Iwasawa theory for quadratic Hilbert modular forms

Loeffler, David A.; Zerbes, Sarah Livia

doi:10.48550/arxiv.2006.14491

Cited by 1 publication

(4 citation statements)

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“…We now apply the axiomatic leading-term argument developed in [LZ20c], leading up to the proof of Theorem 10.3.3 of op.cit.. This shows that, for any choice of the period Ω π , there exists a family of classes…”

Section: Construction Of the P-adic L-functionmentioning

confidence: 92%

“…Such a construction is available if π is cohomological, via lifting to GL 4 (although this is only written up under a restrictive assumption on the levels). Alternatively, if π is a θ-lift from GL 2 /K with K real quadratic, we can use modular symbols over K, as in [LZ20c].…”

Section: Construction Of the P-adic L-functionmentioning

confidence: 99%

“…Proof. This follows by exactly the same method as Theorem 12.2.2 of [LZ20c]; the assumption that our period be optimally normalised means that the comparison term µ min,u appearing in op.cit. is zero.…”

Section: Construction Of the P-adic L-functionmentioning

confidence: 99%

“…see [KL89] for the case of GL 2 -type abelian surfaces). For the case of inductions from real quadratic fields, see [LZ20c]. • The Paramodular Conjecture [BK14] predicts that (2) should imply (3), and many partial results are known; for instance, it is shown in [BCGP18] that (3) holds for infinitely many Q-isomorphism classes of generic abelian surfaces over Q.…”

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On the Birch-Swinnerton-Dyer conjecture for modular abelian surfaces

Loeffler¹,

Zerbes²

2021

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Let A be a modular abelian surface over Q which either has trivial geometric endomorphism ring, or arises as the restriction of scalars of an elliptic curve over an imaginary quadratic field which is modular and is not a Q-curve. In the former case, assume that there exists an odd Dirichlet character χ such that L(A, χ, 1) = 0. We prove the following implication: if L(A, 1) = 0, and the p-adic eigenvariety for GSp 4 is smooth at the point corresponding to A (and some auxiliary technical hypotheses hold), then A(Q) is finite, as predicted by the Birch-Swinnerton-Dyer conjecture, and the p-part of the Tate-Shafarevich group is also finite. We also prove one inclusion of the cyclotomic Iwasawa Main Conjecture for A. Moreover, we also prove analogous results for cohomological automorphic representations of GSp 4 , removing many of the restrictive hypotheses in our earlier work [LZ20a]; for cohomological representations we do not need to assume smoothness of the eigenvariety, since it is automatic in this case. The main ingredient in the proof is the Euler system attached to the spin representations of genus 2 Siegel modular forms constructed in our earlier work with Skinner [LSZ17].Date: last revised 2021-10-27 by davidloeffler, revision 1241.

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Section: Construction Of the P-adic L-functionmentioning

confidence: 92%

Section: Construction Of the P-adic L-functionmentioning

confidence: 99%

Section: Construction Of the P-adic L-functionmentioning

confidence: 99%

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confidence: 99%

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