In this appendix to a paper by B. Williams, we give birational equivalences between the models of... more In this appendix to a paper by B. Williams, we give birational equivalences between the models of the Hilbert modular surfaces for Q(√ 29) and Q(√ 37) given there and those previously found by Elkies and Kumar.
Let R be the ring of S-integers in a number field K. Let B = {β, β * } be the multiset of roots o... more Let R be the ring of S-integers in a number field K. Let B = {β, β * } be the multiset of roots of a nonzero quadratic polynomial over R. There are varieties V (B) N,k defined over R parametrizing periodic continued fractions [b 1 ,. .. , b N , a 1 ,. .. , a k ] for β or β *. We study the R-points on these varieties, finding contrasting behavior according to whether groups of units are infinite or not. If R is the rational integers or the ring of integers in an imaginary quadratic field, we prove that the R-points of V (B) N,k are not Zariski dense. On the other hand, suppose that β ∈ K ∪ {∞}, R × is infinite, and that there are infinitely many units in the (left) order R β of βR + R ⊆ K(β) with norm to K equal to (−1) k. Then we prove that the R-points on V (B) 1,k are Zariski dense for k ≥ 8 and the R-points on V (B) 0,k are Zariski dense for k ≥ 9. We also prove that V (B) 1,k and V (B) 0,k are K-rational irreducible varieties for k sufficiently large.
A collection S = {D 1 ,. .. , Dn} of divisors on a smooth variety X is an arrangement if the inte... more A collection S = {D 1 ,. .. , Dn} of divisors on a smooth variety X is an arrangement if the intersection of every subset of S is smooth. We show that a double cover of X ramified on an arrangement has a crepant resolution under additional hypotheses. Namely, we assume that all intersection components that change the canonical divisor when blown up satisfy are splayed, a property of the tangent spaces of the components first studied by Faber. This strengthens a result of Cynk and Hulek, which requires a stronger hypothesis on the intersection components. Further, we study the singular subscheme of the union of the divisors in S and prove that it has a primary decomposition where the primary components are supported on exactly the subvarieties which are blown up in the course of constructing the crepant resolution of the double cover.
We consider spaces of modular forms attached to definite orthogonal groups of low even rank and n... more We consider spaces of modular forms attached to definite orthogonal groups of low even rank and nontrivial level, equipped with Hecke operators defined by Kneser neighbours. After reviewing algorithms to compute with these spaces, we investigate endoscopy using theta series and a theorem of Rallis. Along the way, we exhibit many examples and pose several conjectures. As a first application, we express counts of Kneser neighbours in terms of coefficients of classical or Siegel modular forms, complementing work of Chenevier–Lannes. As a second application, we prove new instances of Eisenstein congruences of Ramanujan and Kurokawa–Mizumoto type.
We show that, for each n > 0, there is a family of elliptic surfaces which are covered by the squ... more We show that, for each n > 0, there is a family of elliptic surfaces which are covered by the square of a curve of genus 2n + 1, and whose Hodge structures have an action by Q(√ −n). By considering the case n = 3, we show that one particular family of K3 surfaces are covered by the squares of curves of genus 7. Using this, we construct a correspondence between the square of a curve of genus 7 and a general K3 surface in P 4 with 15 ordinary double points up to a map of finite degree of K3 surfaces. This gives an explicit proof of the Kuga-Satake-Deligne correspondence for these K3 surfaces and any K3 surfaces related to them by maps of finite degree, and further, a proof of the Hodge conjecture for the squares of these surfaces. We conclude that the motives of these surfaces are Kimura-finite. Our analysis gives a birational equivalence between a moduli space of curves with additional data and the moduli space of these K3 surfaces with a specific elliptic fibration.
We consider a natural variant of the Erdős-Rényi random graph process in which k vertices are spe... more We consider a natural variant of the Erdős-Rényi random graph process in which k vertices are special and are never put into the same connected component. The model is natural and interesting on its own, but is actually inspired by the combinatorial data fusion problem that itself is connected to a number of important problems in graph theory. We will show that a phase transition occurs when the number of special vertices is roughly n 1/3 , where n is the number of vertices.
