Gross vectors modulo 2 and elliptic curves of prime conductor

Abstract: Let p > 3 be a prime, and let S_p denote the geometric isomorphism classes of supersingular elliptic curves in characteristic p whose j-invariants lie in F_p. For each negative fundamental discriminant -D for which p is inert in Q(sqrt(-D)), let m_i(D), i in S_p, be the integral coefficients of the corresponding Gross vector. We prove that the vectors (m_i(D) mod 2)_{i in S_p} span F_2^{S_p}. The key step reduces the parity of the representation numbers of Gross's ternary lattices to representation by rank-two sublattices perpendicular to Frobenius. Using Ibukiyama's explicit maximal orders, the resulting primitive binary forms are identified with those occurring in the Xiao--Zhou--Deng--Qu parametrization of supersingular elliptic curves over F_p. Class field theory and Chebotarev's theorem then allow the individual supersingular coordinates to be isolated. As a consequence, if E/Q has prime conductor p and positive Mordell--Weil rank, then every coefficient of its Brandt eigenvector indexed by S_p is even, proving a conjecture of Kazalicki and Kohen. Thus an odd coefficient at a rational supersingular class is an algebraic certificate of rank 0. Combining this parity theorem with formulas of Mestre and Gross--Kudla, we also prove that the modular degree of every positive-rank elliptic curve of prime conductor and root number +1 is divisible by 4. Consequently, Watkins' conjecture holds for all such curves of rank 2.

• The paper proves that Gross vectors modulo 2 span the full coordinate space of Frobenius-fixed supersingular elliptic curves for every prime p>3, using binary quadratic forms and Chebotarev isolation.
• The paper reduces ternary representation parities to binary norm forms perpendicular to Frobenius, then uses explicit quaternionic orders, class field theory, and admissible discriminants to isolate individual coordinates.
• The paper shows that positive-rank elliptic curves of prime conductor have even Brandt eigenvector coefficients on all rational supersingular classes and proves 4 divides the modular degree when the root number is +1, including rank-two cases.

Overview and main results

This paper, by Kazalicki and Slijepčević (2608.13371), proves a spanning theorem for the mod-2 reductions of Gross vectors attached to supersingular elliptic curves in characteristic $p$, and derives two arithmetic consequences for elliptic curves of prime conductor: a parity criterion for Brandt eigenvector coefficients (confirming a conjecture of Kazalicki–Kohen) and a divisibility-by-four statement for modular degrees that establishes Watkins' conjecture in rank two.

Fix a prime $p>3$ and let $S$ be the set of geometric isomorphism classes of supersingular elliptic curves over $\overline{\mathbb F}_p$, with $S_p=\{i\in S: j(E_i)\in\mathbb F_p\}$ the Frobenius-fixed classes. The definite quaternion algebra $B_{p,\infty}$ ramified at $p$ and $\infty$ carries the Brandt module with basis $e_i$, $i\in S$. For each negative fundamental discriminant $p>3$0 inert at $p>3$1 (an admissible discriminant), Gross's CM vector has integral coefficients

$p>3$2

where $p>3$3, $p>3$4, and $p>3$5 counts optimal embeddings of $p>3$6 into $p>3$7 modulo conjugacy. The Gross-row space is $p>3$8. The main theorem asserts:

Theorem (Gross-row spanning). For every prime $p>3$9, $S$0.

The proof strategy is structural rather than computational. It reduces ternary representation parities to binary ones via Frobenius, identifies the resulting binary forms through Ibukiyama's explicit maximal orders and the Xiao–Zhou–Deng–Qu parametrization of supersingular curves over $S$1, and then uses class field theory and Chebotarev to construct admissible discriminants whose Gross rows isolate individual coordinates.

From elliptic curves to Gross relations

For an elliptic curve $S$2 of conductor $S$3, Jacquet–Langlands produces a primitive integral Brandt eigenvector $S$4 for the newform $S$5. Two standard ingredients convert positive Mordell–Weil rank into congruences on $S$6 restricted to $S$7.

First, the degree-$S$8 Brandt operator sends $S$9 (the Frobenius conjugate) and corresponds under Jacquet–Langlands to $\overline{\mathbb F}_p$0; since the Atkin–Lehner eigenvalue of a weight-two newform of prime level is $\overline{\mathbb F}_p$1, one obtains the Frobenius sign relation $\overline{\mathbb F}_p$2, so in particular $\overline{\mathbb F}_p$3.

