Abstract
We prove an explicit integral representation—involving the pullback of a suitable Siegel Eisenstein series—for the twisted standard L-function associated to a holomorphic vector-valued Siegel cusp form of degree n and arbitrary level. In contrast to all previously proved pullback formulas in this situation, our formula involves only scalar-valued functions despite being applicable to L-functions of vector-valued Siegel cusp forms. The key new ingredient in our method is a novel choice of local vectors at the archimedean place which allows us to exactly compute the archimedean local integral. By specializing our integral representation to the case n=2 we are able to prove a reciprocity law—predicted by Deligne’s conjecture—for the critical values of the twisted standard L-function for vector-valued Siegel cusp forms of degree 2 and arbitrary level. This arithmetic application generalizes previously proved critical-value results for the full level case. By specializing further to the case of Siegel cusp forms obtained via the Ramakrishnan–Shahidi lift, we obtain a reciprocity law for the critical values of the symmetric fourth L-function of a classical newform.
Highlights
Résumé Nous prouvons une représentation intégrale explicite – faisant intervenir le pullback d’une série de Siegel–Eisenstein appropriée – pour la fonction L standard tordue associée à une forme modulaire vectorielle de Siegel holomorphe cuspidale de degré n et de niveau arbitraire
En spécialisant notre représentation intégrale au cas n = 2 nous sommes en mesure de prouver une loi de réciprocité – prédite par la conjecture de Deligne – pour les valeurs critiques de la fonction L standard tordue pour les formes modulaires vectorielles de Siegel cuspidales de degré 2 et de niveau
En spécialisant de plus au cas des formes modulaires de Siegel cuspidales obtenues par le relèvement de Ramakrishnan–Shahidi, nous obtenons une loi de réciprocité pour les valeurs critiques de la fonction L du quatrième produit symétrique pour une forme primitive (“newform”) classique
Summary
We explain our choice of φv and fv at a real place v, which represents one of the main new ideas of this paper. In all previously proved classical pullback formulas [6,61,62] for L(s, π χ) with π∞ a general discrete series representation, the analogues of F and Ekχ,N (−, s) were vector-valued objects; in contrast, our formula involves only scalar-valued functions This is a key point of the present work. Initial steps towards this application have already been made by us in [40] where we build upon the results of this paper, and prove p-integrality and cuspidality of pullbacks of the Eisenstein series Ekχ,N (Z , s) It seems worth mentioning here the recent work of Zheng Liu [33] who uses the doubling method for vector-valued Siegel modular forms and constructs a p-adic L-function
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