For two varieties of dimension 5 constructed as double covers of P^5 branched along the union of ... more For two varieties of dimension 5 constructed as double covers of P^5 branched along the union of 12 hyperplanes, we prove that the number of points over F_p can be described in terms of Artin symbols and the pth Fourier coefficients of modular forms. Many analogous results are known in dimension < 3, but very few in higher dimension. In addition, we use an idea of Burek to construct quotients of our varieties for which the point counts mod p appear to be expressible in terms of Artin symbols and the coefficients of a single modular form of weight 6.
For an integer n≥ 8 divisible by 4, let R_n=Z[ζ_n,1/2] and let U_2(R_n) be the group of 2× 2 unit... more For an integer n≥ 8 divisible by 4, let R_n=Z[ζ_n,1/2] and let U_2(R_n) be the group of 2× 2 unitary matrices with entries in R_n. Set U_2^ζ(R_n)={γ∈U_2(R_n)|γ∈〈ζ_n〉}. Let G_n⊆U_2^ζ(R_n) be the Clifford-cyclotomic group generated by a Hadamard matrix H=1/2[< s m a l l m a t r i x >] and the gate T=[< s m a l l m a t r i x >]. We prove that G_n=U_2^ζ(R_n) if and only if n=8, 12, 16, 24 and that [U_2^ζ(R_n):G_n]=∞ if U_2^ζ(R_n)≠G_n.
For two varieties of dimension 5 constructed as double covers of ℙ^5 branched along the union of ... more For two varieties of dimension 5 constructed as double covers of ℙ^5 branched along the union of 12 hyperplanes, we prove that the number of points over 𝔽_p can be expressed in terms of Artin symbols and the pth Fourier coefficients of modular forms. Many analogous results are known in dimension ≤ 3, but very few in higher dimension. In addition, we use an idea of Burek to construct quotients of our varieties for which the point counts mod p are expressible in terms of Artin symbols and the coefficients of a single modular form of weight 6.
The corank of a group $G$ is the maximum $k$ such that $G$ surjects onto a free group of rank $k$... more The corank of a group $G$ is the maximum $k$ such that $G$ surjects onto a free group of rank $k$. We study the corank of the groups $\operatorname{PSU}_2$ and $\operatorname{PU}_2$ over cyclotomic rings ${\mathbf Z}[\zeta_{n}, 1/2]$ with $\zeta_n=e^{2\pi i/n}$, $n=2^s$ or $n=3\cdot 2^s$, $n\geq 8$. We do this by studying their actions on Bruhat-Tits trees constructed using definite quaternion algebras. The quotients of the trees by this action are finite graphs whose first Betti number is the corank of the group. Our main result is that for the families $n=2^s$ and $n=3\cdot 2^s$ the corank grows doubly exponentially in $s$ as $s\rightarrow\infty$; it is $0$ precisely when $n= 8,12, 16, 24$. We deduce from this main result a theorem on the Clifford-cyclotomic group in quantum computing.
Paranjape showed that K3 surfaces that are double covers of P^2 branched along six lines are domi... more Paranjape showed that K3 surfaces that are double covers of P^2 branched along six lines are dominated by the square of a curve of genus 5. In this talk, we describe a somewhat analogous construction and use it to show that K3 surfaces in P^4 with 15 nodes are dominated by the square of a curve of genus 7. We will explain a birational equivalence between the moduli space of a related family of K3 surfaces and a moduli space of covers of rational curves with additional data. This is joint work with Colin Ingalls and Owen Patashnick.
Suppose 4|n, n ≥ 8, F = F n = Q(ζ n + ζ n), and there is one prime p = p n above 2 in F n. We stu... more Suppose 4|n, n ≥ 8, F = F n = Q(ζ n + ζ n), and there is one prime p = p n above 2 in F n. We study amalgam presentations for PU 2 (Z[ζ n , 1/2]) and PSU 2 (Z[ζ n , 1/2]) with the Clifford-cyclotomic group in quantum computing as a subgroup. These amalgams arise from an action of these groups on the Bruhat-Tits tree ∆ = ∆ p for SL 2 (F p) constructed via the Hamilton quaternions. We explicitly compute the finite quotient graphs and the resulting amalgams for 8 ≤ n ≤ 48, n = 44, as well as for PU 2 (Z[ζ 60 , 1/2]).
