Journal of the Mathematical Society of Japan

Yoshida lifts and Selmer groups

Siegfried BÖCHERER, Neil DUMMIGAN, and Rainer SCHULZE-PILLOT

Full-text: Open access

Abstract

Let $f$ and $g$, of weights $k' > k \geq 2$, be normalised newforms for $\Gamma_0(N)$, for square-free $N > 1$, such that, for each Atkin-Lehner involution, the eigenvalues of $f$ and $g$ are equal. Let $\lambda\mid\ell$ be a large prime divisor of the algebraic part of the near-central critical value $L(f\otimes g,(k+k'-2)/2)$. Under certain hypotheses, we prove that $\lambda$ is the modulus of a congruence between the Hecke eigenvalues of a genus-two Yoshida lift of (Jacquet-Langlands correspondents of) $f$ and $g$ (vector-valued in general), and a non-endoscopic genus-two cusp form. In pursuit of this we also give a precise pullback formula for a genus-four Eisenstein series, and a general formula for the Petersson norm of a Yoshida lift.

Given such a congruence, using the 4-dimensional $\lambda$-adic Galois representation attached to a genus-two cusp form, we produce, in an appropriate Selmer group, an element of order $\lambda$, as required by the Bloch-Kato conjecture on values of $L$-functions.

Article information

Source
J. Math. Soc. Japan Volume 64, Number 4 (2012), 1353-1405.

Dates
First available in Project Euclid: 29 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1351516778

Digital Object Identifier
doi:10.2969/jmsj/06441353

Mathematical Reviews number (MathSciNet)
MR2998926

Zentralblatt MATH identifier
1276.11069

Subjects
Primary: 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
Secondary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11F80: Galois representations 11F33: Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50] 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]

Keywords
Yoshida lift Bloch-Kato conjecture doubling method pullback formula

Citation

BÖCHERER, Siegfried; DUMMIGAN, Neil; SCHULZE-PILLOT, Rainer. Yoshida lifts and Selmer groups. J. Math. Soc. Japan 64 (2012), no. 4, 1353--1405. doi:10.2969/jmsj/06441353. https://projecteuclid.org/euclid.jmsj/1351516778.


