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Register a new userSato-Tate groups of y^2=x^8+c and y^2=x^7-cx
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We consider the distribution of normalized Frobenius traces for two families of genus 3 hyperelliptic curves over Q that have large automorphism groups: y^2=x^8+c and y^2=x^7-cx with c in Q*. We give efficient algorithms to compute the trace of Frobenius for curves in these families at primes of good reduction. Using data generated by these algorithms, we obtain a heuristic description of the Sato-Tate groups that arise, both generically and for particular values of c. We then prove that these heuristic descriptions are correct by explicitly computing the Sato-Tate groups via the correspondence between Sato-Tate groups and Galois endomorphism types.
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Given an abelian variety over a number field, its Sato-Tate group is a compact Lie group which conjecturally controls the distribution of Euler factors of the L-function of the abelian variety. It was previously shown by Fite, Kedlaya, Rotger, and Sutherland that there are 52 groups (up to conjugation) that occur as Sato-Tate groups of abelian surfaces over number fields; we show here that for abelian threefolds, there are 410 possible Sato-Tate groups, of which 33 are maximal with respect to inclusions of finite index. We enumerate candidate groups using the Hodge-theoretic construction of Sato-Tate groups, the classification of degree-3 finite linear groups by Blichfeldt, Dickson, and Miller, and a careful analysis of Shimuras theory of CM types that rules out 23 candidate groups; we cross-check this using extensive computations in Gap, SageMath, and Magma. To show that these 410 groups all occur, we exhibit explicit examples of abelian threefolds realizing each of the 33 maximal groups; we also compute moments of the corresponding distributions and numerically confirm that they are consistent with the statistics of the associated L-functions.
In this paper, we determine the primitive solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ when $ngeq 2$ and $d=p^b$, $p$ a prime and $pleq 10^4$. The main ingredients are the characterization of primitive divisors on Lehmer sequences and the development of an algorithmic method of proving the non-existence of integer solutions of the equation $f(x)=a^b$, where $f(x)inmathbb Z[x]$, $a$ a positive integer and $b$ an arbitrary positive integer.
We establish the group-theoretic classification of Sato-Tate groups of self-dual motives of weight 3 with rational coefficients and Hodge numbers h^{3,0} = h^{2,1} = h^{1,2} = h^{0,3} = 1. We then describe families of motives that realize some of these Sato-Tate groups, and provide numerical evidence supporting equidistribution. One of these families arises in the middle cohomology of certain Calabi-Yau threefolds appearing in the Dwork quintic pencil; for motives in this family, our evidence suggests that the Sato-Tate group is always equal to the full unitary symplectic group USp(4).
Spin correlations of the frustrated pyrochlore oxide Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$ have been investigated by using inelastic neutron scattering on single crystalline samples ($x=-0.007, 0.000,$ and $0.003$), which have the putative quantum-spin-liquid (QSL) or electric-quadrupolar ground states. Spin correlations, which are notably observed in nominally elastic scattering, show short-ranged correlations around $L$ points [$q = (tfrac{1}{2},tfrac{1}{2},tfrac{1}{2})$], tiny antiferromagnetic Bragg scattering at $L$ and $Gamma$ points, and pinch-point type structures around $Gamma$ points. The short-ranged spin correlations were analyzed using a random phase approximation (RPA) assuming the paramagnetic state and two-spin interactions among Ising spins. These analyses have shown that the RPA scattering intensity well reproduces the experimental data using temperature and $x$ dependent coupling constants of up to 10-th neighbor site pairs. This suggests that no symmetry breaking occurs in the QSL sample, and that a quantum treatment beyond the semi-classical RPA approach is required. Implications of the experimental data and the RPA analyses are discussed.
Suppose that $n$ is a positive integer. In this paper, we show that the exponential Diophantine equation $$(n-1)^{x}+(n+2)^{y}=n^{z}, ngeq 2, xyz eq 0$$ has only the positive integer solutions $(n,x,y,z)=(3,2,1,2), (3,1,2,3)$. The main tools on the proofs are Bakers theory and Bilu-Hanrot-Voutiers result on primitive divisors of Lucas numbers.
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