Abstract A collection A = { D 1 , … , D n } of divisors on a smooth variety X is an arrangement i... more Abstract A collection A = { D 1 , … , D n } of divisors on a smooth variety X is an arrangement if the intersection of every subset of A is smooth. We show that, if X is defined over a field of characteristic not equal to 2, a double cover of X ramified on an arrangement has a crepant resolution under additional hypotheses. Namely, we assume that all intersection components that change the canonical divisor when blown up are splayed, a property of the tangent spaces of the components first studied by Faber. This strengthens a result of Cynk and Hulek, which requires a stronger hypothesis on the intersection components. Further, we study the singular subscheme of the union of the divisors in A and prove that it has a primary decomposition where the primary components are supported on exactly the subvarieties which are blown up in the course of constructing the crepant resolution of the double cover.
For an integer $n\geq 8$ divisible by $4$ , let $R_n={\mathbb Z}[\zeta _n,1/2]$ and let $\operato... more For an integer $n\geq 8$ divisible by $4$ , let $R_n={\mathbb Z}[\zeta _n,1/2]$ and let $\operatorname {\mathrm {U_{2}}}(R_n)$ be the group of $2\times 2$ unitary matrices with entries in $R_n$ . Set $\operatorname {\mathrm {U_2^\zeta }}(R_n)=\{\gamma \in \operatorname {\mathrm {U_{2}}}(R_n)\mid \det \gamma \in \langle \zeta _n\rangle \}$ . Let $\mathcal {G}_n\subseteq \operatorname {\mathrm {U_2^\zeta }}(R_n)$ be the Clifford-cyclotomic group generated by a Hadamard matrix $H=\frac {1}{2}[\begin {smallmatrix} 1+i & 1+i\\1+i &-1-i\end {smallmatrix}]$ and the gate $T_n=[\begin {smallmatrix}1 & 0\\0 & \zeta _n\end {smallmatrix}]$ . We prove that $\mathcal {G}_n=\operatorname {\mathrm {U_2^\zeta }}(R_n)$ if and only if $n=8, 12, 16, 24$ and that $[\operatorname {\mathrm {U_2^\zeta }}(R_n):\mathcal {G}_n]=\infty $ if $\operatorname {\mathrm {U_2^\zeta }}(R_n)\neq \mathcal {G}_n$ . We compute the Euler–Poincaré characteristic of the groups $\operatorname {\mathrm {SU_{2}}}(R_n)$ , $\oper...
Over an algebraically closed field, various finiteness results are known regarding the automorphi... more Over an algebraically closed field, various finiteness results are known regarding the automorphism group of a K3 surface and the action of the automorphisms on the Picard lattice. We formulate and prove versions of these results over arbitrary base fields, and give examples illustrating how behaviour can differ from the algebraically closed case. Keywords K3 surfaces • Automorphism groups • Picard groups • Non-algebraically closed fields Mathematics Subject Classification 14J28 • 14J50 • 14G27 The second author would like to thank the Tutte Institute for Mathematics and Computation for its partial support for a visit to the University of Leiden during which much of this research was done.
Brown and Schnetz found that the number of points over Fp of a graph hypersurface is often relate... more Brown and Schnetz found that the number of points over Fp of a graph hypersurface is often related to the coefficients of a modular form. We set some of the reduction techniques used to discover such relation in a general geometric context. We also prove the relation for one example of a modular form of weight 4 and two of weight 3, refine the statement and suggest a method of proving it for four more of weight 4, and use the one proved example to construct two new rigid Calabi-Yau threefolds that realize Hecke eigenforms of weight 4 (one provably and one conjecturally).
Let E be a modular elliptic curve over a totally real field. Chapter 8 of (Dar2) formulates a con... more Let E be a modular elliptic curve over a totally real field. Chapter 8 of (Dar2) formulates a conjecture allowing the construction of canoni- cal algebraic points on E by suitably integrating the associated Hilbert modular form. The main goal of the present work is to obtain numerical evidence for this conjecture in the first case where it asserts something nontrivial, namely, when E has everywhere good reduction over a real quadratic field.
We present a new method to show that a principal homogeneous space of the Jacobian of a curve of ... more We present a new method to show that a principal homogeneous space of the Jacobian of a curve of genus two is nontrivial. The idea is to exhibit a Brauer-Manin obstruction to the existence of rational points on a quotient of this principal homogeneous space. In an explicit example we apply the method to show that a specific curve has infinitely many quadratic twists whose Jacobians have nontrivial Tate-Shafarevich group.
