claimroutineinferred_dep

Realize prescribed symbols by Dirichlet primes

Realizes prescribed residues and symbols by Dirichlet primes, giving

n

final construction of n

Claim·§3.4 One parameter for both twists·inside lemOne squarefree n serving both twists

Statement (paper)

The prescribed types modulo

12

and the prescribed Legendre symbols among the primes of

B\cup J\cup I\cup\{q_0\}

can be realized by choosing the auxiliary primes successively, each distinct from all previously used primes, so that the resulting product

n=\prod_{q\in B\cup I\cup\{q_0\}}q

has all the asserted properties.

In plain English

One can pick the auxiliary primes one at a time, each new and distinct from earlier ones, so that the required residue types modulo 12 and the required Legendre symbols among the primes of

B\cup J\cup I\cup\{q_0\}

hold. The product

n=\prod_{q\in B\cup I\cup\{q_0\}}q

then has all the properties asserted in the lemma.

Why it's here

This is the final construction step of the lemma. The earlier claims fix which symbols and congruence types are needed, including the bits

[q_0/q_j]

for

j\in J

that make the deleted matrices nonsingular. Here those requirements are met by successive choices of primes, which gives the

n

with

l\equiv3\pmod4

l\equiv2\pmod3

and the matrix target for both

l=n

and

l=n\theta

Classified routine:

Standard successive choice of primes via Dirichlet's theorem on primes in arithmetic progressions combined with quadratic reciprocity and CRT, so each new prime has prescribed residues and Legendre symbols. This is the usual bookkeeping in Selmer-rank/2-descent constructions.

Uses

claimChoosing J bits makes enlarged matrix nonsingular

Used by

Nothing in the paper

Sources

PaperOpen

Section: §3.4 One parameter for both twists
Page: —
Label: —
Line: sections/03-parity.tex:692

Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

info
inferred_dep
outline: "Realize prescribed symbols by Dirichlet primes" rests on claim-par-simultaneous-matrices-7