Historical notes on Serre’s conjecture Sample Clauses

Historical notes on Serre’s conjecture. ‌ The history of the area of number theory that this thesis is a part of, which is nowadays often referred to as the ‘Serre weight conjectures’, all starts with a conjecture made by Xxxx-Xxxxxx Xxxxx in 1973 in a letter to Xxxx Xxxx [ST15, p.451]. The details were worked out by Xxxxx in the follow- ing decade and a half until he published a precise form of his conjecture in 1987 [Ser87]. In his letter of the first of May 1973, Xxxx–Xxxxxx Xxxxx writes the following to Xxxx Xxxx (quoted from [ST15, pp. 451–453]; translation by the author). Dear Xxxx, [...] I would like to tell you about a conjecture on Ga- xxxx extensions of Q whose Galois group is a subgroup of GL2(Fp): I will be careful, and consider only representations ρ : Gal(Q/Q) → GL2(Fq), q = pf , unramified outside of p and satisfying the following condi- tion: det ρ : Gal → Fqx is equal to χk´1 (k even), where χ : Gal → Fqx is the fundamental character modulo p (given by the action on µp). In fact, this condition is equivalent to: the image under ρ of the real Xxxxxxxxx has eigenvalues .1 and —1 (i.e., det ρ is an odd character). ź [...] If l ‰ p, the trace and determinant of ρ(Frobl) have an obvious meaning; denote them by al and lk´1, and take the Dirichlet series (with coefficients in Fq) (1 — all´s . lk´1´2s)´1, l‰p `8 ÿ x` xx Xxxxx; denote it by x xxx´s, and take the formal series f = anqn, n“1 where this time q denotes a variable... a thousand xxxxx- xxxx for having used q for the number of elements of the finite field, but I’m too lazy to start over for so little! Conjecture. The series f is a modular form (modulo p) for SL2(Z) of weight congruent to k mod (p — 1). [...] Enough talking like so. Until another time. J.-X. Xxxxx Let us briefly explain what Xxxxx is proposing in his letter. Since the 1971 paper of Deligne [Del71] it was known that starting with a eigenform f of weight k ě 2, level Γ1(N ) and character χ it was possible to attach to f a continuous irreducible p-adic representation ρf : Gal(Q/Q) → GL2(Qp). The modular form defines the representation uniquely: ρf is unramified at any l - Np and ρf (Frobl) has characteristic polynomial x2—al(f )t.χ(l)lk´1. What Xxxxx describes in his letter is a mod p analogue of this result going the other way, that is, starting with a mod p Galois representation he would like to obtain a mod p modular form. One way of stating the question is as follows. It is not hard to show that the representation ρf will always fix a lattice, and we can reduce...