In this appendix to a paper by B. Williams, we give birational equivalences between the models of... more In this appendix to a paper by B. Williams, we give birational equivalences between the models of the Hilbert modular surfaces for Q(√ 29) and Q(√ 37) given there and those previously found by Elkies and Kumar.
Let R be the ring of S-integers in a number field K. Let B = {β, β * } be the multiset of roots o... more Let R be the ring of S-integers in a number field K. Let B = {β, β * } be the multiset of roots of a nonzero quadratic polynomial over R. There are varieties V (B) N,k defined over R parametrizing periodic continued fractions [b 1 ,. .. , b N , a 1 ,. .. , a k ] for β or β *. We study the R-points on these varieties, finding contrasting behavior according to whether groups of units are infinite or not. If R is the rational integers or the ring of integers in an imaginary quadratic field, we prove that the R-points of V (B) N,k are not Zariski dense. On the other hand, suppose that β ∈ K ∪ {∞}, R × is infinite, and that there are infinitely many units in the (left) order R β of βR + R ⊆ K(β) with norm to K equal to (−1) k. Then we prove that the R-points on V (B) 1,k are Zariski dense for k ≥ 8 and the R-points on V (B) 0,k are Zariski dense for k ≥ 9. We also prove that V (B) 1,k and V (B) 0,k are K-rational irreducible varieties for k sufficiently large.
A collection S = {D 1 ,. .. , Dn} of divisors on a smooth variety X is an arrangement if the inte... more A collection S = {D 1 ,. .. , Dn} of divisors on a smooth variety X is an arrangement if the intersection of every subset of S is smooth. We show that a double cover of X ramified on an arrangement has a crepant resolution under additional hypotheses. Namely, we assume that all intersection components that change the canonical divisor when blown up satisfy are splayed, a property of the tangent spaces of the components first studied by Faber. This strengthens a result of Cynk and Hulek, which requires a stronger hypothesis on the intersection components. Further, we study the singular subscheme of the union of the divisors in S and prove that it has a primary decomposition where the primary components are supported on exactly the subvarieties which are blown up in the course of constructing the crepant resolution of the double cover.
We consider spaces of modular forms attached to definite orthogonal groups of low even rank and n... more We consider spaces of modular forms attached to definite orthogonal groups of low even rank and nontrivial level, equipped with Hecke operators defined by Kneser neighbours. After reviewing algorithms to compute with these spaces, we investigate endoscopy using theta series and a theorem of Rallis. Along the way, we exhibit many examples and pose several conjectures. As a first application, we express counts of Kneser neighbours in terms of coefficients of classical or Siegel modular forms, complementing work of Chenevier–Lannes. As a second application, we prove new instances of Eisenstein congruences of Ramanujan and Kurokawa–Mizumoto type.
We show that, for each n > 0, there is a family of elliptic surfaces which are covered by the squ... more We show that, for each n > 0, there is a family of elliptic surfaces which are covered by the square of a curve of genus 2n + 1, and whose Hodge structures have an action by Q(√ −n). By considering the case n = 3, we show that one particular family of K3 surfaces are covered by the squares of curves of genus 7. Using this, we construct a correspondence between the square of a curve of genus 7 and a general K3 surface in P 4 with 15 ordinary double points up to a map of finite degree of K3 surfaces. This gives an explicit proof of the Kuga-Satake-Deligne correspondence for these K3 surfaces and any K3 surfaces related to them by maps of finite degree, and further, a proof of the Hodge conjecture for the squares of these surfaces. We conclude that the motives of these surfaces are Kimura-finite. Our analysis gives a birational equivalence between a moduli space of curves with additional data and the moduli space of these K3 surfaces with a specific elliptic fibration.
We consider a natural variant of the Erdős-Rényi random graph process in which k vertices are spe... more We consider a natural variant of the Erdős-Rényi random graph process in which k vertices are special and are never put into the same connected component. The model is natural and interesting on its own, but is actually inspired by the combinatorial data fusion problem that itself is connected to a number of important problems in graph theory. We will show that a phase transition occurs when the number of special vertices is roughly n 1/3 , where n is the number of vertices.
