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Articles 1 - 11 of 11
Full-Text Articles in Number Theory
Hypergeometric Motives, David P. Roberts, Fernando Rodriguez Villegas
Hypergeometric Motives, David P. Roberts, Fernando Rodriguez Villegas
Mathematics Publications
No abstract provided.
Pgl2(FL) Number Fields With Rational Companion Forms, David P. Roberts
Pgl2(FL) Number Fields With Rational Companion Forms, David P. Roberts
Mathematics Publications
We give a list of PGL2(Fl) number fields for ℓ ≥ 11 which have rational companion forms. Our list has fifty-three fields and seems likely to be complete. Some of the fields on our list are very lightly ramified for their Galois group.
Polynomials With Prescribed Bad Primes, David P. Roberts
Polynomials With Prescribed Bad Primes, David P. Roberts
Mathematics Publications
We tabulate polynomials in ℚ[t] with a given factorization partition, bad reduction entirely within a given set of primes, and satisfying auxiliary conditions associated to 0, 1, and ∞. We explain how these polynomials are of particular interest because of their role in the construction of nonsolvable number fields of arbitrarily large degree and bounded ramification.
The Tame-Wild Principle For Discriminant Relations For Number Fields, John W. Jones, David P. Roberts
The Tame-Wild Principle For Discriminant Relations For Number Fields, John W. Jones, David P. Roberts
Mathematics Publications
Consider tuples ( K1 , … , Kr ) of separable algebras over a common local or global number field F1, with the Ki related to each other by specified resolvent constructions. Under the assumption that all ramification is tame, simple group-theoretic calculations give best possible divisibility relations among the discriminants of Ki ∕ F . We show that for many resolvent constructions, these divisibility relations continue to hold even in the presence of wild ramification.
Octic 2-Adic Fields, John W. Jones, David P. Roberts
Octic 2-Adic Fields, John W. Jones, David P. Roberts
Mathematics Publications
We compute all octic extensions of Q2 and find that there are 1823 of them up to isomorphism. We compute the associated Galois group of each field, slopes measuring wild ramification, and other quantities. We present summarizing tables here with complete information available at our online database of local fields.
Number Fields Ramified At One Prime, John W. Jones, David P. Roberts
Number Fields Ramified At One Prime, John W. Jones, David P. Roberts
Mathematics Publications
For G a finite group and p a prime, a G-p field is a Galois number field K with Gal(K/Q)≅G and disc(K)=±pa for some a. We study the existence of G-p fields for fixed G and varying p.
Galois Number Fields With Small Root Discriminant, John W. Jones, David P. Roberts
Galois Number Fields With Small Root Discriminant, John W. Jones, David P. Roberts
Mathematics Publications
We pose the problem of identifying the set K(G,Ω) of Galois number fields with given Galois group G and root discriminant less than the Serre constant Ω ≈ 44.7632. We definitively treat the cases G = A4. A5, A6, and S4, S5, S6, finding exactly 59, 78, 5 and 527, 192, 13 fields, respectively. We present other fields with Galois groups SL3(2), A7, S7, PGL2(7), SL2(8), ΣL2(8), PGL2(9), PSL2(11), and …
Wild Partitions And Number Theory, David P. Roberts
Wild Partitions And Number Theory, David P. Roberts
Mathematics Publications
We introduce the notion of wild partition to describe in combinatorial language an important situation in the theory of p-adic fields. For Q a power of p, we get a sequence of numbers λQ,n counting the number of certain wild partitions of n. We give an explicit formula for the corresponding generating function ΛQ(x) = ΣλQ,nxn and use it to show that λ1/n Q,n tends to Q1/(p-1). We apply this asymptotic result to support a finiteness conjecture about number fields. Our finiteness conjecture …
Nonic 3-Adic Fields, John W. Jones, David P. Roberts
Nonic 3-Adic Fields, John W. Jones, David P. Roberts
Mathematics Publications
We compute all nonic extensions of Q3 and find that there are 795 of them up to isomorphism. We describe how to compute the associated Galois group of such a field, and also the slopes measuring wild ramification. We present summarizing tables and a sample application to number fields.
An Abc Construction Of Number Fields, David P. Roberts
An Abc Construction Of Number Fields, David P. Roberts
Mathematics Publications
We describe a general three step method for constructing number fields with Lie-type Galois groups and discriminants factoring into powers of specified primes. The first step involves extremal solutions of the matrix equation ABC = I. The second step involves extremal polynomial solutions of the equation A(x) + B(x) + C(x) = 0. The third step involves integer solutions of the generalized Fermat equation axp + byq + czr = 0. We concentrate here on details associated to the third step and give examples where the field discriminants have the form ±2a3b .
Sextic Number Fields With Discriminant -J2A3B, John W. Jones, David P. Roberts
Sextic Number Fields With Discriminant -J2A3B, John W. Jones, David P. Roberts
Mathematics Publications
. Complete lists of number fields, of given degree n and unramified outside a given finite set S of primes, are both of intrinsic interest and useful in some applications. For degrees n ≤ 5 and S = {∞, 2, 3}, the complete lists have appeared previously; there are in total 85 such fields. Here we give the complete list for n = 6 and S = {∞, 2, 3}, finding in particular exactly 398 such fields. We use a three-pronged approach to obtain this classification: an exhaustive computer search, sextic twinning, and class field theory. Also we completely identify …