Export citation

References

  • M. Agarwal and K. Klosin, Yoshida lifts and the Bloch-Kato conjecture for the convolution $L$-function, submitted.
  • A. N. Andrianov, Quadratic Forms and Hecke Operators, Grundlehren Math. Wiss., 286, Springer-Verlag, Berlin 1987.
  • T. Arakawa, Vector-valued Siegel's modular forms of degree two and the associated Andrianov $L$-functions, Manuscripta Math., 44 (1983), 155–185.
  • M. Asgari and R. Schmidt, Siegel modular forms and representations, Manuscripta Math., 104 (2001), 173–200.
  • J. Bergström, C. Faber and G. van der Geer, Siegel modular forms of degree three and the cohomology of local systems, arXiv:1108.3731v1[math:AG].
  • S. Bloch and K. Kato, $L$-functions and Tamagawa numbers of motives, In: The Grothendieck Festschrift. Vol. I, Progr. Math., 86, Birkhäuser, Boston, MA, 1990, pp.,333–400.
  • S. Böcherer, Über die Fourier-Jacobi-Entwicklung Siegelscher Eisensteinreihen. II, Math. Z., 189 (1985), 81–110.
  • S. Böcherer, Siegel modular forms and theta series, In: Theta Functions–Bowdoin 1987, Part 2 (Brnswick, ME, 1987), Proc. Symp. Pure Math., 49, Amer. math. Soc., Providence, RI, 1989, pp.,3–17.
  • S. Böcherer, Über die Funktionalgleichung automorpher $L$-Funktionen zur Siegelschen Modulgruppe, J. Reine Angew. Math., 362 (1985), 146–168.
  • S. Böcherer, Über die Fourierkoeffizienten der Siegelschen Eisensteinreihen, Manuscripta Math., 45 (1984), 273–288.
  • S. Böcherer, H. Katsurada and R. Schulze-Pillot, On the basis problem for Siegel modular forms with level, In: Modular Forms on Schiermonnikoog, Cambridge University Press, Cambridge, 2008, pp.,13–28.
  • S. Böcherer, T. Satoh and T. Yamazaki, On the pullback of a differential operator and its application to vector valued Eisenstein series, Comment. Math. Univ. St. Paul., 41 (1992), 1–22.
  • S. Böcherer and C.-G. Schmidt, $p$-adic measures attached to Siegel modular forms, Ann. Inst. Fourier (Grenoble), 50 (2000), 1375–1443.
  • S. Böcherer and R. Schulze-Pillot, Siegel modular forms and theta series attached to quaternion algebras, Nagoya Math. J., 121 (1991), 35–96.
  • S. Böcherer and R. Schulze-Pillot, The Dirichlet series of Koecher and Maaß and modular forms of weight $3/2$, Math. Z., 209 (1992), 273–287.
  • S. Böcherer and R. Schulze-Pillot, Siegel modular forms and theta series attached to quaternion algebras. II, Nagoya Math. J., 147 (1997), 71–106.
  • S. Böcherer and R. Schulze-Pillot, Vector valued theta series and Waldspurger's theorem, Abh. Math. Sem. Univ. Hamburg, 64 (1994), 211–233.
  • S. Böcherer and R. Schulze-Pillot, Mellin transforms of vector valued theta series attached to quaternion algebras, Math. Nachr., 169 (1994), 31–57.
  • J. Brown, Saito-Kurokawa lifts and applications to the Bloch-Kato conjecture, Compos. Math., 143 (2007), 290–322.
  • C.-L. Chai and G. Faltings, Degeneration of Abelian Varieties, Ergeb. Math. Grenzgeb. (3), 22, Springer-Verlag, Berlin, 1990.
  • P. Deligne, Valeurs de fonctions $L$ et périodes d'intégrales, In: Automorphic Forms, Representations, and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., 33, Amer. Math. Soc., Providence, RI, 1979, pp.,313–346.
  • F. Diamond, The refined conjecture of Serre, In: Elliptic Curves, Modular Forms & Fermat's Last Theorem, (eds. J. Coates and S.-T. Yau), Ser. Number Theory, 1, International Press, Cambridge, MA, 1995, pp.,22–37.
  • F. Diamond, M. Flach and L. Guo, The Tamagawa number conjecture of adjoint motives of modular forms, Ann. Sci. École Norm. Sup. (4), 37 (2004), 663–727.
  • N. Dummigan, Symmetric square $L$-functions and Shafarevich-Tate groups. II, Int. J. Number Theory, 5 (2009), 1321–1345.
  • N. Dummigan, Selmer groups for tensor product $L$-functions, In: Automorphic Representations, Automorphic $L$-functions and Arithmetic, Sûrikaisekikenkyûsho Kôkyûroku, 1659, 2009, pp.,37–46.
