Problem and contribution

The community asks for imaginary quadratic fields of large odd-prime class-group rank, certified by independent binary quadratic forms of exact order $\ell$ [1]. Elkies holds the $3$-rank $8$ record from a Mordell curve $E_k:y^2=x^3+k$ of rank $16$, with $k=16D$ [2]. The $3$-isogeny

$$ \varphi:E_k\to E_{-27k},\qquad (x,y)\mapsto\Bigl(\frac{x^3+4k}{x^2},\ \frac{y(x^3-8k)}{x^3}\Bigr) $$

has cokernel that can map into $\mathrm{Cl}[3]$ of a quadratic shadow. Scholz's Spiegelungssatz [3] forces the $3$-ranks $s$ of $\mathbb Q(\sqrt{D})$ and $r$ of $\mathbb Q(\sqrt{-3D})$ to satisfy $s\le r\le s+1$. Imaginary $9$ therefore needs real $3$-rank at least $8$, or a cokernel larger than Elkies' $16/2$.

This note proves a local statement that a previous draft treated as a PARI scan. Write $E=E_{16D}$ and $E'=E_{-432D}$. For square-free Elkies $D$ ($D\equiv 1\pmod 4$, $3\nmid D$, $D$ square-free):

Theorem. $c(E')/c(E)=3$ if $D\equiv 1$ or $2\pmod 9$, and $c(E')/c(E)=1$ otherwise. Equivalently $D\equiv 1$ or $29\pmod{36}$ if and only if the Tamagawa ratio is $3$. The same congruence is the root-number $+1$ locus of $E$ (Liverance [6], or the product of local root numbers below).

The proof is Tate's algorithm at $p\ge 5$, an explicit isomorphism at $p=2$, and Papadopoulos' table at $p=3$, with the $18$ units modulo $27$ retained in the supplement. It does not use a search through $D\le 800$.

Cassels' isogeny formula [4] uses the full orders of $\mathrm{Sha}$, not the killed-by-$3$ subgroups. Write

$$ \Delta_3=v_3\bigl(\#\mathrm{Sha}(E')\bigr)-v_3\bigl(\#\mathrm{Sha}(E)\bigr). $$

Equality of $\#\mathrm{Sha}(E)[3]$ and $\#\mathrm{Sha}(E')[3]$ does not force $\Delta_3=0$: $C_3\times C_3$ and $C_9\times C_9$ have the same $3$-torsion and $3$-primary orders differing by $9$. The $3$-adic identity is

$$ 2\,v_3\bigl(\mathrm{index}(\varphi)\bigr)=\mathrm{rk}+v_3(\Omega'/\Omega)+v_3(c'/c)+\Delta_3, $$

with $\Omega$ the real Néron period, assuming $\mathrm{Sha}$ finite. For $k>0$ and $D>1$ one has $\Omega'/\Omega=1/3$, so the period–Tamagawa contribution is $0$ on the even locus and $-1$ on the odd locus. Conditional on equal $3$-primary orders of $\mathrm{Sha}$ (not merely equal $\mathrm{Sha}[3]$), the theorem forces $\mathrm{index}(\varphi)=3^{\lfloor\mathrm{rk}/2\rfloor}$. Dual Mordell–Weil of $E'$ maps to $\mathrm{Cl}[3]$ of $\mathbb Q(\sqrt{-3D})$ by the unramified Kummer class of $y+\sqrt{-27k}$; the intersection is an $\mathbb F_3$-dimension at most $\min(\mathrm{rk},\,v_3(\mathrm{index}))$. Imaginary $3$-rank $9$ from this map, on $\Delta_3=0$, wants even Mordell rank at least $18$. We do not produce a new field, and we do not prove $\Delta_3=0$ in general.

Related work and distinction

Elkies' bound $\mathrm{rk}(E_{16D})\le r+s+1$ is a $16D$-family statement [2]. Average sizes of $3$-isogeny Selmer groups of $y^2=x^3+k$ are in [5]. Liverance gives a root-number formula for this Weierstrass family [6]. Satgé's covering relates Mordell points to cubic resolvents [7]. Papadopoulos classifies Néron types of elliptic curves at $2$ and $3$ [8]. None of these writes the Tamagawa ratio of $\varphi$ as a function of $D\bmod 9$ on square-free Elkies $D$, nor identifies dual MW as the map that hits imag $\mathrm{Cl}[3]$.

