# Twin convention audit for the finite Kurihara certificate

Scope: $E:y^2+y=x^3+x^2-2x$, level $N=389$, coefficient field $k=\mathbb F_{11}$, conductor $n=397\cdot991$. This note fixes conventions before reading the producer's answer. No scalar will be selected to force the archived residue 6.

## Manin symbols and Hecke operators

Write $m(c:d)=g\{0,\infty\}$ for a determinant-one integral matrix $g$ with bottom row $(c,d)$ modulo $N$. Scalar multiples of a nonzero bottom row represent the same projective class. Stein's *Modular Forms, a Computational Approach*, Chapter 3, Proposition 3.10, the Manin presentation theorem, and the subsection “Hecke Operators on Modular Symbols” supply this presentation and the cusp-path Hecke action. Source: [Chapter 3](https://wstein.org/books/modform/modform/weight_two.html), [exact text source](https://wstein.org/books/modform/modform/_sources/weight_two.txt).

For $S=\left(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\right)$ and $R=\left(\begin{smallmatrix}0&-1\\1&-1\end{smallmatrix}\right)$, the row formulas are

$$m(c,d)+m(d,-c)=0,$$
$$m(c,d)+m(d,-c-d)+m(-c-d,c)=0.$$

The equally valid $R=\left(\begin{smallmatrix}0&-1\\1&1\end{smallmatrix}\right)$ gives $(c,d),(d,d-c),(d-c,-c)$ instead. A verifier may use a different triangle convention, since both describe the same integral path relations.

For $q\nmid N$,

$$T_q\{\alpha,\beta\}=\{q\alpha,q\beta\}+\sum_{t=0}^{q-1}\{(\alpha+t)/q,(\beta+t)/q\}.$$

If row $i$ of $H_q$ contains coefficients of $T_qm_i$, the functional-value vector satisfies $H_q\lambda=a_q\lambda$. Determinant-$q$ matrices acting on paths must be reduced by unimodular path decomposition, unless a proved Heilbronn/Merel formula is used; blindly reducing their bottom rows is invalid.

## Plus means the real-part functional

Kim v6, §1.4.1, defines

$$[r]^+=\operatorname{Re}\left(2\pi i\int_\infty^r f(z)\,dz\right)/\Omega_E^+.$$

This is the functional evaluated on $\{\infty,r\}$, not its reversed path. The geometric involution $z\mapsto-\bar z$ maps $m(c,d)$ to $m(-c,d)$. Because the Fourier coefficients are real, its action on $2\pi i\int f$ is complex conjugation. Thus the real-part functional obeys

$$\lambda(c,d)=\lambda(-c,d).$$

**Sign warning:** Stein Chapter 8's *star involution* has an additional minus sign. In weight two it is $m(c,d)\mapsto-m(-c,d)$, and Stein explicitly distinguishes Cremona's convention. Consequently Stein's star-plus label must not be substituted for the geometric plus condition above. This is a convention difference, not a mathematical contradiction. Source: [Chapter 8 text, “Pairing Modular Symbols and Modular Forms,” equations `eqn:star`, `eqn:iota` and the following remark](https://wstein.org/books/modform/modform/_sources/modular_symbols.txt).

## Kurihara sum

Use precisely

$$\delta_n^{(\lambda)}=\sum_{\substack{1\le a<n\\(a,n)=1}}\lambda\{\infty,a/n\}\,\log_5(a\bmod397)\,\log_6(a\bmod991)\pmod{11}.$$

The logs are reduced modulo 11 after specifying primitive roots 5 and 6. Source: [Kim, arXiv:2203.12159v6, §§1.4.1–1.4.3](https://arxiv.org/html/2203.12159v6). Multiplying a nonzero eigenvector by a scalar multiplies the sum by that scalar. The original archive does not disclose its deterministic normalization; matching its bare number 6 is therefore not a valid normalization rule.

## Independent path verification

The producer uses ordinary continued fractions. The independent verifier will instead use balanced Farey parents, a variant of the inductive proof of Stein's Proposition 3.11. For reduced $0<a<b$, set $r=a^{-1}\pmod b$, $q=\min(r,b-r)$, and choose $s\in\{1,-1\}$ so that $aq\equiv s\pmod b$. Put $t=(aq-s)/b$. Then

$$\{\infty,a/b\}=\{\infty,t/q\}+m(sb,q).$$

Indeed the matrix $\left(\begin{smallmatrix}sa&t\\sb&q\end{smallmatrix}\right)$ has determinant one and sends $0,\infty$ to $t/q,a/b$. The denominator decreases by at least a factor two. Integer cusps reduce to $\{\infty,0\}=m(1,0)$ by translation; infinity has zero path. This is an independent proof of the specific balanced recursion used, not an assumption about producer code.

The verifier will also use sparse scalar finite-field elimination rather than NumPy reduction and compute logarithms by projection to the order-11 subgroup rather than walking the full multiplicative group. Altered vector, rank, log, and sum artifacts must be rejected.