Ishii's Conjecture: Elliptic Curves & Surface Singularities

• Ishii's Conjecture is a term describing two distinct frameworks: one in arithmetic that predicts asymptotic behavior of Frobenius trace sums for elliptic curves, and one in geometry that classifies resolutions of surface singularities via moduli spaces.
• In the elliptic curve setting, the conjecture predicts that a logarithmically weighted prime sum converges to -r + ½, directly relating the limit to the analytic rank and underlying BSD expectations.
• In the surface singularity formulation, the conjecture characterizes resolutions of quotient singularities as projective moduli spaces of θ-stable G-constellations, with proofs established for dihedral reflection groups using Bridgeland stability.

Searching arXiv for the specified papers and topic to ground the article in current literature. "Ishii's conjecture" designates two distinct conjectural statements in contemporary mathematics. In the arithmetic of elliptic curves over $\mathbb{Q}$, it denotes the one-curve "Mestre–Nagao/Ishii" prediction that a logarithmically weighted prime sum of Frobenius traces converges to a constant determined by the analytic rank, namely $-r+\tfrac12$, where $r=\operatorname{ord}_{s=1}L_E(s)$; equivalently, an unweighted prime sum should have main term $-r\log\log x$ (Kim et al., 2021). In birational geometry and representation theory, it denotes a statement for quotient surface singularities $\mathbb{C}^2/G$ asserting that a resolution is dominated by the maximal resolution of the pair $(\mathbb{C}^2/G,B)$ if and only if it arises as a projective moduli space $M_\theta$ of $\theta$-stable $G$-constellations for some generic $\theta\in\Theta(G)$; this conjecture has been proved for dihedral reflection groups by a Bridgeland-stability approach (Nimura, 21 May 2026).

1. Terminological scope and disambiguation

The shared name masks two mathematically unrelated settings.

Setting	Core object	Conjectural statement
Elliptic curves over $-r+\tfrac12$0	Prime sums of Frobenius traces $-r+\tfrac12$1	$-r+\tfrac12$2
Quotient surface singularities	Resolutions of $-r+\tfrac12$3 and moduli of $-r+\tfrac12$4-constellations	A resolution is dominated by the maximal resolution iff $-r+\tfrac12$5 for some generic $-r+\tfrac12$6

In the elliptic-curve literature, the conjecture is tied to Birch–Swinnerton-Dyer, explicit-formula methods, and Nagao-type averages. In the surface literature, it is a two-dimensional analogue of the Craw–Ishii conjecture and is formulated in terms of birational models, klt pairs, and moduli of $-r+\tfrac12$7-constellations. The identical name therefore refers to two separate research programs rather than to a single conjecture with multiple formulations.

2. Elliptic-curve formulation over $-r+\tfrac12$8

Let $-r+\tfrac12$9 be an elliptic curve with discriminant $r=\operatorname{ord}_{s=1}L_E(s)$0 and conductor $r=\operatorname{ord}_{s=1}L_E(s)$1. For a prime $r=\operatorname{ord}_{s=1}L_E(s)$2 of good reduction, $r=\operatorname{ord}_{s=1}L_E(s)$3, write

$r=\operatorname{ord}_{s=1}L_E(s)$4

so that $r=\operatorname{ord}_{s=1}L_E(s)$5 satisfies Hasse's bound $r=\operatorname{ord}_{s=1}L_E(s)$6 (Kim et al., 2021). At bad primes $r=\operatorname{ord}_{s=1}L_E(s)$7, the finitely many terms can be omitted from prime sums without affecting the limiting statements.

The associated Hasse–Weil $r=\operatorname{ord}_{s=1}L_E(s)$8-function is

$r=\operatorname{ord}_{s=1}L_E(s)$9

absolutely convergent for $-r\log\log x$0. Writing $-r\log\log x$1 and $-r\log\log x$2 for $-r\log\log x$3, one also has

$-r\log\log x$4

with $-r\log\log x$5. By modularity, $-r\log\log x$6 extends to an entire function and satisfies the functional equation

$-r\log\log x$7

where

$-r\log\log x$8

and $-r\log\log x$9.

The limit studied in the paper is

$\mathbb{C}^2/G$0

Ishii's conjecture in this setting states that

$\mathbb{C}^2/G$1

where $\mathbb{C}^2/G$2 is the analytic rank, expected by BSD to equal $\mathbb{C}^2/G$3. An equivalent formulation, obtained by integration by parts, is

$\mathbb{C}^2/G$4

for some constant $\mathbb{C}^2/G$5.