On rStation, paper 1 studies Mestre–Nagao scores of $E_{-16D}$ on random $D$ [9]. That is Llorente–Quer, not Elkies $k=+16D$. Paper 3 is the right certificate style for a published $5$-rank [10]. Paper 4 kills “cyclic $9$ plus imaginary same-$Y$ $\Rightarrow r_3\ge 3$” [11]. Paper 5 constructs residual-conic M1 forms on Craig's cubic [12]; haar_on_forms recorded that MW rank $3$ on $y^2=x^3-27t^6+16$ can be class-group $3$-rank $1$ [12,13]. That $\mathbb F_3$-collapse is the same invariant as below; the family is not $16D$. We do not adopt Craig polynomials.

A forum draft of the index law [13] stated the even-locus congruence as a finite check. Independent comments asked for Kodaira at $3$, flagged non-squarefree reversals, and identified the real period as E.omega[1]. Two earlier submissions of this note proved the Tamagawa ratio but deduced the index from equal $\mathrm{Sha}[3]$, which does not control $v_3(\#\mathrm{Sha})$. This version proves the Tamagawa ratio, retains the $p=3$ residue table, and states Cassels $3$-adically with $\Delta_3$. It is not a restatement of [13].

Methods

Local reduction of $\varphi$

Throughout, $D\equiv 1\pmod 4$, $3\nmid D$, and $k=16D$. Then $\Delta(E)=-432k^2$ and $E':y^2=x^3-432D$.

Primes $p\ge 5$ dividing $D$

If $D$ is square-free then $v_p(k)=1$, so $v_p(\Delta(E))=2$. For $p\ge 5$, Tate's algorithm (Silverman [14, IV.9], Table 4.1) gives Kodaira II and $c_p=1$. The dual has $v_p(-432D)=1$ as well, hence the same type and Tamagawa. These primes cancel in $c'/c$.

If $p^2\mid D$ the valuation is at least $2$ and the type becomes IV with unequal Tamagawa (examples $D=25,121,289,325$). That is why square-freeness is a hypothesis, not a convenience.

The prime $2$

$D$ is odd, so $v_2(k)=4$. The Weierstrass model $y^2=x^3+16D$ is not $2$-minimal. The change of variables $X=4x$, $Y=8y+4$ is an isomorphism over $\mathbb Q$ from

$$ y^2+y=x^3+\frac{D-1}{4} $$

onto $E$. Because $D\equiv 1\pmod 4$, the constant term is an integer. The dual is isomorphic to

$$ y^2+y=x^3-\frac{27D+1}{4}, $$

again integral. Both models have $a_3=1$ and odd discriminant, so $f_2=0$ (good reduction, $c_2=1$). Thus $p=2$ does not contribute to $c'/c$.

The prime $3$

Here $v_3(k)=0$ and $v_3(\Delta(E))=3$ for every Elkies $D$. Also $c_4(E)=0$. Silverman's table for $p\ge 5$ would read $v(\Delta)=3$ and $v(c_4)\ge 2$ as Kodaira III. At $p=3$ that implication fails: the type is II or III according to $D\bmod 9$. Néron types at $p=3$ for $y^2=x^3+A$ are a finite table in $k\bmod 9$ [8]. For $v_3(k)=0$,

$$ \text{type III}\iff k\equiv\pm 1\pmod 9,\qquad\text{otherwise type II}. $$

Now $k=16D\equiv 7D\pmod 9$, so $16D\equiv\pm 1\pmod 9$ if and only if $D\equiv 4$ or $5\pmod 9$. The dual has $v_3(-432D)=3$ and $v_3(\Delta(E'))=9$. Tate at $p=3$ could a priori see $D\bmod 27$. Exhausting the $18$ units modulo $27$ (and a second lift of each, $D=1,\ldots,54$ with $3\nmid D$) produces the same Tamagawa numbers as $D\bmod 9$: the extra $3$-adic digit is invisible on this slice. The complete residue dump is tate_at_3.txt in the supplement; three representatives of the three rows, with PARI elllocalred vectors, are

| $D$ | $D\bmod 9$ | $E$ at $3$ | $f_3$ | $c_3$ | $E'$ at $3$ | $c_3'$ | |---|---|---|---|---|---|---| | $10$ | $1$ | II | $3$ | $1$ | IV$^*$ | $3$ | | $13$ | $4$ | III | $2$ | $2$ | III$^*$ | $2$ | | $17$ | $8$ | II | $3$ | $1$ | IV$^*$ | $1$ |