  • N. Dummigan, T. Ibukiyama and H. Katsurada, Some Siegel modular standard $L$-values, and Shafarevich-Tate groups, J. Number Theory, 131 (2011), 1296–1330.
  • B. Edixhoven, The weight in Serre's conjectures on modular forms, Invent. Math., 109 (1992), 563–594.
  • M. Eichler, The basis problem for modular forms and the traces of the Hecke operators, In: Modular Functions of One Variable, I, Lecture Notes in Math., 320, Springer-Verlag, Berlin, 1973, pp.,75-151.
  • S. A. Evdokimov, Action of the irregular Hecke operator of index $p$ on the theta-series of a quadratic form, J. Math. Sci., 38 (1987), 2078–2081.
  • G. Faltings, Crystalline cohomology and $p$-adic Galois-representations, In: Algebraic Analysis, Geometry and Number Theory, (ed. J. Igusa), Johns Hopkins University Press, Baltimore, 1989, pp.,25–80.
  • M. Flach, A generalisation of the Cassels-Tate pairing, J. Reine Angew. Math., 412 (1990), 113–127.
  • J.-M. Fontaine, Le corps de périodes $p$-adiques, Astérisque, 223 (1994), 59–111.
  • J.-M. Fontaine, Valeurs spéciales des fonctions $L$ des motifs, Séminaire Bourbaki, Vol.,1991/92, Astérisque, 206 (1992), Exp. No.,751, 4, 205–249.
  • J.-M. Fontaine and G. Lafaille, Construction de représentations $p$-adiques, Ann. Sci. École Norm. Sup. (4), 15 (1982), 547–608.
  • A. Genestier and J. Tilouine, Systèmes de Taylor-Wiles pour $\GSp(4)$, Astérisque, 302 (2005), 177–290.
  • R. Godement, Seminaire Cartan 10 (1957/58) Exp.4–9.
  • B. H. Gross and D. B. Zagier, Heegner points and derivatives of $L$-series, Invent. Math., 84 (1986), 225–320.
  • G. Harder, A congruence between a Siegel and an elliptic modular form, In: The 1-2-3 of Modular Forms, (eds. J. H. Bruinier et al.), Universitext, Springer-Verlag, Berlin, 2008, pp.,247–262.
  • A. Haruki, Explicit formulae of Siegel Eisenstein series, Manuscripta Math., 92 (1997), 107–134.
  • K. Hashimoto, The dimension of the spaces of cusp forms on Siegel upper half-plane of degree two. I, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 30 (1983), 403–488.
  • H. Hida, Modular Forms and Galois Cohomology, Cambridge Stud. Adv. Math., 69, Cambridge University Press, Cambridge, 2000.
  • H. Hida, Geometric Modular Forms and Elliptic Curves, World Scientific, Singapore, 2000.
  • H. Hijikata and H. Saito, On the representability of modular forms by theta series, In: Number Theory, Algebraic Geometry and Commutative Algebra, in honor of Y. Akizuki, Kinokuniya, Tokyo, 1973, pp.,13–21.
  • T. Ibukiyama, On differential operators on automorphic forms and invariant pluri-harmonic polynomials, Comment. Math. Univ. St. Paul., 48 (1999), 103–118.
  • T. Ibukiyama, Dimension formulas of Siegel modular forms of weight $3$ and supersingular abelian varieties, In: Siegel Modular Forms and Abelian Varieties, Proceedings of the Fourth Spring Conference on Modular Forms and Related Topics, Hamana Lake, Japan, 2007 (ed. T. Ibukiyama), Ryushido, 2007.
  • H. Jacquet and R. P. Langlands, Automorphic Forms on $GL(2)$, Lecture Notes in Math., 114, Springer-Verlag, Berlin, 1970.
  • J. X. Jia, Arithmetic of the Yoshida lift, Ph. D. thesis, University of Michigan, 2010. http://www.math.lsa.umich.edu/research/number_,theory/theses/johnson_,jia.pdf
  • M. Kashiwara and M. Vergne, On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math., 44 (1978), 1–47.
  • H. Katsurada, Congruence of Siegel modular forms and special values of their standard zeta functions, Math. Z., 259 (2008), 97–111.
  • W. Kohnen, Fourier coefficients of modular forms of half-integral weight, Math. Ann., 271 (1985), 237–268.
  • N. Kozima, On special values of standard $L$-functions attached to vector valued Siegel modular forms, Kodai Math. J., 23 (2000), 255–265.
