Oddness of residually reducible Galois representations

Berger, Tobias

doi:10.1142/s1793042118500835

Cited by 5 publications

(5 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These fundamental p-adic properties of Gauss sums may have crucial consequences in various domains since Vandiver's conjecture is often required; for instance: In [8] about the Galois cohomology of Fermat curves, in [47] for the root numbers of the Jacobian varieties of Fermat curves, then in several papers on Galois p-ramification theory as in [36,44,45,46], or [53,54] in relation with modular forms, then in numerous papers and books on the theory of deformations of Galois representations as in [2,37], Iwasawa's theory context and cyclotomy, as in [7] on Ihara series, [5] for µ-invariants in Hida families, [32] for the main conjecture of the Iwasawa theory).…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Test of Vandiver's conjecture with Gauss sums -- Heuristics

Gras

2018

Preprint

View full text Add to dashboard Cite

The link between Vandiver's conjecture and Gauss sums is well known since the papers of Iwasawa (1975), Thaine (1995 and Anglès-Nuccio (2010). This conjecture is required in many subjects and we shall give such examples of relevant references. In this paper, we recall our interpretation of Vandiver's conjecture in terms of minus part of the torsion of the Galois group of the maximal abelian p-ramified pro-p-extension of the pth cyclotomic field (1984). Then we provide a specific use of Gauss sums of characters of order p of F × ℓ and prove new criteria for Vandiver's conjecture to hold (Theorem 1.2 (a) using both the sets of exponents of p-irregularity and of p-primarity of suitable twists of the Gauss sums, and Theorem 1.2 (b) which does not need the knowledge of Bernoulli numbers or cyclotomic units). We propose in § 5.2 new heuristics showing that any counterexample to the conjecture leads to excessive constraints modulo p on the above twists as ℓ varies and suggests analytical approaches to evidence. We perform numerical experiments to strengthen our arguments in direction of the very probable truth of Vandiver's conjecture. All the calculations are given with their PARI/GP programs.

show abstract

Section: Discussionmentioning

confidence: 99%

“…2,2,1,5,7,6,2,2,1,5,5,5,4,4,3,3,4,5,4,5,6,5,5,5,3,6,1,6,3,5,4,5, 0,2,3,5,7,3,3,3,2,4,5,7,6,6,5,6,1,7,4,7] …”

mentioning

confidence: 99%

Test of Vandiver's conjecture with Gauss sums -- Heuristics

Gras

2018

Preprint

View full text Add to dashboard Cite

show abstract

“…When π is regular, Theorem 1.3 is the main result of [BC11]. For an application of the oddness of these Galois representations see [Ber18], in particular Remark 2.7.…”

Section: The Sign Of An Essentially Conjugate Self-dual Representationmentioning

confidence: 99%

Lafforgue pseudocharacters and parities of limits of Galois representations

Berger

Weiss

2021

manuscripta math.

Self Cite

View full text Add to dashboard Cite

Let F be a CM field with totally real subfield F + and let π be a C-algebraic cuspidal automorphic representation of the unitary group U(a, b)(A F + ), whose archimedean components are discrete series or non-degenerate limit of discrete series representations. We attach to π a Galois representation Rπ : Gal(F /F + ) → C U(a, b)(Q ℓ ) such that, for any complex conjugation element c, Rπ(c) is as predicted by the Buzzard-Gee conjecture [BG14]. As a corollary, we deduce that the Galois representations attached to certain irregular, C-algebraic essentially conjugate self-dual cuspidal automorphic representations of GLn(AF ) are odd in the sense of Bellaïche-Chenevier [BC11].

show abstract

“…whose isomorphism class is independent of the choice of h. For any quadratic extension M/K of fields with Gal(M/K) = ι and a Galois representation ρ of G M , the representation ρ ⊗ ι ρ of G K is said to be the twisted tensor product of ρ (cf. Section 2.1 of [7]).…”

Section: An Identification Betweenmentioning

confidence: 99%

Automorphy of mod 2 Galois representations associated to the quintic Dwork family and reciprocity of some quintic trinomials

Tsuzuki¹,

Yamauchi²

2020

Preprint

View full text Add to dashboard Cite

In this paper, we determine mod 2 Galois representations ρ ψ,2 : GK := Gal(K/K) −→ GSp 4 (F2) associated to the mirror motives of rank 4 with pure weight 3 coming from the Dwork quintic family X 5 0 + X 5 1 + X 5 2 + X 5 3 + X 5 4 − 5ψX0X1X2X3X4 = 0, ψ ∈ K defined over a number field K under the irreducibility condition of the quintic trinomial f ψ below.Applying this result, when K = F is a totally real field, for some at most qaudratic totally real extension M/F , we prove that ρ ψ,2 |G M is associated to a Hilbert-Siegel modular Hecke eigen cusp form for GSp 4 (AM ) of parallel weight three.In the course of the proof, we observe that the image of such a mod 2 representation is governed by reciprocity of the quintic trinomialwhose decomposition field is generically of type 5-th symmetric group S5. This enable us to use results on the modularity of 2-dimensional, totally odd Artin representations of Gal(F /F ) due to Shu Sasaki and several Langlands functorial lifts for Hilbert cusp forms. Then, it guarantees the existence of a desired Hilbert-Siegel modular cusp form of parallel weight three matching with the Hodge type of the compatible system in question.

show abstract

Oddness of residually reducible Galois representations

Cited by 5 publications

References 24 publications

Test of Vandiver's conjecture with Gauss sums -- Heuristics

Test of Vandiver's conjecture with Gauss sums -- Heuristics

Lafforgue pseudocharacters and parities of limits of Galois representations

Automorphy of mod 2 Galois representations associated to the quintic Dwork family and reciprocity of some quintic trinomials

Contact Info

Product

Resources

About