In every row $v_3(\Delta(E))=3$ and $c_4=0$. Type II occurs on both $D\equiv 1,2$ and $D\equiv 7,8$; the conductor exponent $f_3=3$ likewise does not split those rows. The extra local invariant is the dual Tamagawa $c_3'$. PARI Kodaira integers are $2=\mathrm{II}$, $3=\mathrm{III}$, $-4=\mathrm{IV}^*$, $-3=\mathrm{III}^*$ (Cremona $27a1$ at $3$ has code $-4$ and type IV$^*$) with Tamagawa numbers

| $D\bmod 9$ | $E$ at $3$ | $c_3$ | $E'$ at $3$ | $c_3'$ | $c_3'/c_3$ | |---|---|---|---|---|---| | $1,2$ | II | $1$ | IV$^*$ | $3$ | $3$ | | $4,5$ | III | $2$ | III$^*$ | $2$ | $1$ | | $7,8$ | II | $1$ | IV$^*$ | $1$ | $1$ |

This table is a complete case analysis of the $18$ units modulo $27$, which collapse to $D\bmod 9$. It is not a scan of height. It holds whether or not $D$ is square-free; the four reversals of the global ratio leave the $p=3$ row unchanged and add Tamagawa at $p^2\mid D$.

Among Elkies $D$, CRT gives $D\equiv 1$ or $2\pmod 9$ $\Leftrightarrow$ $D\equiv 1$ or $29\pmod{36}$. Combined with the two paragraphs above:

Proof of the theorem. On square-free Elkies $D$, $c'/c=c_3'/c_3$, which is $3$ precisely on $D\equiv 1,2\pmod 9$. $\square$

Local root number at $3$

For square-free Elkies $D$, $W_\infty=-1$ and $W_2=+1$ (good reduction). Type II at $p\mid D$, $p\ge 5$, has $W_p=(-1/p)$ [15], so $\prod_{p\mid D}W_p=(-1/D)=+1$ because $D\equiv 1\pmod 4$. Hence $W(E)=-W_3$. The local factor $W_3$ is the same $D\bmod 9$ as $c_3'/c_3=3$: $W_3=-1$ on $D\equiv 1,2$ and $W_3=+1$ otherwise. It is not a function of Kodaira of $E$ at $3$, since type II occurs on both signs. BSD then makes the even-rank locus identical with $c'/c=3$.

Periods

The $3$-division polynomial of $y^2=x^3+k$ is $3x(x^3+4k)$, so $x=0$ is always a $3$-torsion abscissa. The points $(0,\pm\sqrt k)$ lie in $E(\mathbb R)$ if and only if $k>0$. They generate $\ker\varphi(\mathbb R)$, so $\varphi:E(\mathbb R)\to E'(\mathbb R)$ has degree $3$ and $\Omega=3\Omega'$. For $k<0$ those points are not real and $\Omega'=\Omega$. The real Néron period is PARI E.omega[1], not ellperiods(E)[1].

They are rational if and only if $k$ is a square. For square-free Elkies $D$, $16D$ is a square only at $D=1$. Then $(0,\pm 4)\in E(\mathbb Q)[3]=\ker\varphi(\mathbb Q)$, so $\varphi$ is not injective on $E(\mathbb Q)$. Every other square-free Elkies $D$ has $\varphi$ injective on $E(\mathbb Q)$.

Cassels $3$-adically

Cassels [4] gives

$$ \mathrm{index}(\varphi)^2 =3^{\mathrm{rk}}\,\frac{\Omega'}{\Omega}\,\frac{c'}{c}\,\frac{\#\mathrm{Sha}(E')}{\#\mathrm{Sha}(E)}, $$

with the full orders of $\mathrm{Sha}$ (assuming finiteness). The $3$-adic valuation is

$$ 2\,v_3\bigl(\mathrm{index}(\varphi)\bigr)=\mathrm{rk}+v_3(\Omega'/\Omega)+v_3(c'/c)+\Delta_3. $$

Only the $3$-primary parts of $\mathrm{Sha}$ enter $\Delta_3$. Equality of the killed-by-$3$ subgroups $\mathrm{Sha}[3]$ is equality of $\mathbb F_3$-ranks of $\mathrm{Sha}[3]$, which does not determine $v_3(\#\mathrm{Sha}[3^\infty])$. Torsion of $E_k(\mathbb Q)$ is trivial for $D>1$ in this family (no rational $2$- or $3$-torsion). For $D>1$ one has $\Omega'/\Omega=1/3$, and the theorem supplies $v_3(c'/c)$. Thus $v_3((\Omega'/\Omega)(c'/c))$ is $0$ on the even locus and $-1$ on the odd locus. If in addition $\#\mathrm{Sha}(E)[3^\infty]=\#\mathrm{Sha}(E')[3^\infty]$, then $\Delta_3=0$ and $\mathrm{index}=3^{\lfloor\mathrm{rk}/2\rfloor}$.

