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Register a new userThe square root law and structure of finite rings
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Let $R$ be a finite ring and define the hyperbola $H={(x,y) in R times R: xy=1 }$. Suppose that for a sequence of finite odd order rings of size tending to infinity, the following square root law bound holds with a constant $C>0$ for all non-trivial characters $chi$ on $R^2$: [ left| sum_{(x,y)in H}chi(x,y)right|leq Csqrt{|H|}. ] Then, with a finite number of exceptions, those rings are fields. For rings of even order we show that there are other infinite families given by Boolean rings and Boolean twists which satisfy this square-root law behavior. We classify the extremal rings, those for which the left hand side of the expression above satisfies the worst possible estimate. We also describe applications of our results to problems in graph theory and geometric combinatorics. These results provide a quantitative connection between the square root law in number theory, Salem sets, Kloosterman sums, geometric combinatorics, and the arithmetic structure of the underlying rings.
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Let $E$ be an elliptic curve over $Q$. It is well known that the ring of endomorphisms of $E_p$, the reduction of $E$ modulo a prime $p$ of ordinary reduction, is an order of the quadratic imaginary field $Q(pi_p)$ generated by the Frobenius element $pi_p$. When the curve has complex multiplication (CM), this is always a fixed field as the prime varies. However, when the curve has no CM, very little is known, not only about the order, but about the fields that might appear as algebra of endomorphisms varying the prime. The ring of endomorphisms is obviously related with the arithmetic of $a_p^2-4p$, the discriminant of the characteristic polynomial of the Frobenius element. In this paper, we are interested in the function $pi_{E,r,h}(x)$ counting the number of primes $p$ up to $x$ such that $a_p^2-4p$ is square-free and in the congruence class $r$ modulo $h$. We give in this paper the precise asymptotic for $pi_{E,r,h}(x)$ when averaging over elliptic curves defined over the rationals, and we discuss the relation of this result with the Lang-Trotter conjecture, and with some other problems related to the curve modulo $p$.
Let R and S be two irreducible root systems spanning the same vector space and having the same Weyl group W, such that S (but not necessarily R) is reduced. For each such pair (R,S) we construct a family of W-invariant orthogonal polynomials in several variables, whose coefficients are rational functions of parameters $q,t_1,t_2,...,t_r$, where r (=1,2 or 3) is the number of W-orbits in R. For particular values of these parameters, these polynomials give the values of zonal spherical functions on real and p-adic symmetric spaces. Also when R=S is of type $A_n$, they conincide with the symmetric polynomials described in I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Oxford University Press (1995), Chapter VI.
We explore whether a root lattice may be similar to the lattice $mathscr O$ of integers of a number field $K$ endowed with the inner product $(x, y):={rm Trace}_{K/mathbb Q}(xcdottheta(y))$, where $theta$ is an involution of $K$. We classify all pairs $K$, $theta$ such that $mathscr O$ is similar to either an even root lattice or the root lattice $mathbb Z^{[K:mathbb Q]}$. We also classify all pairs $K$, $theta$ such that $mathscr O$ is a root lattice. In addition to this, we show that $mathscr O$ is never similar to a positive-definite even unimodular lattice of rank $leqslant 48$, in particular, $mathscr O$ is not similar to the Leech lattice. In appendix, we give a general cyclicity criterion for the primary components of the discriminant group of $mathscr O$.
Many independent studies on stocks and futures contracts have established that market impact is proportional to the square-root of the executed volume. Is market impact quantitatively similar for option markets as well? In order to answer this question, we have analyzed the impact of a large proprietary data set of option trades. We find that the square-root law indeed holds in that case. This finding supports the argument for a universal underlying mechanism.
We show how the minimal free resolution of a set of $n$ points in general position in projective space of dimension $n-2$ explicitly determines structure constants for a ring of rank $n$. This generalises previously known constructions of Levi-Delone-Faddeev and Bhargava in the cases $n=3,4,5$.
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