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The use of quadratic residues to construct matrices with specific determinant values is a familiar problem with connections to many areas of mathematics and statistics. Our research has focused on using cubic residues to construct matrices with interesting and predictable determinants.
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We introduce mod 3 triple Milnor invariants and triple cubic residue symbols for certain primes of the Eisenstein number field $mathbb{Q}(sqrt{-3})$, following the analogies between knots and primes. Our triple symbol generalizes both the cubic residue symbol and R{e}deis triple symbol, and describes the decomposition law of a prime in a mod 3 Heisenberg extension of degree 27 over $mathbb{Q}(sqrt{-3})$ with restricted ramification, which we construct concretely in the form similar to R{e}deis dihedral extension over $mathbb{Q}$. We also give a cohomological interpretation of our symbols by triple Massey products in Galois cohomology.
Let $(R, mathfrak{m})$ be a complete discrete valuation ring with the finite residue field $R/mathfrak{m} = mathbb{F}_{q}$. Given a monic polynomial $P(t) in R[t]$ whose reduction modulo $mathfrak{m}$ gives an irreducible polynomial $bar{P}(t) in mathbb{F}_{q}[t]$, we initiate the investigation of the distribution of $mathrm{coker}(P(A))$, where $A in mathrm{Mat}_{n}(R)$ is randomly chosen with respect to the Haar probability measure on the additive group $mathrm{Mat}_{n}(R)$ of $n times n$ $R$-matrices. One of our main results generalizes two results of Friedman and Washington. Our other results are related to the distribution of the $bar{P}$-part of a random matrix $bar{A} in mathrm{Mat}_{n}(mathbb{F}_{q})$ with respect to the uniform distribution, and one of them generalizes a result of Fulman. We heuristically relate our results to a celebrated conjecture of Cohen and Lenstra, which predicts that given an odd prime $p$, any finite abelian $p$-group (i.e., $mathbb{Z}_{p}$-module) $H$ occurs as the $p$-part of the class group of a random imaginary quadratic field extension of $mathbb{Q}$ with a probability inversely proportional to $|mathrm{Aut}_{mathbb{Z}}(H)|$. We review three different heuristics for the conjecture of Cohen and Lenstra, and they are all related to special cases of our main conjecture, which we prove as our main theorems. For proofs, we use some concrete combinatorial connections between $mathrm{Mat}_{n}(R)$ and $mathrm{Mat}_{n}(mathbb{F}_{q})$ to translate our problems about a Haar-random matrix in $mathrm{Mat}_{n}(R)$ into problems about a random matrix in $mathrm{Mat}_{n}(mathbb{F}_{q})$ with respect to the uniform distribution.
We investigate the average number of solutions of certain quadratic congruences. As an application, we establish Manins conjecture for a cubic surface whose singularity type is A_5+A_1.
Manins conjecture predicts the asymptotic behavior of the number of rational points of bounded height on algebraic varieties. For toric varieties, it was proved by Batyrev and Tschinkel via height zeta functions and an application of the Poisson formula. An alternative approach to Manins conjecture via universal torsors was used so far mainly over the field Q of rational numbers. In this note, we give a proof of Manins conjecture over the Gaussian rational numbers Q(i) and over other imaginary quadratic number fields with class number 1 for the singular toric cubic surface defined by t^3=xyz.
Let $F$ be a quadratic real field, $p$ be a rational prime inert in $F$. In this paper, we prove that an overconvergent $p$-adic Hilbert eigenform for $F$ of small slope is actually a classical Hilbert modular form.
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