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A classification of the extensions of degree p 2 over ℚ p whose normal closure is a p -extension

Luca Caputo (2007)

Journal de Théorie des Nombres de Bordeaux

Let k be a finite extension of ℚ p and ℰ k be the set of the extensions of degree p 2 over k whose normal closure is a p -extension. For a fixed discriminant, we show how many extensions there are in ℰ ℚ p with such discriminant, and we give the discriminant and the Galois group (together with its filtration of the ramification groups) of their normal closure. We show how this method can be generalized to get a classification of the extensions in ℰ k .

A combinatorial approach to partitions with parts in the gaps

Dennis Eichhorn (1998)

Acta Arithmetica

Many links exist between ordinary partitions and partitions with parts in the “gaps”. In this paper, we explore combinatorial explanations for some of these links, along with some natural generalizations. In particular, if we let p k , m ( j , n ) be the number of partitions of n into j parts where each part is ≡ k (mod m), 1 ≤ k ≤ m, and we let p * k , m ( j , n ) be the number of partitions of n into j parts where each part is ≡ k (mod m) with parts of size k in the gaps, then p * k , m ( j , n ) = p k , m ( j , n ) .

A combinatorial interpretation of Serre's conjecture on modular Galois representations

Adriaan Herremans (2003)

Annales de l’institut Fourier

We state a conjecture concerning modular absolutely irreducible odd 2-dimensional representations of the absolute Galois group over finite fields which is purely combinatorial (without using modular forms) and proof that it is equivalent to Serre’s strong conjecture. The main idea is to replace modular forms with coefficients in a finite field of characteristic p , by their counterparts in the theory of modular symbols.

Currently displaying 81 – 100 of 1964