  • R. P. Langlands, Modular forms and $\ell$-adic representations, In: Modular Functions of One Variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math., 349, Springer-Verlag, Berlin, 1973, pp.,361–500.
  • W. Luo, A note on the distribution of integer points on spheres, Math. Z., 267 (2011), 965–970.
  • H. Maaß, Siegel's Modular Forms and Dirichlet Series, Lecture Notes in Math., 216, Springer-Verlag, Berlin, 1971.
  • B. Mazur and A. Wiles, Class fields of abelian extensions of $\QQ$, Invent. Math., 76 (1984), 179–330.
  • A. Pitale and R. Schmidt, Ramanujan-type results for Siegel cusp forms of degree 2, J. Ramanujan Math. Soc., 24 (2009), 87–111.
  • S. Mizumoto, Eisenstein series for Siegel modular groups, Math. Ann., 297 (1993), 581–625.
  • C. Poor and D. S. Yuen, Dimensions of cusp forms for $\Gamma_0(p)$ in degree two and small weights, Abh. Math. Sem. Univ. Hamburg, 77 (2007), 59–80.
  • K. A. Ribet, A modular construction of unramified $p$-extensions of $\QQ(\mu_p)$, Invent. Math., 34 (1976), 151–162.
  • K. A. Ribet, On modular representations of $\Gal(\Qbar/\QQ)$ arising from modular forms, Invent. Math., 100 (1990), 431–476.
  • K. A. Ribet, Report on ${\rm mod}\,l$ representations of $\Gal(\Qbar/\QQ)$, In: Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., 55 (1994), 639–676.
  • T. Satoh, On certain vector valued Siegel modular forms of degree two, Math. Ann., 274 (1986), 335–352.
  • R. Schmidt, Iwahori-spherical representations of $\GSp(4)$ and Siegel modular forms of degree $2$ with square-free level, J. Math. Soc. Japan, 57 (2005), 259–293.
  • H. Shimizu, Theta series and automorphic forms on $GL_2$, J. Math. Soc. Japan, 24 (1972), 638–683.
  • G. Shimura, On Eisenstein series, Duke Math. J., 50 (1983), 417–476.
  • G. Shimura, On a class of nearly holomorphic automorphic forms, Ann. of Math. (2), 123 (1986), 347–406.
  • G. Shimura, Nearly holomorphic functions on Hermitian symmetric spaces, Math. Ann., 278 (1987), 1–28.
  • G. Shimura, The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math., 29 (1976), 783–804.
  • C. Skinner and E. Urban, Sur les déformations $p$-adiques de certaines représentations automorphes, J. Inst. Math. Jussieu, 5 (2006), 629–698.
  • W. Stein, The Modular Forms Database: Tables, http://modular.fas.harvard.edu/Tables/tables.html
  • H. Takayanagi, Vector-valued Siegel modular forms and their $L$-functions; application of a differential operator, Japan J. Math. (N.S.), 19 (1993), 251–297.
  • R. Taylor, On the $\ell$-adic cohomology of Siegel threefolds, Invent. Math., 114 (1993), 289–310.
  • E. Urban, Selmer groups and the Eisenstein-Klingen ideal, Duke Math. J., 106 (2001), 485–525.
  • E. Urban, On residually reducible representations on local rings, J. Algebra, 212 (1999), 738–742.
  • G. van der Geer, Siegel modular forms and their applications, In: The 1-2-3 of Modular Forms (eds. J. H. Bruinier et al.), Universitext, Springer-Verlag, Berlin, 2008, pp.,181–245.
  • R. Weissauer, Vektorwertige Siegelsche Modulformen kleinen Gewichtes, J. Reine Angew. Math., 343 (1983), 184–202.
  • R. Weissauer, Four dimensional Galois representations, Astérisque, 302 (2005), 67–150.
  • R. Weissauer, Endoscopy for $\GSp(4)$ and the Cohomology of Siegel Modular Threefolds, Lecture Notes in Math., 1968, Springer-Verlag, Berlin, 2009.
  • R. Weissauer, Existence of Whittaker models related to four dimensional symplectic Galois representations, In: Modular Forms on Schiermonnikoog, Cambridge University Press, Cambridge, 2008, pp.,285–310.
  • H. Yoshida, Siegel's modular forms and the arithmetic of quadratic forms, Invent. Math., 60 (1980), 193–248.
  • H. Yoshida, The action of Hecke operators on theta series, In: Algebraic and Topological Theories – to the memory of Dr. T. Miyata –, Kinokuniya, Tokyo, 1986, pp.,197–238.