Second, Gross's central-value formula gives, for admissible $\overline{\mathbb F}_p$4,

$\overline{\mathbb F}_p$5

By Kolyvagin–Logachev, positive rank forces $\overline{\mathbb F}_p$6, hence $\overline{\mathbb F}_p$7 exactly. Since $\overline{\mathbb F}_p$8 by Frobenius transport of optimal embeddings, nonrational coordinates cancel in pairs modulo 2, leaving the rational relation

$\overline{\mathbb F}_p$9

for every admissible $S_p=\{i\in S: j(E_i)\in\mathbb F_p\}$0. Thus $S_p=\{i\in S: j(E_i)\in\mathbb F_p\}$1, and the spanning theorem immediately yields the parity corollary: if $S_p=\{i\in S: j(E_i)\in\mathbb F_p\}$2 has prime conductor $S_p=\{i\in S: j(E_i)\in\mathbb F_p\}$3 and positive rank, then $S_p=\{i\in S: j(E_i)\in\mathbb F_p\}$4 for all $S_p=\{i\in S: j(E_i)\in\mathbb F_p\}$5. Consequently, a single odd coefficient at a rational supersingular class certifies $S_p=\{i\in S: j(E_i)\in\mathbb F_p\}$6. This is explicitly a one-sided criterion; the paper notes the converse fails, e.g. the rank-zero conductor-37 curve $S_p=\{i\in S: j(E_i)\in\mathbb F_p\}$7 has all coefficients even on $S_p=\{i\in S: j(E_i)\in\mathbb F_p\}$8.

Two exceptional coordinates require no class field theory: when $S_p=\{i\in S: j(E_i)\in\mathbb F_p\}$9 is supersingular ($B_{p,\infty}$0), the row $B_{p,\infty}$1; when $B_{p,\infty}$2 is supersingular ($B_{p,\infty}$3), $B_{p,\infty}$4. These come from direct embedding counts, e.g. $B_{p,\infty}$5.

Frobenius reduction to binary quadratic forms

The technical heart is the observation that, at a rational supersingular class, the parity of $B_{p,\infty}$6 is governed by a binary lattice. Fix $B_{p,\infty}$7 with an $B_{p,\infty}$8-model whose Frobenius $B_{p,\infty}$9 satisfies $p$0, $p$1. Inside Gross's trace-zero ternary lattice $p$2, consider the Frobenius-perpendicular lattice $p$3.

Conjugation by $p$4 preserves $p$5 and the reduced norm; its $p$6-eigenspace is $p$7 and its $p$8-eigenspace is $p$9. A norm-$\infty$0 vector with $\infty$1 cannot lie on the Frobenius line (else $\infty$2 has odd $\infty$3-adic valuation). Vectors outside both eigenspaces therefore group into four-element orbits under $\infty$4 and negation, giving $\infty$5. Combined with the normalization $\infty$6, this yields the key congruence

$\infty$7

The paper frames this as an instance of a general mod-4 theta-series factorization $\infty$8 for lattices with norm-preserving involutions. The upshot is that Frobenius converts a ternary representation problem into a binary one — the decisive simplification underlying everything that follows.

Explicit forms via Ibukiyama models

Identifying the norm form on $\infty$9 requires tracking not just the abstract order $e_i$0 but the quadratic order generated by Frobenius, $e_i$1, which is either $e_i$2 or $e_i$3. Using Ibukiyama's explicit maximal orders in $e_i$4 built from an auxiliary prime $e_i$5 inert at $e_i$6, a Skolem–Noether argument shows the marked pair $e_i$7 is inner-conjugate to $e_i$8 or $e_i$9 according to which case holds.

Direct computation of perpendicular bases gives two branches:

Branch	Condition	Norm form	Discriminant
$i\in S$0	always	$i\in S$1	$i\in S$2
$i\in S$3	$i\in S$4	$i\in S$5	$i\in S$6

Both primitive parts are primitive positive-definite binary quadratic forms. The factor 4 in the second branch matters: in the $i\in S$7 branch, $i\in S$8 represents only multiples of 4, so odd $i\in S$9 contribute zero parity there.

Comparing with the Xiao–Zhou–Deng–Qu parametrization of endomorphism rings of supersingular curves over $p>3$00 shows that the proper class $p>3$01 of the form attached to $p>3$02 maps injectively to the geometric class via the inverse orbit $p>3$03 (inverse classes correspond to quadratic twists over $p>3$04), and that the $p>3$05 branch lands in the nonprincipal genus. This yields a clean dictionary: for admissible $p>3$06, $p>3$07 in the $p>3$08 branch, while in the $p>3$09 branch the same holds with $p>3$10 replaced by $p>3$11.

A genus-theoretic lemma supplies admissibility: if $p>3$12 is squarefree, coprime to $p>3$13, and represented by a class in the nonprincipal genus of discriminant $p>3$14, then $p>3$15 and $p>3$16, so $p>3$17 is automatically admissible.

