Explicit Hecke series for symplectic group of genus 4

Vankov, Kirill

Journal de théorie des nombres de Bordeaux, Tome 23 (2011) no. 1, pp. 279-298.

Shimura a conjecturé la rationalité de la série de Hecke des groupes symplectiques de genre $n$. La conjecture a été prouvée par Andrianov pour un genre arbitraire mais une forme explicite n’était connue que pour les cas des genres 1, 2 et 3. Dans l’article, la forme explicite des polynômes rationnels pour la somme de la série génératrice de Hecke dans le groupe symplectiques de genre 4 a été présentée. Le calcul est basé sur l’isomorphisme de Satake, qui permet de réaliser toutes les opérations dans l’algèbre des polynômes à plusieurs variables. Nous avons aussi calculé les séries génératrices dans le cas spécial du choix des paramètres de Satake.

Shimura conjectured the rationality of the generating series for Hecke operators for the symplectic group of genus $n$. This conjecture was proved by Andrianov for arbitrary genus $n$, but the explicit expression was out of reach for genus higher than 3. For genus $n=4$, we explicitly compute the rational fraction in this conjecture. Using formulas for images of double cosets under the Satake spherical map, we first compute the sum of the generating series, which is a rational fraction with polynomial coefficients. Then we recover the coefficients of this fraction as elements of the Hecke algebra using polynomial representation of basis Hecke operators under the spherical map. Numerical examples of these fractions for special choice of Satake parameters are given.

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