# 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

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topHarron, 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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- Eric Urban, On the rank of Selmer groups for elliptic curves over $\mathbb{Q}$, Automorphic representations and -functions 22 (2013), 651-680, Tata Inst. Fund. Res., Mumbai Zbl06322118MR3156865

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