For two varieties of dimension 5 constructed as double covers of P^5 branched along the union of ... more For two varieties of dimension 5 constructed as double covers of P^5 branched along the union of 12 hyperplanes, we prove that the number of points over F_p can be described in terms of Artin symbols and the pth Fourier coefficients of modular forms. Many analogous results are known in dimension < 3, but very few in higher dimension. In addition, we use an idea of Burek to construct quotients of our varieties for which the point counts mod p appear to be expressible in terms of Artin symbols and the coefficients of a single modular form of weight 6.
For an integer n≥ 8 divisible by 4, let R_n=Z[ζ_n,1/2] and let U_2(R_n) be the group of 2× 2 unit... more For an integer n≥ 8 divisible by 4, let R_n=Z[ζ_n,1/2] and let U_2(R_n) be the group of 2× 2 unitary matrices with entries in R_n. Set U_2^ζ(R_n)={γ∈U_2(R_n)|γ∈〈ζ_n〉}. Let G_n⊆U_2^ζ(R_n) be the Clifford-cyclotomic group generated by a Hadamard matrix H=1/2[< s m a l l m a t r i x >] and the gate T=[< s m a l l m a t r i x >]. We prove that G_n=U_2^ζ(R_n) if and only if n=8, 12, 16, 24 and that [U_2^ζ(R_n):G_n]=∞ if U_2^ζ(R_n)≠G_n.
For two varieties of dimension 5 constructed as double covers of ℙ^5 branched along the union of ... more For two varieties of dimension 5 constructed as double covers of ℙ^5 branched along the union of 12 hyperplanes, we prove that the number of points over 𝔽_p can be expressed in terms of Artin symbols and the pth Fourier coefficients of modular forms. Many analogous results are known in dimension ≤ 3, but very few in higher dimension. In addition, we use an idea of Burek to construct quotients of our varieties for which the point counts mod p are expressible in terms of Artin symbols and the coefficients of a single modular form of weight 6.
The corank of a group $G$ is the maximum $k$ such that $G$ surjects onto a free group of rank $k$... more The corank of a group $G$ is the maximum $k$ such that $G$ surjects onto a free group of rank $k$. We study the corank of the groups $\operatorname{PSU}_2$ and $\operatorname{PU}_2$ over cyclotomic rings ${\mathbf Z}[\zeta_{n}, 1/2]$ with $\zeta_n=e^{2\pi i/n}$, $n=2^s$ or $n=3\cdot 2^s$, $n\geq 8$. We do this by studying their actions on Bruhat-Tits trees constructed using definite quaternion algebras. The quotients of the trees by this action are finite graphs whose first Betti number is the corank of the group. Our main result is that for the families $n=2^s$ and $n=3\cdot 2^s$ the corank grows doubly exponentially in $s$ as $s\rightarrow\infty$; it is $0$ precisely when $n= 8,12, 16, 24$. We deduce from this main result a theorem on the Clifford-cyclotomic group in quantum computing.
Paranjape showed that K3 surfaces that are double covers of P^2 branched along six lines are domi... more Paranjape showed that K3 surfaces that are double covers of P^2 branched along six lines are dominated by the square of a curve of genus 5. In this talk, we describe a somewhat analogous construction and use it to show that K3 surfaces in P^4 with 15 nodes are dominated by the square of a curve of genus 7. We will explain a birational equivalence between the moduli space of a related family of K3 surfaces and a moduli space of covers of rational curves with additional data. This is joint work with Colin Ingalls and Owen Patashnick.
Suppose 4|n, n ≥ 8, F = F n = Q(ζ n + ζ n), and there is one prime p = p n above 2 in F n. We stu... more Suppose 4|n, n ≥ 8, F = F n = Q(ζ n + ζ n), and there is one prime p = p n above 2 in F n. We study amalgam presentations for PU 2 (Z[ζ n , 1/2]) and PSU 2 (Z[ζ n , 1/2]) with the Clifford-cyclotomic group in quantum computing as a subgroup. These amalgams arise from an action of these groups on the Bruhat-Tits tree ∆ = ∆ p for SL 2 (F p) constructed via the Hamilton quaternions. We explicitly compute the finite quotient graphs and the resulting amalgams for 8 ≤ n ≤ 48, n = 44, as well as for PU 2 (Z[ζ 60 , 1/2]).
