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Mod-2 Hecke algebras of level 3 and 5

  • Shaunak V. Deo [2] ; Anna Medvedovsky [1]
    1. [1] University of Arizona

      University of Arizona

      Estados Unidos

    2. [2] Department of Mathematics, Indian Institute of Science, India
  • Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 5, 2024
  • Idioma: inglés
  • DOI: 10.1007/s00029-024-00961-3
  • Enlaces
  • Resumen

    • We use deformation theory to study the big Hecke algebra acting on mod-2 modular forms of prime level N and all weights, especially its local component at the trivial representation. For N=3,5, we prove that the maximal reduced quotient of this big Hecke algebra is isomorphic to the maximal reduced quotient of the corresponding universal deformation ring. Then we completely determine the structure of this big Hecke algebra. We also describe a natural grading on mod-p Hecke algebras, and prove an R=T theorem for the partially full version of our mod-2 Hecke algebra.

  • Referencias bibliográficas
    • Anni, S., Ghitza, A., Medvedovsky, A.: Elementary symmetric functions and deep power-sum congruences. To appear in Integers. Available at...
    • Atkin, A.O.L., Lehner, J.: Hecke operators on 0(m). Math. Ann. 185, 134–160 (1970)
    • Atiyah, M. F., Macdonald, I. G.: Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont.,...
    • Bellaïche, J.: Une représentation galoisienne universelle attachée aux formes modulaires modulo 2. C. R. Math. Acad. Sci. Paris 350(9–10),...
    • Bellaïche, J.: The Eigenbook: eigenvarieties, families of Galois representations, p-adic L-functions. Pathways in Mathematics, Birkhauser-Springer,...
    • Bellaïche, J.: Image of pseudo-representations and coefficients of modular forms modulo p. Adv. Math. 353, 647–721 (2019)
    • Bellaïche, J., Chenevier, G.: Families of Galois representations and Selmer groups. Asterisque 324 (2009)
    • Bellaïche, J., Khare, C.: Level 1 Hecke algebras of modular forms modulo p. Compos. Math. 151(3), 397–415 (2015)
    • Bergdall, J., Pollack, R.: Slopes of modular forms and the Ghost conjecture II. Trans. Amer. Math. Soc. 372(1), 357–388 (2019)
    • Calegari, F.: Congruences between modular forms. Notes from Arizona Winter School (2013). Available at http://www.math.uchicago.edu/~fcale/papers/AWS.pdf
    • Calegari, F., Specter, J.: Pseudorepresentations of weight one are unramified. Algebra Number Theory 13(7), 1583–1596 (2019)
    • Carayol, H.: Sur les représentations -adiques associées aux formes modulaires de Hilbert. Ann. Sci. École Norm. Sup. (4) 19, 409–468 (1986)
    • Carayol, H.: Sur les représentations galoisienne modulo attachées aux formes modulaires. Duke Math. J. 59, 785–801 (1989)
    • Chenevier, G.: The p-adic analytic space of pseudocharacters of a profinite group and pseudorepresentations over arbitrary rings. Automorphic...
    • Deo, S.V.: Structure of Hecke algebras of modular forms modulo p. Algebra Number Theory 11(1), 1–38 (2017)
    • Deo, S.V., Medvedovsky, A.: Newforms mod p in squarefree level with applications to Monsky’s Hecke-stable filtration (with an appendix by...
    • Diamond, F., Shurman, J.: A first course in modular forms. Graduate Texts in Mathematics, 228, Springer-Verlag, New York, (2005), xvi+436
    • Emerton, M.: p-adic families of modular forms (after Hida, Coleman and Mazur). Séminaire Bourbaki, Vol. 2009/2010, Exposés 1012–1026, Astérisque...
    • Emerton, M., Pollack, R., Weston, T.: Variation of Iwasawa invariants in Hida families. Invent. Math. 163(3), 523–580 (2006)
    • Gerbelli-Gauthier, M.: The order of nilpotence of Hecke operators mod 2: a new proof. Res. Number Theory (2016). https://doi.org/10.1007/s40993-016-0038-6
    • Gouvêa, F.: Deformations of Galois representations. IAS, Park City Math. Ser., 9, Arithmetic algebraic geometry (Park City, UT, 233–406, p....
    • Gouvêa, F., Mazur, B.: On the density of modular representations. Computational perspectives on number theory (Chicago, IL, 1995), 127-142,...
    • Hatada, K.: On classical and -adic modular forms of levels N m and N. J. Number Theory 87, 1–14 (2001)
