Abelian surfaces over totally real fields are potentially modular

Boxer, George; Calegari, Frank; Gee, Toby; Pilloni, Vincent

doi:10.1007/s10240-021-00128-2

Cited by 29 publications

(63 citation statements)

References 199 publications

(260 reference statements)

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“…Boxer and Pilloni conjectured the existence of Hida and Coleman theories in all cohomological degrees for all Shimura varieties, confirming this prediction in the simplest case of GL 2 in the recent work [BP20b] and started developing Coleman theory more generally in [BP20a]. In [Pil20] and [BCGP18], the integral control theorem for ordinary families is obtained assuming the weights are big enough. The control theorem for more general weights is obtained at the cost of inverting p and losing control of torsion, using Coleman theory: the authors of op.…”

Section: Introductionmentioning

confidence: 65%

“…The recent pioneering works [Pil20, BCGP18, BP20b, BP20a] have developed analogous theories for higher degree coherent cohomology. In [Pil20], Pilloni introduced higher Hida and Coleman theory for automorphic forms for GSp 4 /Q and these ideas were later generalised in [BCGP18] for Res F/Q GSp 4 , where F is a totally real field in which the prime p totally splits, and used to prove potential modularity of abelian surfaces over F . Boxer and Pilloni conjectured the existence of Hida and Coleman theories in all cohomological degrees for all Shimura varieties, confirming this prediction in the simplest case of GL 2 in the recent work [BP20b] and started developing Coleman theory more generally in [BP20a].…”

Section: Introductionmentioning

confidence: 99%

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Higher Hida theory for Hilbert modular varieties in the totally split case

Grossi¹

2021

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We study p-adic properties of the coherent cohomology of some automorphic sheaves on the Hilbert modular variety X for a totally real field F in the case where the prime p is totally split in F . More precisely, we develop higher Hida theory à la Pilloni, constructing, for 0 ≤ q ≤ [F : Q], some modules M q which p-adically interpolate the ordinary part of the cohomology groups H q (X, ω κ ), varying the weight κ of the automorphic sheaf. Contents 1. Introduction 1 2. Preliminaries 4 2.1. Hilbert modular varieties and moduli interpretation 4 2.2. Compactifications 6 2.3. Automorphic vector bundles 6 3. Hecke operators and Hasse invariants 10 3.1. Hecke operators 10 3.2. Partial Hasse invariants and the Goren-Oort stratification 12 4. Higher Hida theory 13 4.1. Mod p theory 13 4.2. Characteristic zero theory 17 4.3. Duality 27 References 29

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Section: Introductionmentioning

confidence: 65%

Section: Introductionmentioning

confidence: 99%

Higher Hida theory for Hilbert modular varieties in the totally split case

Grossi¹

2021

Preprint

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“…Potential modularity of elliptic curves is known if F is totally real or quadratic over a totally real field. We refer to [7][1.1] for a discussion of these results and for the original references. In [7] potential modularity is also shown for abelian surfaces over totally real fields F and hence Conjecture 1.3 should hold for regular models of genus 2 curves over totally real fields F (since this involves the local Langlands correspondence for GSp 4 /F we are unsure whether the conductor in the functional equation is indeed the Artin conductor).…”

Section: Recall the Definition Of The Completedmentioning

confidence: 99%

“…We call the real line det R RΓ(X (C), C) + the de Rham real structure of det C RΓ(X (C), C) and the real line det R RΓ(X (C), R(n)) the Betti real structure of det C RΓ(X (C), C). By (7) we have…”

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confidence: 99%

Compatibility of Special value conjectures with the functional equation of Zeta functions

Flach¹,

Morin²

2020

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We prove that the special value conjecture for the Zeta function ζ(X , s) of a proper, regular arithmetic scheme X that we formulated in [8][Conj. 5.12] is compatible with the functional equation of ζ(X , s) provided that the factor C(X , n) we were not able to compute in loc. cit. has the simple explicit form suggested in [9].We now explain the significance of this result. Let ζ(X , s)Conjecture 1.3. (Functional Equation) Let X be a regular scheme of dimension d, proper and flat over Spec(Z). Then ζ(X , s) has a meromorphic continuation to all s ∈ C and By [8][Prop. 5.29], the exact triangle (3) and Weil-étale duality [8][Thm. 3.22] induce a canonical isomorphismcompatible with ξ ∞ and the trivializations (2) of ∆(X /Z, n) and ∆(X /Z, d − n). We obtain Theorem 1.4. Assume that ζ(X , s) satisfies Conjecture 1.3. Then Conjecture 1.1 for (X , n) is equivalent to Conjecture 1.1 for (X , d − n).