Chebotarev isolation of coordinates

The form–ideal correspondence identifies $p>3$18 with the number of proper ideals of norm $p>3$19 in the ideal class corresponding to $p>3$20. For split primes this is rigid: a prime $p>3$21 splitting in $p>3$22 has exactly two proper ideals of norm $p>3$23, with inverse classes. Chebotarev applied to the ring class field, using that the conjugacy class of $p>3$24 in the dihedral group $p>3$25 is $p>3$26, produces infinitely many such primes realizing any prescribed inverse orbit.

The proof of the spanning theorem then proceeds in two stages. For $p>3$27 in the $p>3$28 branch, choose $p>3$29 whose ideal classes are $p>3$30; the genus condition makes $p>3$31 admissible, and the corresponding Gross row restricts to $p>3$32 on the nonexceptional $p>3$33 coordinates (and vanishes on the $p>3$34 branch since $p>3$35 is odd). For $p>3$36 in the $p>3$37 branch (only when $p>3$38), a refined Chebotarev argument — showing $p>3$39 because the Hilbert class field is unramified at finite primes while $p>3$40 ramifies above 2 — produces a prime $p>3$41 splitting in $p>3$42 with prescribed ideal class; then $p>3$43 is admissible and its Gross row isolates $p>3$44 modulo the already-generated $p>3$45 subspace. Together with the exceptional rows for $p>3$46, this exhausts all of $p>3$47.

Beyond what the theorem requires, the paper records a semiprime support phenomenon: since any two classes in the same genus satisfy $p>3$48, one can write $p>3$49, $p>3$50 and realize $p>3$51 by primes above distinct rationals $p>3$52; the four ideals of norm $p>3$53 then have classes $p>3$54, giving an admissible Gross row with support exactly $p>3$55 modulo exceptional coordinates. Every pair of nonexceptional $p>3$56-branch coordinates is thus realized by a single semiprime row.

Consequences for modular degree and Watkins' conjecture

Watkins conjectured that $p>3$57, where $p>3$58 is the minimal degree of a modular parametrization $p>3$59. Combining the parity corollary with Mestre's norm formula and the Gross–Kudla cubic identity (following the earlier argument of Kazalicki–Kohen), the paper proves:

Theorem. If $p>3$60 has prime conductor $p>3$61, positive rank, and root number $p>3$62, then $p>3$63. In particular, Watkins' conjecture holds for such curves of rank 2.

The passage from the $p>3$64-optimal curve to arbitrary curves in the isogeny class uses Calegari–Emerton's observation that a translated parametrization factors through the optimal curve, so $p>3$65. Note the root-number restriction: the divisibility statement is proved only for $p>3$66; the parity theorem itself applies regardless of root number, but the Mestre–Gross–Kudla mechanism used here requires it.

The example $p>3$67

The prime 83 illustrates both branches concretely. Here $p>3$68 with $p>3$69, the $p>3$70 branch contributes $p>3$71 and the $p>3$72 branch $p>3$73. The row at $p>3$74 isolates coordinate 28 (only $p>3$75 represents 7 among the relevant classes), and the row at $p>3$76 has nonexceptional support $p>3$77, generating $p>3$78 once $p>3$79 is known. Semiprime rows realize all three pairs: $p>3$80 gives $p>3$81, $p>3$82 gives $p>3$83, and $p>3$84 gives $p>3$85, matching the general proposition.

Limitations and open questions

Several restrictions bound the scope of the results. The spanning theorem and parity corollary concern prime conductor only; the extension to composite conductors, where the Brandt module decomposes less simply and the Frobenius sign argument has no direct analogue, remains open. The modular-degree application additionally requires root number $p>3$86; whether the Mestre–Gross–Kudla route can be adapted to root number $p>3$87 curves is not addressed. The parity criterion is one-sided — evenness of all $p>3$88 does not imply positive rank — and the paper offers no characterization of the rank-zero curves failing the converse. Finally, the semiprime support phenomenon is established only for the $p>3$89 branch; an analogous statement for the $p>3$90 branch is not developed.

Conclusion

The paper establishes that Gross vectors modulo 2 span the full coordinate space on Frobenius-fixed supersingular classes, by reducing ternary lattice parities to binary forms perpendicular to Frobenius and controlling those forms through ring class fields and Chebotarev. This yields a clean algebraic certificate of Mordell–Weil rank zero for prime-conductor elliptic curves, confirms the Kazalicki–Kohen conjecture without its previous discriminant and 2-torsion hypotheses, and proves Watkins' conjecture for rank-two curves of prime conductor and root number $p>3$91.