Abstract A collection A = { D 1 , … , D n } of divisors on a smooth variety X is an arrangement i... more Abstract A collection A = { D 1 , … , D n } of divisors on a smooth variety X is an arrangement if the intersection of every subset of A is smooth. We show that, if X is defined over a field of characteristic not equal to 2, a double cover of X ramified on an arrangement has a crepant resolution under additional hypotheses. Namely, we assume that all intersection components that change the canonical divisor when blown up are splayed, a property of the tangent spaces of the components first studied by Faber. This strengthens a result of Cynk and Hulek, which requires a stronger hypothesis on the intersection components. Further, we study the singular subscheme of the union of the divisors in A and prove that it has a primary decomposition where the primary components are supported on exactly the subvarieties which are blown up in the course of constructing the crepant resolution of the double cover.
For an integer $n\geq 8$ divisible by $4$ , let $R_n={\mathbb Z}[\zeta _n,1/2]$ and let $\operato... more For an integer $n\geq 8$ divisible by $4$ , let $R_n={\mathbb Z}[\zeta _n,1/2]$ and let $\operatorname {\mathrm {U_{2}}}(R_n)$ be the group of $2\times 2$ unitary matrices with entries in $R_n$ . Set $\operatorname {\mathrm {U_2^\zeta }}(R_n)=\{\gamma \in \operatorname {\mathrm {U_{2}}}(R_n)\mid \det \gamma \in \langle \zeta _n\rangle \}$ . Let $\mathcal {G}_n\subseteq \operatorname {\mathrm {U_2^\zeta }}(R_n)$ be the Clifford-cyclotomic group generated by a Hadamard matrix $H=\frac {1}{2}[\begin {smallmatrix} 1+i & 1+i\\1+i &-1-i\end {smallmatrix}]$ and the gate $T_n=[\begin {smallmatrix}1 & 0\\0 & \zeta _n\end {smallmatrix}]$ . We prove that $\mathcal {G}_n=\operatorname {\mathrm {U_2^\zeta }}(R_n)$ if and only if $n=8, 12, 16, 24$ and that $[\operatorname {\mathrm {U_2^\zeta }}(R_n):\mathcal {G}_n]=\infty $ if $\operatorname {\mathrm {U_2^\zeta }}(R_n)\neq \mathcal {G}_n$ . We compute the Euler–Poincaré characteristic of the groups $\operatorname {\mathrm {SU_{2}}}(R_n)$ , $\oper...
Over an algebraically closed field, various finiteness results are known regarding the automorphi... more Over an algebraically closed field, various finiteness results are known regarding the automorphism group of a K3 surface and the action of the automorphisms on the Picard lattice. We formulate and prove versions of these results over arbitrary base fields, and give examples illustrating how behaviour can differ from the algebraically closed case. Keywords K3 surfaces • Automorphism groups • Picard groups • Non-algebraically closed fields Mathematics Subject Classification 14J28 • 14J50 • 14G27 The second author would like to thank the Tutte Institute for Mathematics and Computation for its partial support for a visit to the University of Leiden during which much of this research was done.
Brown and Schnetz found that the number of points over Fp of a graph hypersurface is often relate... more Brown and Schnetz found that the number of points over Fp of a graph hypersurface is often related to the coefficients of a modular form. We set some of the reduction techniques used to discover such relation in a general geometric context. We also prove the relation for one example of a modular form of weight 4 and two of weight 3, refine the statement and suggest a method of proving it for four more of weight 4, and use the one proved example to construct two new rigid Calabi-Yau threefolds that realize Hecke eigenforms of weight 4 (one provably and one conjecturally).
Let E be a modular elliptic curve over a totally real field. Chapter 8 of (Dar2) formulates a con... more Let E be a modular elliptic curve over a totally real field. Chapter 8 of (Dar2) formulates a conjecture allowing the construction of canoni- cal algebraic points on E by suitably integrating the associated Hilbert modular form. The main goal of the present work is to obtain numerical evidence for this conjecture in the first case where it asserts something nontrivial, namely, when E has everywhere good reduction over a real quadratic field.
We present a new method to show that a principal homogeneous space of the Jacobian of a curve of ... more We present a new method to show that a principal homogeneous space of the Jacobian of a curve of genus two is nontrivial. The idea is to exhibit a Brauer-Manin obstruction to the existence of rational points on a quotient of this principal homogeneous space. In an explicit example we apply the method to show that a specific curve has infinitely many quadratic twists whose Jacobians have nontrivial Tate-Shafarevich group.
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