claimroutine

Reduction orders r+1±2a and common factors

Gives orders r+1±2ar+1\pm2a with odd common factors only l≡1(mod4)l\equiv1\pmod4
order conditions for the prime pattern
Claim·§6.1 Reduction orders and their possible common factors·inside lemPrime patterns with controlled reduction orders

Statement (paper)

Let rr be a sufficiently large rational prime with r≡1(mod8)r\equiv1\pmod8 and r≡3(mod7)r\equiv3\pmod7. Then rr splits in FF, and at either place of FF above rr the reduction orders of E,EσE,E^\sigma over Fr\F_r are, in some order, d1,d2=r+1±2ad_1,d_2=r+1\pm2a with a,b∈Za,b\in\Z, a2+b2=ra^2+b^2=r, a≠0a\ne0. Both did_i are divisible by 44, the 22-part of gcd⁡(d1,d2)\gcd(d_1,d_2) is exactly 44, and r∤d1d2r\nmid d_1d_2. Moreover, if an odd prime ll divides both d1d_1 and d2d_2, then l∣r+1l\mid r+1, a≡0(modl)a\equiv0\pmod l and b2≡−1(modl)b^2\equiv-1\pmod l; in particular l≡1(mod4)l\equiv1\pmod4.

In plain English

For large primes r≡1(mod8)r\equiv1\pmod8 with r≡3(mod7)r\equiv3\pmod7, rr splits in FF and the reduction orders of EE and EσE^\sigma at a place above rr are r+1±2ar+1\pm2a where a2+b2=ra^2+b^2=r and a≠0a\ne0. Both orders are divisible by 4, their gcd has 2-part exactly 4, and rr divides neither. Any odd prime ll dividing both orders satisfies l∣r+1l\mid r+1, l∣al\mid a, b2≡−1(modl)b^2\equiv-1\pmod l, so l≡1(mod4)l\equiv1\pmod4.

Why it's here

This claim gives the arithmetic input for the order conditions gcd⁡(d1,d2)∣4\gcd(d_1,d_2)\mid4 and r∤d1d2r\nmid d_1d_2 in the parent lemma. It confines any common factor of the two orders to odd primes l≡1(mod4)l\equiv1\pmod4 dividing both r+1r+1 and aa, with b2≡−1(modl)b^2\equiv-1\pmod l. The later claims can therefore control such primes (those above a bound LL) by a counting argument and discard them, leaving only the 2-part to handle, which is exactly 4.
Classified routine:
Elementary computation of the Frobenius traces of the CM-type curves E,EσE,E^\sigma at split primes, writing r=a2+b2r=a^2+b^2 and doing gcd/congruence bookkeeping mod 22 and mod odd ll. This is a standard calculation for CM elliptic curves rather than a new idea.

Uses

Nothing else in the paper

Used by

claimFinite multiplier set and initial residue classclaimFew primes with large common order factorclaimSurviving prime pattern satisfies order conditions

Sources

PaperOpen
Section
§6.1 Reduction orders and their possible common factors
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sections/06-primes.tex:127
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