Gauss–Manin connections for p -adic families of nearly overconvergent modular forms

Robert Harron[1]; Liang Xiao[2]

  • [1] Department of Mathematics, Keller Hall, University of Hawai‘i at Mānoa, Honolulu, HI 96822, USA
  • [2] Department of Mathematics, Mathematical Sciences Building, University of Connecticut, Storrs, Storrs, CT 06269, USA

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 6, page 2449-2464
  • ISSN: 0373-0956

Abstract

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We interpolate the Gauss–Manin connection in p -adic families of nearly overconvergent modular forms. This gives a family of Maass–Shimura type differential operators from the space of nearly overconvergent modular forms of type r to the space of nearly overconvergent modular forms of type r + 1 with p -adic weight shifted by 2 . Our construction is purely geometric, using Andreatta–Iovita–Stevens and Pilloni’s geometric construction of eigencurves, and should thus generalize to higher rank groups.

How to cite

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Harron, Robert, and Xiao, Liang. "Gauss–Manin connections for $p$-adic families of nearly overconvergent modular forms." Annales de l’institut Fourier 64.6 (2014): 2449-2464. <http://eudml.org/doc/275416>.

@article{Harron2014,
abstract = {We interpolate the Gauss–Manin connection in $p$-adic families of nearly overconvergent modular forms. This gives a family of Maass–Shimura type differential operators from the space of nearly overconvergent modular forms of type $r$ to the space of nearly overconvergent modular forms of type $r+1$ with $p$-adic weight shifted by $2$. Our construction is purely geometric, using Andreatta–Iovita–Stevens and Pilloni’s geometric construction of eigencurves, and should thus generalize to higher rank groups.},
affiliation = {Department of Mathematics, Keller Hall, University of Hawai‘i at Mānoa, Honolulu, HI 96822, USA; Department of Mathematics, Mathematical Sciences Building, University of Connecticut, Storrs, Storrs, CT 06269, USA},
author = {Harron, Robert, Xiao, Liang},
journal = {Annales de l’institut Fourier},
keywords = {Gauss–Manin connections; Nearly overconvergent modular forms; Eigencurves; Families of $p$-adic modular forms; Gauss-Manin connections; nearly overconvergent modular forms; eigencurves; families of -adic modular forms; Hodge filtration},
language = {eng},
number = {6},
pages = {2449-2464},
publisher = {Association des Annales de l’institut Fourier},
title = {Gauss–Manin connections for $p$-adic families of nearly overconvergent modular forms},
url = {http://eudml.org/doc/275416},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Harron, Robert
AU - Xiao, Liang
TI - Gauss–Manin connections for $p$-adic families of nearly overconvergent modular forms
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 6
SP - 2449
EP - 2464
AB - We interpolate the Gauss–Manin connection in $p$-adic families of nearly overconvergent modular forms. This gives a family of Maass–Shimura type differential operators from the space of nearly overconvergent modular forms of type $r$ to the space of nearly overconvergent modular forms of type $r+1$ with $p$-adic weight shifted by $2$. Our construction is purely geometric, using Andreatta–Iovita–Stevens and Pilloni’s geometric construction of eigencurves, and should thus generalize to higher rank groups.
LA - eng
KW - Gauss–Manin connections; Nearly overconvergent modular forms; Eigencurves; Families of $p$-adic modular forms; Gauss-Manin connections; nearly overconvergent modular forms; eigencurves; families of -adic modular forms; Hodge filtration
UR - http://eudml.org/doc/275416
ER -

References

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  1. F. Andreatta, A. Iovita, V. Pilloni, On overconvergent modular sheaves and modular forms for G L 2 / F  
  2. F. Andreatta, A. Iovita, G. Stevens, p -adic families of Siegel modular cuspforms 
  3. K. Buzzard, Eigenvarieties, -functions and Galois representations 320 (2007), 59-120, Cambridge Univ. Press, Cambridge Zbl1230.11054MR2367390
  4. R. Coleman, F. Gouvêa, N. Jochnowitz, E 2 , Θ , and overconvergence, Int. Math. Res. Not. (1995), 23-41 Zbl0846.11027MR1317641
  5. H. Darmon, V. Rotger, Diagonal cycles and Euler systems I: A p -adic Gross–Zagier formula Zbl06380120
  6. L. Fargues, La filtration de Harder–Narasimhan des schémas en groupes finis et plats, J. Reine Angew. Math. 645 (2010), 1-39 Zbl1199.14015MR2673421
  7. N. Katz, p -adic properties of modular schemes and modular forms, Modular functions of one variable, III 350 (1973), 69-190, Springer, Berlin Zbl0271.10033MR447119
  8. V. Pilloni, Overconvergent modular forms, Ann. Inst. Fourier 63 (2013), 219-239 Zbl1316.11034MR3097946
  9. E. Urban, Nearly overconvergent modular forms Zbl1328.11052
  10. Eric Urban, On the rank of Selmer groups for elliptic curves over ℚ , Automorphic representations and -functions 22 (2013), 651-680, Tata Inst. Fund. Res., Mumbai Zbl06322118MR3156865

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