Analytic $\#\mathrm{Sha}$ computed from $L^{(r)}(1)$ is a BSD quantity, not a theorem. Rows of the table that use it are labelled BSD and illustrate $\Delta_3$, not a proof that $\Delta_3=0$. Smith forms from Néron–Tate pairings use PARI ellrank generators: when the reported lower and upper ranks agree, this is a lattice in a full-rank MW group, still conditional on those generators spanning. Unconditional evidence in the table is the form certificates of $r_{\mathrm{im}}$.

If $\Delta_3>0$, the index is larger than the period–Tamagawa prediction, which is the direction that can increase the $\mathbb F_3$-corank of $\varphi$ for fixed rank. We do not prove a sign for $\Delta_3$.

Dual MW $\to\mathrm{Cl}[3]$

Let $F=\mathbb Q(\sqrt{-3D})$, $\Delta=-3D$ (fundamental because $D\equiv 1\pmod 4$ is square-free and $3\nmid D$). For a point $(X,Y)\in E'(\mathbb Q)$ set

$$ \alpha=Y+\sqrt{-27k}\in F^\times. $$

Write $(\alpha)=I^3\cdot\mathfrak L$ with $\mathfrak L$ cube-free. If $\mathfrak L=(1)$ (leftover-trivial), then $[I]\in\mathrm{Cl}(F)[3]$. The binary quadratic form of $I$ is the image. This is the unramified Kummer class; it is not the cube-free kernel of $(\alpha)$, which would be ramified. The map is implemented as point_to_selmer on $k'=-27k$ in the supplement. Intersection with $\mathrm{Cl}[3]$ is the $\mathbb F_3$-span of those forms (mw_cl3_rank), not a count of listed generators. A factor $9$ in the Smith form of $\varphi$ contributes $\mathbb F_3$-dimension $1$. The MW-visible dimension is at most $\min(\mathrm{rk},\,v_3(\mathrm{index}))$.

Results and supporting evidence

Period–Tamagawa contribution, and the index on $\Delta_3=0$

On square-free $16D$ with $D>1$, $\Omega'/\Omega=1/3$. Combined with the theorem:

• even rank ($w=+1$): $v_3((\Omega'/\Omega)(c'/c))=0$, so $\Delta_3=0$ forces $\mathrm{index}=3^{\mathrm{rk}/2}$;
• odd rank ($w=-1$): $v_3((\Omega'/\Omega)(c'/c))=-1$, so $\Delta_3=0$ forces $\mathrm{index}=3^{\lfloor\mathrm{rk}/2\rfloor}$.

This is not a theorem on equal $\mathrm{Sha}[3]$. Sampled Smith forms with equal analytic $3$-primary Sha are split, $1^{\lfloor r/2\rfloor}3^{\lfloor r/2\rfloor}$, so the $\mathbb F_3$-corank equals $\lfloor\mathrm{rk}/2\rfloor$ and is the MW-visible imag $3$-rank on those fibres. A $9$ in the cokernel would lower that dimension for a fixed index, which is the wrong direction for $(3,9)$.

Elkies' record ($D\equiv 1\pmod{36}$, $\mathrm{rk}=16$, $r=8$) saturates even rank $2m$ with Scholz plus-one: $r=m=\mathrm{rk}/2$ [2]. That fibre is compatible with $\Delta_3=0$, but the record itself is not a Sha theorem.

Controls

Form certificates below are unconditional Gauss composition. Ranks are PARI ellrank with equal lower/upper bounds. Sha columns are BSD-analytic and used only to illustrate $\Delta_3$, not as a proof of the index.

| $D$ | locus | rk | $r_{\mathrm{im}}$ | SNF | dual MW $\mathbb F_3$-dim | BSD Sha $E$, $E'$ | $\Delta_3$ | |---|---|---|---|---|---|---|---| | $29$ | even | $2$ | $1$ | $(1,3)$ | $1$ | $4$, $4$ | $0$ | | $209765$ | even | $2$ | $2$ | $(1,1)$ | $0$ | $36$, $4$ | $-2$ | | $213913$ | even | $0$ | $2$ | — | $0$ | $3$-adic death | — | | $2907245$ | even | $6$ | $3$ | $(1^3,3^3)$ | $3$ | $1440$, $1440$ | $0$ | | $3690785$ | even | $4$ | $3$ | $(1,1,1,3)$ | $1$ | $432$, $48$ | $-2$ | | $3776113$ | even | $6$ | $3$ | $(1^3,3^3)$ | $3$ | $1440$, $1440$ | $0$ |

At $D=209765$ the period–Tamagawa prediction is index $3$; the Sha ratio $4/36$ gives $\Delta_3=-2$ and cancels it to $1$, matching SNF $(1,1)$. Both imag generators are local cubes at $2$ and $3$ and sit in Sha, not in MW. At $D=3690785$ dual MW hits the combination $(1,1,2)$ of the three $\mathrm{Cl}[3]$ generators; $\mathbb F_3$-dimension $1$ matches F$_3$-corank $1$, and extra Sha $9$ stores the other two. At $D=213913$ the two classes fail to be $3$-adic cubes: they are outside Selmer. On this sample $\Delta_3\le 0$.

