HIRZEBRUCH–ZAGIER CYCLES IN <i>p</i> -ADIC FAMILIES AND ADJOINT <i>L</i> -VALUES

Abstract Let <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>E</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>F</mml:mi> </mml:math> $E/F$ upper E divided by upper F be a quadratic extension of totally real number fields. We show that the generalized Hirzebruch–Zagier cycles arising from the associated Hilbert modular varieties can be put in p -adic families. As an application, using the theory of base change, we give a geometric construction of the multivariable p -adic adjoint L -function twisted by the Hecke character of <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>E</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>F</mml:mi> </mml:math> $E/F$ upper E divided by upper F , attached to Hida families of Hilbert modular forms over F .