Ring of integers

About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.

Uniform boundedness in terms of ramification

Abstract: Let $$d\ge 1$$ be fixed. Let F be a number field of degree d, and let E / F be an elliptic curve. Let $$E(F)_{\text {tors}}$$ be the torsion subgroup of E(F). In 1996, Merel proved the uniform boundedness conjecture, i.e., there is a constant B(d), which depends on d but not on the chosen field F or on the curve E / F, such that the size of $$E(F)_{\text {tors}}$$ is bounded by B(d). Moreover, Merel gave a bound (exponential in d) for the largest prime that may be a divisor of the order of $$E(F)_{\text {tors}}$$ . In 1996, Parent proved a bound (also exponential in d) for the largest p-power order of a torsion point that may appear in $$E(F)_{\text {tors}}$$ . It has been conjectured, however, that there is a bound for the size of $$E(F)_{\text {tors}}$$ that is polynomial in d. In this article we show that under certain hypotheses there is a linear bound for the largest p-power order of a torsion point defined over F, which in fact is linear in the maximum ramification index of a prime ideal of the ring of integers F over (p).

Lower bounds on representing Boolean functions as polynomials in Z/sub m/

Abstract: The MOD/sub m/-degree of Boolean function F is defined to be the smallest degree of any polynomial P, over the ring of integers modulo m, such that for all 0-1 assignments x, F(x)=0 iff P(x)=0. By exploring the periodic property of the binomial coefficients module m, two new lower bounds on the MOD/sub m/-degree of the MOD/sub l/ and not-MOD/sub m/ functions are proved, where m is any composite integer and l has a prime factor not dividing m. Both bounds improve from n/sup Omega (1)/ in D.A.M. Barrington et al. (1992) to Omega (n). A lower bound, n/2, for the majority function and a lower bound, square root n, for the MidBit function are also proved. >

On norms of integers in a full module of an algebraic number field and the distribution of values of binary integral quadratic forms

Abstract: Let K be an algebraic number field. By a. full module in K [l,p.83] we mean a finitely-generated (necessarily free) subgroup M of the additive group of K whose rank is equal to the degree [ K : ℚ ] of K over the rational field ℚ. The intersection of M with ℤ K , the ring of integers of K , is also a full module I , and we shall concern ourselves chiefly with the latter, in that we wish to count the number of rational integers in a given interval which can be expressed as the norms of elements of I . More precisely, we shall adapt the methods of [2] to prove the following

Hecke operators and Hilbert modular forms

Abstract: Let F be a real quadratic field with ring of integers O andwith class number 1. Let Γ be a congruence subgroup of GL2(O). Wedescribe a technique to compute the action of the Hecke operators on thecohomology H3(Γ;C). For F real quadratic this cohomology group containsthe cuspidal cohomology corresponding to cuspidal Hilbert modularforms of parallel weight 2. Hence this technique gives a way to computethe Hecke action on these Hilbert modular forms.