A recurring source of confusion is sign normalization. Some authors define the weighted sum with an overall minus sign or normalize differently; the 2021 paper fixes the convention above and therefore states the conjectural limit as $\mathbb{C}^2/G$6 rather than $\mathbb{C}^2/G$7. The two presentations are equivalent after replacing $\mathbb{C}^2/G$8 by $\mathbb{C}^2/G$9.

3. Explicit-formula mechanism and the origin of the $(\mathbb{C}^2/G,B)$0

The analytic mechanism behind the conjecture is unusually transparent. One sets

$(\mathbb{C}^2/G,B)$1

with $(\mathbb{C}^2/G,B)$2 the von Mangoldt function and $(\mathbb{C}^2/G,B)$3 for $(\mathbb{C}^2/G,B)$4; for $(\mathbb{C}^2/G,B)$5, $(\mathbb{C}^2/G,B)$6 (Kim et al., 2021). Introducing

$(\mathbb{C}^2/G,B)$7

and shifting contours produces residues at $(\mathbb{C}^2/G,B)$8 and $(\mathbb{C}^2/G,B)$9 together with contributions from nontrivial zeros. From the Laurent expansions of $M_\theta$0 at these points one obtains

$M_\theta$1

hence

$M_\theta$2

The decisive step is the separation of prime and prime-square terms:

$M_\theta$3

Using $M_\theta$4, this becomes

$M_\theta$5

By Rankin–Selberg theory, the first sum is $M_\theta$6, while the second is $M_\theta$7. Therefore

$M_\theta$8

Substituting this into the preceding identity yields the fundamental relation

$M_\theta$9

The $\theta$0 is thus not an ad hoc correction term. It is a secondary main term coming from prime squares, i.e. the $\theta$1 contribution in the Euler product. More precisely, it arises from

$\theta$2

after isolating the prime-square part of the logarithmic derivative.

4. Consequences, conditional results, and relation to Nagao's conjecture

The 2021 paper establishes three principal facts about the elliptic-curve conjecture (Kim et al., 2021). First, under the original 1965 Birch–Swinnerton-Dyer conjecture (OBSD), the limit follows. OBSD predicts

$\theta$3

Kuo–Murty and Conrad showed that this is equivalent to

$\theta$4

and splitting off the $\theta$5 and $\theta$6 terms yields

$\theta$7

Second, there is an unconditional rigidity statement: if $\theta$8 exists, then its value is necessarily $\theta$9, and the existence of the limit forces the Generalized Riemann Hypothesis for $G$0. In this sense, the existence of the full limit is a very strong assertion, not a soft consequence of known analytic estimates.

Third, assuming GRH for $G$1, there exists a sequence $G$2 with $G$3 such that

$G$4

The proof uses the explicit formula for

$G$5

together with a Cramér-type mean-square argument to find arbitrarily large $G$6 in dyadic intervals with $G$7.

The paper also relates the single-curve constant $G$8 to Nagao's conjecture for elliptic surfaces

$G$9

With

$\theta\in\Theta(G)$0

Nagao's conjecture predicts

$\theta\in\Theta(G)$1

By heuristically inserting the single-fiber asymptotic, the authors propose a modified Nagao identity

$\theta\in\Theta(G)$2

Using Abel summation, this is equivalent to an average-rank statement

$\theta\in\Theta(G)$3

up to lower-order terms. The paper further refines this with parity bias considerations and proposes an average rank of $\theta\in\Theta(G)$4, where $\theta\in\Theta(G)$5 is a parity-density.

The numerical appendix by A. V. Sutherland plots $\theta\in\Theta(G)$6 for many curves of ranks $\theta\in\Theta(G)$7 up to $\theta\in\Theta(G)$8, and for families with $\theta\in\Theta(G)$9, showing behavior consistent with convergence to a rank-dependent constant. This does not prove the conjecture, but it is compatible with the predicted dependence on rank and with the sign normalization discussed above.

