires additional point orders or structural information.

At a good prime, reduction is compatible with addition. Prime-to-p torsion injects into the reduced group because the formal reduction kernel has no torsion of order prime to p. This supplies useful torsion restrictions but says little directly about the free rank: infinitely many rational points can share a finite residue class. Local counts become coefficients of the global L-function later, where their relation to rank is subtler than identifying a single finite group.