On the modularity of endomorphism algebras

• Autorzy: Brunault François

Identyfikatory

DOI

10.4064/ba230310-29-3

• Języki publikacji: EN

Abstrakty

EN

We show that any homomorphism between Jacobians of modular curves arises from a linear combination of Hecke modular correspondences. The proof uses the adelic language and is based on a study of the actions of GL2 and Galois on the étale cohomology of the tower of modular curves. We also make this result explicit for Ribet’s twisting operators on modular abelian varieties.

Słowa kluczowe

EN

modular curves; Hecke correspondences; endomorphism algebras; automorphic representations; Galois representations

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2023
• Tom: Vol. 71, no 1
• Strony: 22--33
• Opis fizyczny: Bibliogr. 9 poz.

Twórcy

autor

Brunault François

• ÉNS Lyon Unité de mathématiques pures et appliquées, 69007 Lyon, France

• https://orcid.org/0000-0002-6882-8694

Bibliografia

• [1] N. Bourbaki, Éléments de mathématique. Algèbre. Chapitre 8. Modules et anneaux semi-simples, 2nd rev. ed. of the 1958 edition, Springer, Berlin, 2012.
• [2] M. Dickson and M. Neururer, Products of Eisenstein series and Fourier expansions of modular forms at cusps, J. Number Theory 188 (2018), 137-164.
• [3] E. Kani, Endomorphisms of Jacobians of modular curves, Arch. Math. (Basel) 91 (2008), 226-237.
• [4] R. P. Langlands, Modular forms and ℓ-adic representations, in: Modular Functions of One Variable, II (Antwerp, 1972), Lecture Notes in Math. 349, Springer, Berlin, 1973, 361-500.
• [5] B. Mazur, Modular curves and the Eisenstein ideal, Publ. Math. Inst. Hautes Études Sci. 47 (1977), 33-186.
• [6] K. A. Ribet, Endomorphisms of semi-stable abelian varieties over number fields, Ann. of Math. (2) 101 (1975), 555-562.
• [7] K. A. Ribet, Twists of modular forms and endomorphisms of abelian varieties, Math. Ann. 253 (1980), 43-62.
• [8] A. J. Scholl, Motives for modular forms, Invent. Math. 100 (1990),419-430.
• [9] A. J. Scholl, Integral elements in K-theory and products of modular curves, in: The Arithmetic and Geometry of Algebraic Cycles (Banff, AB, 1998), NATO Sci. Ser. C Math. Phys. Sci. 548, Kluwer, Dordrecht, 2000, 467-489.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).