60 research outputs found
The field of moduli of quaternionic multiplication on abelian varieties
We consider principally polarized abelian varieties with quaternionic
multiplication over number fields and we study the field of moduli of their
endomorphisms in relation to the set of rational points on suitable Shimura
varieties.Comment: This version corrects Proposition 4.2 and some misprints in section 4
that appeared in the published versio
Stark points and Hida-Rankin p-adic L-function
This article is devoted to the elliptic Stark conjecture formulated by
Darmon, Lauder and Rotger [DLR], which proposes a formula for the
transcendental part of a -adic avatar of the leading term at of the
Hasse-Weil-Artin -series of an elliptic
curve twisted by the tensor product of two odd
-dimensional Artin representations, when the order of vanishing is two. The
main ingredient of this formula is a -adic regulator involving
the -adic formal group logarithm of suitable Stark points on . This
conjecture was proved in [DLR] in the setting where and
are induced from characters of the same imaginary quadratic field . In this
note we prove a refinement of this result, that was discovered experimentally
in Remark 3.4 of [DLR] in a few examples. Namely, we are able to determine the
algebraic constant up to which the main theorem of [DLR] holds in a particular
setting where the Hida-Rankin -adic -function associated to a pair of
Hida families can be exploited to provide an alternative proof of the same
result. This constant encodes local and global invariants of both and
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