--- title: 'Cremona–Mazur Construction: Visible Sha Varieties' url: https://www.emergentmind.com/topics/cremona-mazur-construction type: topic --- # Cremona–Mazur Construction: Visible Sha Varieties The Cremona–Mazur construction produces explicit abelian varieties in which nontrivial elements of the Shafarevich–Tate group (Sha) of an elliptic curve over $\mathbb{Q}$ are “visible” in the sense defined by Mazur’s visibility philosophy. Given a nonzero $σ\in\mathrm{Sha}(E/\mathbb{Q})$, this method constructs geometric objects—typically Jacobians of genus $2$ or $4$ curves—into which the twist of $E$ by $σ$ embeds as a subvariety and $σ$ becomes trivial in their cohomology. The construction originally gave existence proofs but recent work has rendered all steps algorithmically explicit in the cases when $\operatorname{ord}(σ)=2$ or $3$, revealing conditions for minimality and illuminating the structure of the associated visibility category $\mathcal{V}(E;σ)$ [2601.21519]. ## 1. Foundational Principles of the Cremona–Mazur Construction Let $E/\mathbb{Q}$ be an elliptic curve, and let $σ$ be a nonzero cohomology class of order $n$ in $\mathrm{Sha}(E/\mathbb{Q})\subseteq H^1(\mathbb{Q}, E)$. The key principle is to construct an abelian variety $A$ containing $E$ such that the twist $C_\sigma$ (interpreted as a torsor under $E$ and a curve covering $E$) embeds in $A$ and $σ$ becomes trivial in $H^1(\mathbb{Q},A)$. The existence argument rests upon: - Representing $σ$ as a central simple Azumaya algebra $A_σ$ of dimension $n^2$ over $K=\mathbb{Q}(E)$. - Selecting a maximal commutative subalgebra $L\subset A_σ$ of degree $n$ over $K$, corresponding to a degree-$n$ cover $C\to E$ over $\mathbb{Q}$. - Showing that $C\to E$ is totally ramified over at least one point, ensuring injectivity of the pull-back $J(C)\leftarrow E$. - As $A_σ\otimes_K L$ is split, $σ$ becomes trivial over $L$, hence $σ$ is visible in $J(C)$. Therefore, $A := J(C)$ is an abelian variety of dimension at least $n$ containing $E$ and visualizing $σ$. The original construction in [Cremona–Mazur ’00] established existence without explicit geometry; recent work makes all steps fully concrete for $n=2$ and $n=3$ [2601.21519]. This process is general, but explicit models are essential for applications and for certifying minimality. ## 2. Explicit Construction for Sha Classes of Order Two For $\operatorname{ord}(σ)=2$, any nonzero class $σ\in\mathrm{Sha}(E)[2]$ is represented by a binary quartic form $f(x,z)=ax^4 + bx^3z + cx^2z^2 + dxz^3 + ez^4$. The associated genus-$1$ cover $C$ is given by $y^2=f(x,1)$, corresponding to the torsor $C_\sigma$. Haile–Han’s results provide an explicit quaternion algebra $\mathsf{Cliff}(f)$ (Clifford algebra) over $\mathbb{Q}(E)$, with generators and quadratic relations reflecting the coefficients of $f$. The central element $\Delta - 4a \xi$, where $\Delta=b^2-4ac$ and $\xi = s^2-c$, yields a maximal commutative subfield $L=\mathbb{Q}(E)(\sqrt{\Delta - 4a \xi})$. The genus-$2$ Cremona–Mazur curve is then written as $$C: Y^2 = f\left(\frac{\Delta - X^2}{4a}\right),$$ where $X$ arises from a coordinate renaming on $E$. In short Weierstrass form $E: y^2= x^3 + \alpha x^2 + \beta x + \gamma$, the same construction delivers an explicit genus-$2$ curve $C$ whose Jacobian contains $E$ and trivializes $σ$. If $E(\mathbb{Q})/2E(\mathbb{Q})=0$, then $\mathrm{Jac}(C) \simeq (E \times F)/\Delta$ for some complementary curve $F$, and $F(\mathbb{Q})$ has positive rank, confirming that $\mathrm{Jac}(C)$ is minimal in the $\sigma$-visibility category. ## 3. Explicit Construction for Sha Classes of Order Three For $\operatorname{ord}(σ)=3$, each $σ\in\mathrm{Sha}(E)[3]$ is represented by a non-diagonal ternary cubic $f(x,y,z)=a x^3 + b y^3 + c z^3 + a_2 x^2y + b_1 xy^2 + c_1 xz^2 + c_2 yz^2 + m xyz$. The associated Azumaya algebra (by Kuo and Fisher) admits two generators with cubic relations. Making the Cremona–Mazur cover explicit involves selecting a function $g\in \mathsf{Azumaya}(f)$ whose divisor on $E$ is $(P)+(\tau P)+(\tau^2P)-3(0)$ for $P\in E(L)$ in a specific cyclic cubic extension $L/\mathbb{Q}(E)$. The degree-$3$ cover $C\to E$ is given by the affine equation $z^3 = g(\xi, \eta)$, where $(\xi, \eta)$ are coordinates on $E$. The genus of $C$ is $4$ by Riemann–Hurwitz. By construction, the pull-back map $\mathrm{Jac}(C)\leftarrow E$ is injective and trivializes $σ$, making $\mathrm{Jac}(C)$ a dimension-$4$ abelian variety visualizing $σ$. In practice, computational runs across Fisher’s database confirm that in over $99\%$ of $123,544$ non-diagonal cases, the construction yields a minimal Jacobian. ## 4. Minimality Criteria for Visualizing Abelian Varieties A visualizing abelian variety $A$ is minimal for $σ$ if no strictly smaller abelian subvariety $A'\subset A$ containing $E$ renders $σ$ visible. Two criteria confirm minimality: 1. **Simplicity Criterion:** If $A/E$ is $K$-simple, $A$ is minimal. For $n=2$, the complementary factor $F$ is an elliptic curve; for $n=3$, the Weil restriction $\mathrm{Res}_{L/\mathbb{Q}}(E)$ decomposes as $E\times B$ with $B$ simple of dimension $2$ due to irreducibility of the standard $S_3$ representation. 2. **Cohomological (Fisher’s Theorem 2.2):** Realize $A = (E \times F)/\Delta$, gluing along $n$-torsion subgroups. If $E'(\mathbb{Q})/\varphi(E(\mathbb{Q}))=0$, both $E$ and $F$ have Tamagawa numbers prime to $n$, and $A$ has good reduction at all primes dividing $n$, then $$\mathrm{Vis}_A \mathrm{Sha}(E) \simeq F'(\mathbb{Q})/\psi(F(\mathbb{Q})),$$ with the size yielding the exact number of $\sigma$ visible in $A$. Minimality holds when this quotient is as small as possible, typically isomorphic to $\mathbb{Z}/n\mathbb{Z}$. ## 5. Explicit Examples ### Order Two Example Let $E/\mathbb{Q}: y^2 = x^3 - x^2 - 169321x - 28379327$ (LMFDB 161472.bz.1), with $\mathrm{Sha}(E)[2]\simeq (\mathbb{Z}/2)^2$ and a nonzero class $\psi_1(x,z) = -4x^4 - 60x^3z - 232x^2z^2 - 52xz^3 - 3z^4$. Compute $\Delta = b^2 - 4ac = (-60)^2 - 4(-4)(-232) = 3600-3712 = -112$. The genus-$2$ Cremona–Mazur curve: $$C_1: Y^2 = \psi_1\left(\frac{\Delta - X^2}{4a}\right) = \psi_1\left(\frac{-112-X^2}{-16}\right),$$ expands to $Y^2 = (1/64) X^6 - (211/16) X^4 - (29/4) X^2 - 1/1$. Via Magma, $\mathrm{Jac}(C_1)\simeq(E\times F)/\Delta$ with $F: y^2 = x^3 -1204416x -508759920$, $F(\mathbb{Q})\{2\}=0$, $F(\mathbb{Q})$ has rank $1$, and $F'(\mathbb{Q})/2F(\mathbb{Q})\simeq\mathbb{Z}/2$ exactly once. By Fisher’s criterion, $C_1$ minimally visualizes the given order-$2$ class. ### Order Three Example Take $E/\mathbb{Q}$ with nontrivial $3$-torsion $σ\in\mathrm{Sha}(E)[3]$ represented by a non-diagonal ternary cubic $f(x,y,z)$. With formulas from Fisher, select $g(\xi, \eta)\in\mathbb{Q}(E)$ with divisor $(P)+(\tau P)+(\tau^2P)-3(0)$ and define $C: z^3 = g(\xi,\eta)$. Looping through Fisher’s database and applying Sage/Magma, one tests primes $p<1000$ for good reduction; in $>$99\% of $123,544$ non-diagonal cases, the Jacobian $\mathrm{Jac}(C)$ is certifiably minimal. ## 6. Connections to Related Work and Computational Advances Parallel constructions by Agashe–Stein employ explicit Weil restriction arguments that guarantee minimal dimension $n$ for visualizing abelian varieties for $n=2,3$. Large torsor databases have been compiled by Bruin and Fisher, facilitating algorithmic testing of Cremona–Mazur constructions on order-$3$ Sha classes [Agashe–Stein ’02; Bruin–Fisher ’18; Fisher ’14/’16]. Mazur established the foundational visibility philosophy; Voight’s book provides background on Azumaya and Clifford algebra theory. The explicit nature of Banwait–Caro–Chidambaram's refinements has enabled comprehensive computational evidence, demonstrating that the Cremona–Mazur construction produces minimal visualizations for the majority of Sha classes of orders $2$ and $3$ under natural arithmetic conditions, answering a question of Mazur regarding the existence and dimensionality of minimal visualizing abelian varieties [2601.21519]. Source: https://www.emergentmind.com/topics/cremona-mazur-construction