--- title: 'Ishii''s Conjecture: Elliptic Curves & Surface Singularities' url: https://www.emergentmind.com/topics/ishii-s-conjecture type: topic --- # Ishii's Conjecture: Elliptic Curves & Surface Singularities 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 $-r\log\log x$ [2105.10805]. 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 [2605.22474]. ## 1. Terminological scope and disambiguation The shared name masks two mathematically unrelated settings. | Setting | Core object | Conjectural statement | |---|---|---| | Elliptic curves over $\mathbb{Q}$ | Prime sums of Frobenius traces $a_p$ | $\displaystyle \lim_{x\to\infty}\frac{1}{\log x}\sum_{p\le x,\ p\nmid \Delta_E}\frac{a_p\log p}{p}=-r+\frac12$ | | Quotient surface singularities | Resolutions of $\mathbb{C}^2/G$ and moduli of $G$-constellations | A resolution is dominated by the maximal resolution iff $X=M_\theta(v)$ for some generic $\theta\in\Theta(G)$ | 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 $G$-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 $\mathbb{Q}$ Let $E/\mathbb{Q}$ be an elliptic curve with discriminant $\Delta_E$ and conductor $N_E$. For a prime $p$ of good reduction, $p\nmid \Delta_E$, write $$ N_p=\#E(\mathbb{F}_p)=p+1-a_p, $$ so that $a_p=p+1-N_p$ satisfies Hasse's bound $|a_p|\le 2\sqrt p$ [2105.10805]. At bad primes $p\mid \Delta_E$, the finitely many terms can be omitted from prime sums without affecting the limiting statements. The associated Hasse–Weil $L$-function is $$ L_E(s)=\prod_{p\nmid \Delta_E}(1-a_pp^{-s}+p^{1-2s})^{-1}\cdot \prod_{p\mid \Delta_E}(1-a_pp^{-s})^{-1}, $$ absolutely convergent for $\Re s>3/2$. Writing $\alpha_p+\beta_p=a_p$ and $\alpha_p\beta_p=p$ for $p\nmid \Delta_E$, one also has $$ L_E(s)=\prod_{p\nmid \Delta_E}(1-\alpha_pp^{-s})^{-1}(1-\beta_pp^{-s})^{-1}\times \prod_{p\mid \Delta_E}(1-a_pp^{-s})^{-1}, $$ with $|\alpha_p|=|\beta_p|=\sqrt p$. By modularity, $L_E(s)$ extends to an entire function and satisfies the functional equation $$ \Lambda_E(s)=w_E\Lambda_E(2-s), $$ where $$ \Lambda_E(s)=N_E^{s/2}(2\pi)^{-s}\Gamma(s)L_E(s), $$ and $w_E\in\{\pm1\}$. The limit studied in the paper is $$ S_E(x):=\frac{1}{\log x}\sum_{\substack{p\le x\\ p\nmid \Delta_E}}\frac{a_p\log p}{p}. $$ Ishii's conjecture in this setting states that $$ \lim_{x\to\infty}S_E(x)=-r+\frac12, $$ where $r=\operatorname{ord}_{s=1}L_E(s)$ is the analytic rank, expected by BSD to equal $\operatorname{rank}E(\mathbb{Q})$. An equivalent formulation, obtained by integration by parts, is $$ \sum_{\substack{p\le x\\ p\nmid \Delta_E}}\frac{a_p}{p}=-r\log\log x+C_E+o(1) $$ for some constant $C_E$. 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 $-r+\tfrac12$ rather than $r-\tfrac12$. The two presentations are equivalent after replacing $S_E(x)$ by $-S_E(x)$. ## 3. Explicit-formula mechanism and the origin of the $\tfrac12$ The analytic mechanism behind the conjecture is unusually transparent. One sets $$ -\frac{L'_E(s)}{L_E(s)}=\sum_{n\ge 1} c_n\Lambda(n)n^{-s}, $$ with $\Lambda$ the von Mangoldt function and $c_{p^m}=\alpha_p^m+\beta_p^m$ for $p\nmid\Delta_E$; for $p\mid \Delta_E$, $c_{p^m}=a_p^m$ [2105.10805]. Introducing $$ I(x):=\frac{1}{2\pi i}\int_{(2)}\Bigl[-\frac{L'_E(s)}{L_E(s)}\Bigr]\frac{x^s}{s(s-1)}\,ds $$ and shifting contours produces residues at $s=0$ and $s=1$ together with contributions from nontrivial zeros. From the Laurent expansions of $-L'_E/L_E$ at these points one obtains $$ -rx\log x+O(x)=x\sum_{n\le x}\frac{c_n\Lambda(n)}{n}-\sum_{n\le x}c_n\Lambda(n), $$ hence $$ \sum_{n\le x}\frac{c_n\Lambda(n)}{n} = -r\log x+\frac{\sum_{n\le x}c_n\Lambda(n)}{x}+O(1). $$ The decisive step is the separation of prime and prime-square terms: $$ \sum_{n\le x}\frac{c_n\Lambda(n)}{n} = \sum_{p\le x}\frac{a_p\log p}{p} + \sum_{p^2\le x}\frac{(\alpha_p^2+\beta_p^2)\log p}{p^2} +o(1). $$ Using $\alpha_p^2+\beta_p^2=a_p^2-2p$, this becomes $$ \sum_{p^2\le x}\frac{(\alpha_p^2+\beta_p^2)\log p}{p^2} = \sum_{p\le \sqrt x}\frac{a_p^2\log p}{p^2} - 2\sum_{p\le \sqrt x}\frac{\log p}{p}. $$ By Rankin–Selberg theory, the first sum is $o(\log x)$, while the second is $\frac12\log x+O(1)$. Therefore $$ \sum_{p^2\le x}\frac{(\alpha_p^2+\beta_p^2)\log p}{p^2} = -\frac12\log x+o(\log x). $$ Substituting this into the preceding identity yields the fundamental relation $$ \sum_{p\le x}\frac{a_p\log p}{p} = \Bigl(-r+\frac12\Bigr)\log x+\frac{\sum_{n\le x}c_n\Lambda(n)}{x}+O(1). $$ The $\tfrac12$ is thus not an ad hoc correction term. It is a secondary main term coming from prime squares, i.e. the $k=2$ contribution in the Euler product. More precisely, it arises from $$ -2\sum_{p\le \sqrt x}\frac{\log p}{p}= -\frac12\log x+O(1) $$ 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 [2105.10805]. First, under the original 1965 Birch–Swinnerton-Dyer conjecture (OBSD), the limit follows. OBSD predicts $$ \prod_{p Let $G\subset GL_2(\mathbb{C})$ be any finite subgroup. Then, a resolution of $\mathbb{C}^2/G$ is dominated by the maximal resolution of the pair $(\mathbb{C}^2/G,B)$ if and only if $X=M_\theta$ for some generic $\theta\in\Theta(G)$. Equivalently, for a resolution of singularities $f:X\to \mathbb{C}^2/G$, there is a generic $\theta\in\Theta(G)$ such that $X=M_\theta(v)$ if and only if there is a morphism $$ Y\to X\to \mathbb{C}^2/G, $$ where $Y$ is the maximal resolution of $(\mathbb{C}^2/G,B)$. Here $\Theta(G)$ is King's stability parameter space for $G$-constellations, and $M_\theta(v)$ is the projective moduli space of $\theta$-stable $G$-constellations with the class of the regular representation $v=[\mathbb{C}[G]\otimes O_0]$. The phrase "dominated by the maximal resolution" means exactly that such a factorization through $Y$ exists. For dihedral reflection groups, the geometric input is especially explicit. The quotient $X_0:=\mathbb{C}^2/G$ is a normal surface with klt singularities. If $G$ contains pseudo-reflections, the quotient map $\nu:\mathbb{C}^2\to X_0$ is branched along a discriminant divisor $D\subset X_0$, defined by $$ \nu^*(K_{X_0}+B)=K_{\mathbb{C}^2} \quad\text{with}\quad B=D. $$ In the dihedral reflection case, the local isotropy along $D$ is cyclic of order $2$, so the natural stacky enhancement is a root stack of order $2$ along the strict transform of the discriminant. The paper fixes $$ G=\langle \alpha,\beta\rangle\subset GL(2,\mathbb{C}), \qquad \alpha=\operatorname{diag}(\zeta_n,\zeta_n^{-1}), \qquad \zeta_n=\exp(2\pi i/n), \qquad \beta=\begin{bmatrix}0&1\\1&0\end{bmatrix}, $$ with $H:=G\cap SL_2(\mathbb{C})=\langle \alpha\rangle$ cyclic of order $n$ and $G/H\cong \mathbb{Z}_2$. The minimal resolution $Y_H\to \mathbb{A}^2/H$ has exceptional chain $E_1,\dots,E_{n-1}$ of type $A_{n-1}$, and $G/H$ exchanges $E_i$ with $E_{n-i}$. Its fixed locus $R\subset Y_H$ is a smooth divisor, and the quotient $$ Y:=Y_H/(G/H) $$ is smooth; the image $T(R)\subset Y$ is the strict transform of the discriminant divisor. The quotient stack $$ \mathcal{Y}:=[Y_H/\mathbb{Z}_2] $$ is the second root stack of $Y$ along $T(R)$, i.e. $Y_{(T(R),2)}$, with inertia $\mu_2$ along the divisor. A common misunderstanding is to read the conjecture as a statement about arbitrary resolutions of $\mathbb{C}^2/G$. The formulation in the paper is narrower and more precise: the relevant birational models are exactly those dominated by the maximal resolution of the klt pair $(\mathbb{C}^2/G,B)$. ## 6. Bridgeland stability proof for dihedral reflection groups The 2026 paper proves Ishii's conjecture for all dihedral reflection groups by combining a derived McKay correspondence with a geometric construction of Bridgeland stability conditions on the root stack $\mathcal{Y}$ [2605.22474]. The principal conclusion is: > For any projective birational morphism $X\to \mathbb{C}^2/G$ dominated by the maximal resolution $Y$ of the pair $(\mathbb{C}^2/G,B)$, there is a generic stability $\theta\in\Theta(G)$ such that $M_\theta(v)\cong X$. The proof is organized around two structural results. Theorem A associates to any smooth contraction $Y\to X$ a connected open subset $$ U(X)\subset \operatorname{Stab}_n(\mathcal{Y}) $$ such that for any $\sigma\in U(X)$, the Bridgeland moduli space $M_\sigma([T^*O_y])$ equals $X$, and if $X$ and $X'$ are related by a single blowup, then $U(X)\cap U(X')$ is nonempty and has real codimension one. Theorem B identifies the geometric local section of the stability manifold on $\mathcal{Y}$ with the algebraic local section on $[\mathbb{C}^2/G]$ after transport by the derived equivalence and a rotation action. The derived equivalence is $$ \Phi:D_c([Y_H/\mathbb{Z}_2])\xrightarrow{\sim} D_c([\mathbb{C}^2/G]), $$ and is induced from the universal $H$-cluster on $Y_H$ together with the $\mathbb{Z}_2$-action. On the root-stack side, the paper uses a semiorthogonal decomposition of $D_c([Y_H/\mathbb{Z}_2])$ into contributions from $D_c(R)$ and $D_c(Y)$, then defines an orbifold Néron–Severi space $$ NS_{\mathrm{orb}}(\mathcal{Y})_{\mathbb{R}}:=NS(Y)_{\mathbb{R}}\oplus K_0(R)_{\mathbb{R}}. $$ For parameters $((\omega,t),(B,s))\in NS_{\mathrm{orb}}(\mathcal{Y})_{\mathbb{C}}$, the normalized central charge is $$ Z_{\omega,B;t,s}:=(Z_{\omega,B}+Z_{t,s})\circ \vartheta, $$ where $Z_{\omega,B}$ is the standard surface central charge $$ Z_{\omega,B}(E)=\operatorname{ch}_1(E)\cdot B-\operatorname{ch}_2(E)+i\,\operatorname{ch}_1(E)\cdot \omega, $$ and the twisted-sector contribution on each connected component $R_i$ is determined by $$ Z_{t,s}(O_{p_i})=-\frac12+s_i+it_i, \qquad Z_{t,s}(\operatorname{sgn}\otimes O_{p_i})=\frac12-s_i-it_i. $$ For each contraction $f:Y\to X$, a noetherian heart $$ \mathcal{A}_\omega(\mathcal{Y}/X)=\mathcal{C}_{\mathcal{Y}/X}*Lf^*\mathcal{A}_\omega $$ is obtained by gluing a compact-support subcategory with the pullback of a surface heart on $X$. Positivity of the imaginary part on an explicit finite generating set defines a geometric chamber $\mathcal{A}^+(X)$ and hence an open subset $$ U(X)=H_X(\mathcal{A}^+(X)\times \mathfrak{B})\subset \operatorname{Stab}_n(\mathcal{Y}). $$ The moduli statement $$ M_\sigma([T^*O_y])\cong X $$ is then proved for all $\sigma\in U(X)$. On the algebraic side, Bayer–Craw–Zhang construct a region of $\operatorname{Stab}([\mathbb{C}^2/G])$ parameterized by King stability and an auxiliary positive vector: $$ \Theta(G)=\{\theta\in\operatorname{Hom}(K_{\mathrm{num}},\mathbb{R})\mid \theta((\dim\rho)\rho)=0\}, $$ $$ V_1=\{\lambda\in\operatorname{Hom}(K_{\mathrm{num}},\mathbb{R})\mid \lambda((\dim\rho)\rho)=1\}, $$ $$ A_1=\{\lambda\in V_1\mid \lambda(\rho)>0\ \text{for all irreducible}\ \rho\in \operatorname{Irr}G\}, $$ with central charge $$ Z_{\theta,\lambda}(M)=\theta(M)+i\,\lambda(M). $$ For objects of class $v$, $\sigma_{\theta,\lambda}$-semistability matches King $\theta$-semistability, and the moduli $M_{\sigma_{\theta,\lambda}}(v)$ equals $M_\theta(v)$. 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 $\Theta(G)$. 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 $\mathcal{Y}$ and as King moduli of $G$-constellations on $[\mathbb{C}^2/G]$. ## 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+\tfrac12$, and existence already implies GRH for $L_E(s)$ [2105.10805]. 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 $\mathbb{Q}(T)$ and average-rank considerations, it suggests that for a $g$-dimensional abelian variety an analogous constant $g/2$ 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 [2605.22474]. For dihedral reflection groups, the $\mu_2$ 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 $Y$ as wall-crossing in $\operatorname{Stab}_n(\mathcal{Y})$. A final terminological caution is therefore essential. In arithmetic, "Ishii's conjecture" usually means the asymptotic $$ \lim_{x\to\infty}\frac{1}{\log x}\sum_{p\le x}\frac{a_p\log p}{p}=-\operatorname{rank}E(\mathbb{Q})+\frac12 $$ 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 $G$-constellations. Any encyclopedic use of the term requires this disambiguation. Source: https://www.emergentmind.com/topics/ishii-s-conjecture