    • Iwaniec, H.: Topics in classical automorphic forms. Graduate Studies in Mathematics, 17. American Mathematical Society, Providence, RI, (1997)
    • Jochnowitz, N.: Congruences between systems of eigenvalues of modular forms. Trans. Amer. Math. Soc. 270(1), 269–285 (1982)
    • Jochnowitz, N.: A study of the local components of the Hecke algebra mod l. Trans. Amer. Math. Soc. 270(1), 253–267 (1982)
    • Katz, N.: p-Adic properties of modular schemes and modular forms. In Modular functions of one variable, III (Proc. Internat. Summer School,...
    • Katz, N.: Higher congruences between modular forms. Ann. Math. 101, 332–367 (1975)
    • Katz, N.: A result on modular forms in characteristic p. In J.-P. Serre, D. B. Zagier (eds.), Modular functions of one variable V, Lecture...
    • Khare, C.: Mod p modular forms. Number theory (Tiruchirapalli, Contemp. Math., 210, Amer. Math. Soc. Providence, RI 1998, 135–149 (1996)
    • Markshaitis, G.: On p-extensions with single critical number. Izvestiya akademii nauk SSSR 27, 463– 466 (1963). (In Russian)
    • Medvedovsky, A.: Lower bounds on dimensions of mod p Hecke algebras: the nilpotence method. Ph.D. Thesis (2015), Brandeis University, https://www.math.brown.edu/~medved/Mathwriting/ DissertationMedvedovsky.pdf
    • Medvedovsky, A.: An explicit Galois representation on the mod-3 Hecke algebra. In preparation, https://math.bu.edu/people/medved/Mathwriting/Mod3galoisrep.pdf
    • Medvedovsky, A.: Nilpotence order growth of recursion operators in characteristic p. Algebra Number Theory 12(3), 693–722 (2018)
    • Meier, L.: Additive decompositions for rings of modular forms. Doc. Math. 27, 427–488 (2022)
    • Monsky, P.: A Hecke algebra attached to mod 2 modular forms of level 3: arXiv:1508.07523
    • Monsky, P.: A Hecke algebra attached to mod 2 modular forms of level 5: arXiv:1610.07058
    • Monsky, P.: Generators and relations for the shallow mod-2 Hecke algebra in levels 0(3) and 0(5): arXiv:1703.04193
    • Monsky, P.: Variations on a lemma of Nicolas and Serre: arXiv:1604.02622
    • Moon, H., Tanigachi, Y.: The non-existence of certain mod 2 Galois representations of some small quadratic fields. Proc. Jpn. Acad. Ser. A...
    • Nicolas, J.-L., Serre, J.-P.: L’ordre de nilpotence des opérateurs de Hecke modulo 2. C. R. Math. Acad. Sci. Paris 350(7–8), 343–348 (2012)
    • Nicolas, J.-L., Serre, J.-P.: Formes modulaires modulo 2: structure de l’algèbre de Hecke. C. R. Math. Acad. Sci. Paris 350(9–10), 449–454...
    • Nyssen, L.: Pseudo-représentations. Math. Ann. 306, 257–283 (1996)
    • Rouquier, R.: Caractérisation des caractères et pseudo-caractères. J. Algebra 180(2), 571–586 (1992)
    • Sengun, M.H.: The nonexistence of certain representations of the absolute Galois group of quadratic fields. Proc. Amer. Math. Soc. 137, 27–35...
    • de Smit, B., Lenstra, H.W., Jr.: Explicit construction of universal deformation rings. In: Modular forms and Fermat’s last theorem (Boston....
    • Swinnerton-Dyer, H.P.F.: On l-adic representations and congruences for coefficients of modular forms. In Modular functions of one variable,...
    • SageMath, the Sage Mathematics Software System, https://www.sagemath.org
    • The Stacks project: https://stacks.math.columbia.edu, 2021
    • Serre, J-P.: Congruences et formes modulaires [d’après H. P. F. Swinnerton-Dyer]. Séminaire Bourbaki, 24e année (1971/1972), Exp. No. 416....
    • Serre, J-P.: Formes modulaires et fonctions zêta p-adiques. Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp,...
    • Tate, J.: The non-existence of certain Galois extensions of Q unramified outside 2. Arithmetic geometry (Tempe, AZ, 1993), 153–156, Contemp....
    • Taylor, R.: Galois representations associated to Siegel modular forms of low weight. Duke Math. J. 63, 281–332 (1991)
    • Wake, P., Wang-Erickson, C.: The Eisenstein ideal with squarefree level. Adv. Math. 380, 107543 (2021)
    • Weston, T.: Explicit unobstructed primes for modular deformation problems of squarefree level. J. Number Theory 110, 199–218 (2005)
    • Wiles, A.: On ordinary λ-adic representations associated to modular forms. Invent. Math. 94, 529–573 (1988)
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