Non-existence of certain semistable abelian varieties

We prove the existence in many cases of minimally ramified p-adic lifts of 2-dimensional continuous, odd, absolutely irreducible, mod p representations ρ of the absolute Galois group of Q. It is predicted by Serre's conjecture that such representations arise from newforms of optimal level and weight.Using these minimal lifts, and arguments using compatible systems, we prove some cases of Serre's conjectures in low levels and weights. For instance we prove that there are no irreducible (p, p) type group schemes over Z. We prove that aρ as above of Artin conductor 1 and Serre weight 12 arises from the Ramanujan Delta-function.In the last part of the paper we present arguments that reduce Serre's conjecture to proving generalisations of modularity lifting theorems of the type pioneered by Wiles.

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“…(iv) Results of Fontaine, Brumer-Kramer and Schoof [20], [11], [39] which determine semistable abelian varieties over Q of small conductor.…”

Section: Introductionmentioning

confidence: 99%

On Serre’s conjecture for 2-dimensional mod p representations of Gal(ℚ∕ℚ)

Khare¹,

Wintenberger

2009

Ann. Math.

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“…Since inertia at 2 and 3 acts through a cyclic subgroup of order 5, we also have ramification bounds at 2 and 3. As in Schoof [12] and Brumer-Kramer [1], we obtain the following estimate of the root discriminant…”

Section: Lemma 210mentioning

confidence: 88%

“…Our technique for proving these results is linked strongly to the ideas in Brumer-Kramer [1] and Schoof [12], and thus we consider it important to briefly recall the main ideas of these papers now. Schoof's approach is similar in spirit to Fontaine's.…”

Section: Introductionmentioning

confidence: 99%

“…The calculation of the discriminant was included as a check against typographical errors in the defining polynomials 3 . allocatemem() allocatemem() allocatemem() allocatemem() nf=nfinit(poly defining K); factor(nf [3]) bnf=bnfinit(nf [1],1); pd=idealprimedec(nf,5); pd1=idealhnf(nf,pd [1]); idealnorm(nf,pd1) pd2=idealmul(nf,pd1,pd1); bnrclassno(bnf,pd1) bnrclassno(bnf,pd2)…”

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confidence: 99%

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Semistable abelian varieties over Q

Calegari

2004

manuscripta mathematica

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We prove that for N = 6 and N = 10, there do not exist any non-zero semistable abelian varieties over Q with good reduction outside primes dividing N . Our results are contingent on the GRH discriminant bounds of Odlyzko. Combined with recent results of Brumer-Kramer and of Schoof, this result is best possible: if N is squarefree, there exists a non-zero semistable abelian variety over Q with good reduction outside primes dividing N precisely when N / ∈ {1, 2, 3, 5, 6, 7, 10, 13}. 1

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“…In particular, above Theorem implies that there are no such (nontrivial) abelian varieties Y (first proved in [13,27]). Our result also eliminates a great deal of other varieties, e.g.…”

Section: Introductionmentioning

confidence: 95%

A semi-stable case of the Shafarevich Conjecture

Abrashkin

Automorphic Forms and Galois Representations

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Non-existence of certain semistable abelian varieties

Cited by 22 publications

References 14 publications

On Serre’s conjecture for 2-dimensional mod p representations of Gal(ℚ∕ℚ)

On Serre’s conjecture for 2-dimensional mod p representations of Gal(ℚ∕ℚ)

Semistable abelian varieties over Q

A semi-stable case of the Shafarevich Conjecture

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