5. Surface-singularity formulation: maximal resolutions and $-r+\tfrac12$00-constellation moduli

In algebraic geometry, Ishii's conjecture concerns quotient surface singularities $-r+\tfrac12$01 for finite subgroups $-r+\tfrac12$02 (Nimura, 21 May 2026). In the form stated in the 2026 paper, it reads:

Let $-r+\tfrac12$03 be any finite subgroup. Then, a resolution of $-r+\tfrac12$04 is dominated by the maximal resolution of the pair $-r+\tfrac12$05 if and only if $-r+\tfrac12$06 for some generic $-r+\tfrac12$07.

Equivalently, for a resolution of singularities $-r+\tfrac12$08, there is a generic $-r+\tfrac12$09 such that $-r+\tfrac12$10 if and only if there is a morphism

$-r+\tfrac12$11

where $-r+\tfrac12$12 is the maximal resolution of $-r+\tfrac12$13.

Here $-r+\tfrac12$14 is King's stability parameter space for $-r+\tfrac12$15-constellations, and $-r+\tfrac12$16 is the projective moduli space of $-r+\tfrac12$17-stable $-r+\tfrac12$18-constellations with the class of the regular representation $-r+\tfrac12$19. The phrase "dominated by the maximal resolution" means exactly that such a factorization through $-r+\tfrac12$20 exists.

For dihedral reflection groups, the geometric input is especially explicit. The quotient $-r+\tfrac12$21 is a normal surface with klt singularities. If $-r+\tfrac12$22 contains pseudo-reflections, the quotient map $-r+\tfrac12$23 is branched along a discriminant divisor $-r+\tfrac12$24, defined by

$-r+\tfrac12$25

In the dihedral reflection case, the local isotropy along $-r+\tfrac12$26 is cyclic of order $-r+\tfrac12$27, so the natural stacky enhancement is a root stack of order $-r+\tfrac12$28 along the strict transform of the discriminant.

The paper fixes

$-r+\tfrac12$29

with $-r+\tfrac12$30 cyclic of order $-r+\tfrac12$31 and $-r+\tfrac12$32. The minimal resolution $-r+\tfrac12$33 has exceptional chain $-r+\tfrac12$34 of type $-r+\tfrac12$35, and $-r+\tfrac12$36 exchanges $-r+\tfrac12$37 with $-r+\tfrac12$38. Its fixed locus $-r+\tfrac12$39 is a smooth divisor, and the quotient

$-r+\tfrac12$40

is smooth; the image $-r+\tfrac12$41 is the strict transform of the discriminant divisor. The quotient stack

$-r+\tfrac12$42

is the second root stack of $-r+\tfrac12$43 along $-r+\tfrac12$44, i.e. $-r+\tfrac12$45, with inertia $-r+\tfrac12$46 along the divisor.

A common misunderstanding is to read the conjecture as a statement about arbitrary resolutions of $-r+\tfrac12$47. The formulation in the paper is narrower and more precise: the relevant birational models are exactly those dominated by the maximal resolution of the klt pair $-r+\tfrac12$48.

6. Bridgeland stability proof for dihedral reflection groups

The 2026 paper proves Ishii's conjecture for all dihedral reflection groups by combining a derived McKay correspondence with a geometric construction of Bridgeland stability conditions on the root stack $-r+\tfrac12$49 (Nimura, 21 May 2026). The principal conclusion is:

For any projective birational morphism $-r+\tfrac12$50 dominated by the maximal resolution $-r+\tfrac12$51 of the pair $-r+\tfrac12$52, there is a generic stability $-r+\tfrac12$53 such that $-r+\tfrac12$54.

The proof is organized around two structural results. Theorem A associates to any smooth contraction $-r+\tfrac12$55 a connected open subset

$-r+\tfrac12$56

such that for any $-r+\tfrac12$57, the Bridgeland moduli space $-r+\tfrac12$58 equals $-r+\tfrac12$59, and if $-r+\tfrac12$60 and $-r+\tfrac12$61 are related by a single blowup, then $-r+\tfrac12$62 is nonempty and has real codimension one. Theorem B identifies the geometric local section of the stability manifold on $-r+\tfrac12$63 with the algebraic local section on $-r+\tfrac12$64 after transport by the derived equivalence and a rotation action.