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“…We have an isomorphism ι : S 6 ∼ − → Sp 4 (F 2 ), where S 6 is the symmetric group on 6 letters, which we make explicit in the following manner. Let U := F 6 2 , and equip U with the coordinate action of S 6 and the standard nondegenerate alternating (equivalently, symmetric) bilinear form x, y = 6 i=1 x i y i visibly compatible with the S 6 -action. Let U 0 ⊂ U be the trace 0 hyperplane, let L be the F 2 -span of (1, .…”

Section: Group Theory and Galois Theory Formentioning

confidence: 99%

On the paramodularity of typical abelian surfaces

et al. 2019

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Generalizing the method of Faltings-Serre, we rigorously verify that certain abelian surfaces without extra endomorphisms are paramodular. To compute the required Hecke eigenvalues, we develop a method of specialization of Siegel paramodular forms to modular curves.Date: August 10, 2018. 2010 Mathematics Subject Classification. 11F46, 11Y40.1 is made precise by a conjecture of Brumer-Kramer [8, Conjecture 1.1], restricted here for simplicity.Conjecture 1.1.1 (Brumer-Kramer). To every abelian surface A over Q of conductor N with End(A) = Z, there exists a cuspidal, nonlift Siegel paramodular newform f of degree 2, weight 2, and level N with rational Hecke eigenvalues, such thatMoreover, f is unique up to (nonzero) scaling and depends only on the isogeny class of A; and if N is squarefree, then this association is bijective.Conjecture 1.1.1 is often referred to as the paramodular conjecture. As pointed out by Frank Calegari, in general it is necessary to include abelian fourfolds with quaternionic multiplication for the converse assertion: for a precise statement for arbitrary N and further discussion, see Brumer-Kramer [7, Section 8].Extensive experimental evidence [8,47] supports Conjecture 1.1.1. There is also theoretical evidence for this conjecture when the abelian surface A is potentially of GL 2 -type, acquiring extra endomorphisms over a quadratic field: see Johnson-Leung-Roberts [34] for real quadratic fields and Berger-Dembélé-Pacetti-Şengün [5] for imaginary quadratic fields. For a complete treatment of the many possibilities for the association of modular forms to abelian surfaces with potentially extra endomorphisms, see work of Booker-Sijsling-Sutherland-Voight-Yasaki [11]. What remains is the case where End(A Q al ) = Z, which is to say that A has minimal endomorphisms defined over the algebraic closure Q al ; we say then that A is typical. (We do not say generic, since it is not a Zariski open condition on the moduli space.)Recently, there has been dramatic progress in modularity lifting theorems for nonlift Siegel modular forms (i.e., forms not of endoscopic type): see Pilloni [44] for p-adic overconvergent modularity lifting, as well as recent work by Calegari-Geraghty [12, §1.2], Berger-Klosin with Poor-Shurman-Yuen [2] establishing modularity in the reducible case when certain congruences are provided, and a paper in preparation by Boxer-Calegari-Gee-Pilloni [6] establishing potential modularity over totally real fields. 1.2. Main result. For all prime levels N < 277, the paramodular conjecture is known: there are no paramodular forms of the specified type by work of Poor-Yuen [47, Theorem 1.2], and correspondingly there are no abelian surfaces by work of Brumer-Kramer [8, Proposition 1.5]. At level N = 277, there exists a cuspidal, nonlift Siegel paramodular cusp form, unique up to scalar multiple, by work of Poor-Yuen [47, Theorem 1.3]: this form is given explicitly as a rational function in Gritsenko lifts of ten weight 2 theta blocks-see (6.2.2).Our main result is as follows. Theorem 1.2.1. ...

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Abelian surfaces over totally real fields are potentially modular

Cited by 29 publications

References 199 publications

Higher Hida theory for Hilbert modular varieties in the totally split case

Higher Hida theory for Hilbert modular varieties in the totally split case

Compatibility of Special value conjectures with the functional equation of Zeta functions

On the paramodularity of typical abelian surfaces

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