Abelian varieties over number fields, tame ramification and big Galois image

Abstract

Abstract. Given a natural number n ≥ 1 and a number field K, we show the existence of an integer 0 such that for any prime number ≥ 0

Large Galois images for Jacobian varieties of genus 3 curves

“…Note that the above result constitutes an explicit version of Proposition 4.6 of [4] in the case of principally polarised 3-dimensional abelian varieties. We can explicitly give the size of the neighbourhoods where surjectivity of ρ A,ℓ is preserved; in other words, we can give the powers of the auxiliary primes p and q such that any other curve defined by congruence conditions modulo these powers gives rise to a Jacobian variety with surjective ℓ-torsion representation.…”

“…q-adic, neighbourhoods in moduli spaces of principally polarised g-dimensional abelian varieties with full level structure) also has a surjective ℓ-torsion Galois representation ρ B,ℓ . This is a consequence of Kisin's results on local constancy in p-families of Galois representations; the reader can find a detailed explanation of this aspect in [4,Section 4.2].…”

“…P q (X) ≡ (X −α)(X − q α )Q(X), where α = q/α and the irreducible factor Q(X) is the reduction of a degree 4 Weil polynomial; 3. P q (X) is the product of three distinct irreducible quadratic polynomials, i.e., P q (X) ≡ (X 2 − CX + q)Q(X) where X 2 − CX + q is irreducible and Q(X) is the reduction of a degree 4 Weil polynomial which has two distinct irreducible factors, both of which are distinct from X 2 − CX + q.…”

Women in Numbers Europe

“…In fact, by considering compatible systems of Galois representations attached to certain automorphic forms, we know (cf. Wiese 2008; Dieulefait and Wiese 2011;Khare et al 2008;Arias-de-Reyna et al 2013) that these groups are Galois groups over Q, for infinitely many integers r and infinitely many primes`. More precisely, we have:…”

“…Moreover, for each n 2, there is a set of primes`of positive density for which either PGSp 2n .F`r / or PSp 2n .F`r / are Galois groups over Q (cf. Dieulefait and Wiese 2011;Arias-de-Reyna et al 2013). …”