The derived equivalence is

$-r+\tfrac12$65

and is induced from the universal $-r+\tfrac12$66-cluster on $-r+\tfrac12$67 together with the $-r+\tfrac12$68-action. On the root-stack side, the paper uses a semiorthogonal decomposition of $-r+\tfrac12$69 into contributions from $-r+\tfrac12$70 and $-r+\tfrac12$71, then defines an orbifold Néron–Severi space

$-r+\tfrac12$72

For parameters $-r+\tfrac12$73, the normalized central charge is

$-r+\tfrac12$74

where $-r+\tfrac12$75 is the standard surface central charge

$-r+\tfrac12$76

and the twisted-sector contribution on each connected component $-r+\tfrac12$77 is determined by

$-r+\tfrac12$78

For each contraction $-r+\tfrac12$79, a noetherian heart

$-r+\tfrac12$80

is obtained by gluing a compact-support subcategory with the pullback of a surface heart on $-r+\tfrac12$81. Positivity of the imaginary part on an explicit finite generating set defines a geometric chamber $-r+\tfrac12$82 and hence an open subset

$-r+\tfrac12$83

The moduli statement

$-r+\tfrac12$84

is then proved for all $-r+\tfrac12$85.

On the algebraic side, Bayer–Craw–Zhang construct a region of $-r+\tfrac12$86 parameterized by King stability and an auxiliary positive vector:

$-r+\tfrac12$87

$-r+\tfrac12$88

$-r+\tfrac12$89

with central charge

$-r+\tfrac12$90

For objects of class $-r+\tfrac12$91, $-r+\tfrac12$92-semistability matches King $-r+\tfrac12$93-semistability, and the moduli $-r+\tfrac12$94 equals $-r+\tfrac12$95. The geometric and algebraic local sections are then glued via affine isomorphisms between the orbifold Néron–Severi parameters and the King/GIT parameters. As a consequence, the chamber structure in Bridgeland stability matches the GIT wall-and-chamber structure in $-r+\tfrac12$96.

This proof recovers Capellan's theorem for dihedral reflection groups, but it is conceptually different in emphasis: contractions of the maximal resolution are realized simultaneously as Bridgeland moduli on the root stack $-r+\tfrac12$97 and as King moduli of $-r+\tfrac12$98-constellations on $-r+\tfrac12$99.

7. Conceptual significance and related directions

The two conjectures share a name but occupy different parts of mathematics. The elliptic-curve version is an explicit-formula statement about Frobenius traces, analytic rank, and prime sums. The surface version is a birational-moduli statement about quotient singularities, maximal resolutions, and wall-crossing in stability manifolds. Their common feature is structural rather than substantive: both identify a priori complicated geometric or arithmetic behavior with a sharply constrained asymptotic or moduli-theoretic pattern.

In the elliptic-curve setting, the strongest currently established message is rigidity. The 2021 work shows that if the Ishii limit exists at all, then it must equal $r=\operatorname{ord}_{s=1}L_E(s)$00, and existence already implies GRH for $r=\operatorname{ord}_{s=1}L_E(s)$01 (Kim et al., 2021). This places the conjecture much closer to deep zero-distribution problems than to a routine averaging phenomenon. The same paper also suggests broader analogies: using Wazir's generalization to abelian varieties over $r=\operatorname{ord}_{s=1}L_E(s)$02 and average-rank considerations, it suggests that for a $r=\operatorname{ord}_{s=1}L_E(s)$03-dimensional abelian variety an analogous constant $r=\operatorname{ord}_{s=1}L_E(s)$04 may appear in a single-curve analogue of Ishii's limit. This is presented as a guiding principle rather than as a theorem.

In the surface setting, the 2026 result clarifies the role of root stacks and orbifold derived categories in non-crepant birational geometry (Nimura, 21 May 2026). For dihedral reflection groups, the $r=\operatorname{ord}_{s=1}L_E(s)$05 inertia along the strict transform of the discriminant is not an auxiliary embellishment but the mechanism that makes the derived McKay correspondence and the Bridgeland chamber analysis compatible with the geometry of the maximal resolution. The proof also encodes the surface MMP for $r=\operatorname{ord}_{s=1}L_E(s)$06 as wall-crossing in $r=\operatorname{ord}_{s=1}L_E(s)$07.

A final terminological caution is therefore essential. In arithmetic, "Ishii's conjecture" usually means the asymptotic

$r=\operatorname{ord}_{s=1}L_E(s)$08

under the expected identification of analytic and Mordell–Weil ranks. In birational geometry, it means the characterization of resolutions dominated by the maximal resolution as moduli spaces of stable $r=\operatorname{ord}_{s=1}L_E(s)$09-constellations. Any encyclopedic use of the term requires this disambiguation.