By Gerhard Frey

The most subject of the amount is to increase effective algorithms wherein possible ascertain Artin's conjecture for strange two-dimensional representations in a pretty wide variety. to do that, one has to figure out the variety of all representations with given Artin conductor and determinant and to compute the measurement of a corresponding house of cusp kinds of weight 1 that's performed by way of exploiting the categorical wisdom of the operation of Hecke operators on modular symbols. it truly is was hoping that the algorithms constructed within the quantity will be of use for plenty of different difficulties concerning modular kinds.

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**Additional resources for On Artin’s Conjecture for Odd 2-Dimensional Representations**

**Example text**

Now let’s explore some properties of norms and traces. 15 Suppose α ∈ K . Then N K /Q (α) and TK /Q (α) are both in Q. Proof This simply follows because they are the trace and determinant of a matrix with entries in Q. This is a rather abstract definition of the trace and norm of an element, but we can make it a little more concrete. 16 Write σ1 , . . , σn for the embeddings of K into C. If α ∈ K , then n N K /Q (α) = n σk (α) k=1 and TK /Q (α) = σk (α). k=1 Proof Let g denote the minimal polynomial of α over Q.

Further, b itself cannot be in Z; otherwise A2 − b2 d ∈ / Z. 4 Thus B is an odd integer. 7 Rings of Integers of Number Fields 37 A2 − B 2 d ≡ 0 (mod 4) with A and B odd integers. But the squares of odd numbers are all 1 (mod 4). Thus 1 − d ≡ 0 (mod 4). If d ≡ 1 (mod 4), the second case can arise, and the integers are √ {a + b d | either a, b ∈ Z, or both a and b are halves of odd integers}, a set which is easily seen to be the same as that of the √ statement. On the other hand, if d ̸≡ 1 (mod 4), then the only integers are {a + b d | a, b ∈ Z} as claimed.

N ∈ K such that every element of K can be written as a linear combination x1 α1 + x2 α2 + · · · + xn αn where x1 , . . , xn ∈ Q. 1, a special case of the above, where our basis has a particular form. We can ask exactly analogous questions about the ring of integers Z K . 1. Do there exist elements α1 , . . , αn ∈ Z K such that every element of Z K is of the form x1 α1 + x2 α2 + · · · + xn αn for some xi ∈ Z? 2. 1, for some xi ∈ Z? It will turn out that the first question has a positive answer, but the second does not, in general.