claimroutineinferred_dep

Auxiliary primes give required congruences

Auxiliary primes give l≡3(mod4)l\equiv3\pmod4 and l≡2(mod3)l\equiv2\pmod3
fixes prime types for the congruence conditions
Claim·§3.4 One parameter for both twists·inside lemOne squarefree n serving both twists

Statement (paper)

Let q0q_0 be an auxiliary prime of type (s0,λ0)=(1,1)(s_0,\lambda_0)=(1,1) and let II be a set of ∣B∣−1|B|-1 auxiliary primes of type (0,0)(0,0), with types imposed by the residue classes 77 and 11 modulo 1212 respectively. Then on each of the sets Q0=B∪I∪{q0}Q_0=B\cup I\cup\{q_0\} and Q1=Q0∪JQ_1=Q_0\cup J the total ss-sum is 11 and the total λ\lambda-sum is 00, so that the product ll of the corresponding primes satisfies l≡3(mod4)l\equiv3\pmod4 and l≡2(mod3)l\equiv2\pmod3.

In plain English

Take one auxiliary prime q0q_0 of type (1,1)(1,1) (residue class 7 mod 12) and ∣B∣−1|B|-1 auxiliary primes of type (0,0)(0,0) (residue class 1 mod 12). Then on both Q0=B∪I∪{q0}Q_0=B\cup I\cup\{q_0\} and Q1=Q0∪JQ_1=Q_0\cup J, the total ss-sum is 11 and the total λ\lambda-sum is 00, so the product ll of these primes satisfies l≡3(mod4)l\equiv3\pmod4 and l≡2(mod3)l\equiv2\pmod3.

Why it's here

The fixed-prime sums from the previous claim (ss-sums zero on BB and JJ, λ\lambda-sum 11 on BB and 00 on JJ) show that q0q_0 alone supplies the missing ss-sum of 11. Its λ\lambda-contribution of 11 cancels the 11 from BB, while the type-(0,0)(0,0) primes in II contribute nothing. This yields the congruence conditions on ll for both twists l=nl=n and l=nθl=n\theta, and fixes the auxiliary prime types for the later matrix-kernel argument.
Classified routine:
Direct parity bookkeeping from the fixed symbol sums of the previous claim: adding one prime of type (1,1) and several of type (0,0) gives total s-sum 1 and λ-sum 0, which translates into the stated congruences mod 4 and mod 3 via the residue classes mod 12.

Uses

claimFixed symbol sums for B and J

Used by

claimMatrix target reduces to kernel of K-sharp

Sources

PaperOpen
Section
§3.4 One parameter for both twists
Page
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Label
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sections/03-parity.tex:570
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    outline: "Auxiliary prime types give required congruences" rests on claim-par-simultaneous-matrices-1