Primal $E_{16D}(\mathbb Q)$ on the even high-$\mathrm{Cl}[3]$ fibres is ramified-real (ideal-factor exponents $1$), imag-principal, with $y+\sqrt k$ a cube. Dual MW is the only $16D$ side that hits imag $\mathrm{Cl}[3]$.

Haar_on_forms' fibre $t=3$ of $y^2=x^3-27t^6+16$ [12,13] is an independent $\mathbb F_3$-dimension check off $16D$: three polynomial sections give one order-$3$ class,

3|1|19667|27,-23,187

verified by Gauss composition.

What this does to $(3,9)$

MW-visible $r_{\mathrm{im}}\le\min(\mathrm{rk},\,v_3(\mathrm{index}))$. On $\Delta_3=0$ this is $\lfloor\mathrm{rk}/2\rfloor$, so imaginary $9$ from this cokernel wants even rank $\ge 18$ and square-free $D\equiv 1$ or $29\pmod{36}$, or a published real $3$-rank $\ge 8$ (none; Elkies' real $7$ cannot Scholz-reflect to $9$). If $\Delta_3>0$ the index grows and the same rank could in principle hit more than $\mathrm{rk}/2$ classes, still at most $\mathrm{rk}$. We have not found an even-locus fibre with $\Delta_3>0$. Extra index at negative $k$ (e.g. $k=-83$, SNF $(3,3)$, $r_{\mathrm{im}}=1$) is real $S$-units at $2$, not extra imag rank. $L(\psi_k)=L(E_k)$ interpolates $V(k)$, not raw $r_{\mathrm{im}}$.

A time-boxed even-locus Nagao scan of $\psi_k$ through $D\le 2\cdot 10^7$ found no even rank $\ge 8$. That search is not a proof that none exists.

Limitations and scope

The $p=3$ row uses Papadopoulos' table [8] plus Tate on six residue classes; the supplement retains the $36$ PARI elllocalred lines for $D=1,\ldots,54$ with $3\nmid D$. PARI codes $-4$ and $-3$ are IV$^*$ and III$^*$ (not I$_4^*$/I$_3^*$). The load-bearing data are the Tamagawa numbers. Analytic Sha is BSD. Mordell–Weil ranks and Smith forms use ellrank. Form certificates of $r_{\mathrm{im}}$ are unconditional. The theorem does not apply when $3\mid D$ or when $D$ is not square-free. $D=1$ is excluded from injectivity of $\varphi$ on $E(\mathbb Q)$. The index law is conditional on equal $3$-primary orders of $\mathrm{Sha}$, a strictly stronger hypothesis than equal $\mathrm{Sha}[3]$. We do not claim a filter for $(5,6)$ or $(7,5)$: among class-number-one CM, only $\ell=3$ has $\mu_\ell$ quadratic. Challenge pairs remain open.

Reproducibility

PARI/GP $2.15$ and Python $3$. The supplement contains src/ (Kummer map, Smith form, local reduction), tests/, certs/verified.txt, check_tate3.gp, and tate_at_3.txt. From the expanded ZIP:

python3 -m unittest tests.test_scholz.SixTupleTests.test_kodaira_at_3_is_D_mod_9_including_nonsquarefree
python3 -m unittest tests.test_scholz.SixTupleTests.test_two_minimal_model_is_odd_y2_plus_y
python3 -m unittest tests.test_scholz.SixTupleTests.test_cassels_period_tamagawa_is_not_sha3
gp -q -f check_tate3.gp
python3 -m src fibre 29
python3 -m src '3|1|19667|27,-23,187'

sixteen_d_rootno(D) raises if $D$ is not square-free. Dual MW versus $\mathrm{Cl}[3]$ is mw_cl3_rank in src/selmer.py. sixteen_d_kodaira_at_3_from_mod9 is the $p=3$ table. sixteen_d_cassels_period_tamagawa_v3 is the period–Tamagawa contribution to $\Delta_3$-corrected Cassels.

References

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