Abstract Modular Symbols

2021 ◽  
pp. 97-118
Author(s):  
Joël Bellaïche
Keyword(s):  
Modular Symbols

scholarly journals Λ-adic modular symbols over totally real fields

2011 ◽  
pp. 841-865 ◽  
Author(s):  
Baskar Balasubramanyam ◽  
Matteo Longo
Keyword(s):  
Modular Symbols ◽  
Totally Real ◽  
Totally Real Fields

scholarly journals Modular Symbols have a Normal Distribution

2004 ◽  
Vol 14 (5) ◽  
pp. 1013-1043 ◽  
Author(s):  
Y. N. Petridis ◽  
M. S. Risager
Keyword(s):  
Normal Distribution ◽  
Modular Symbols

scholarly journals The Eisenstein elements of modular symbols for level product of two distinct odd primes

2016 ◽  
Vol 281 (2) ◽  
pp. 257-285 ◽  
Author(s):  
Debargha Banerjee ◽  
Srilakshmi Krishnamoorthy
Keyword(s):  
Modular Symbols

scholarly journals Eisenstein series twisted by modular symbols for the group $\SL _{n}$

2000 ◽  
Vol 7 (6) ◽  
pp. 747-756 ◽  
Author(s):  
Dorian Goldfeld ◽  
Paul E. Gunnells
Keyword(s):  
Eisenstein Series ◽  
Modular Symbols

scholarly journals Overconvergent modular symbols and $p$-adic $L$-functions

2011 ◽  
Vol 44 (1) ◽  
pp. 1-42 ◽  
Author(s):  
Robert Pollack ◽  
Glenn Stevens
Keyword(s):  
Modular Symbols

Non-random behavior in sums of modular symbols

2021 ◽  
pp. 1-25
Author(s):  
Alex Cowan
Keyword(s):  
Eisenstein Series ◽  
Spectral Decomposition ◽  
Fourier Coefficients ◽  
Dirichlet Character ◽  
Common Phenomenon ◽  
Modular Symbols ◽  
Dirichlet Characters ◽  
Random Behavior ◽  
The Common ◽  
Automorphic Functions

We give explicit expressions for the Fourier coefficients of Eisenstein series twisted by Dirichlet characters and modular symbols on [Formula: see text] in the case where [Formula: see text] is prime and equal to the conductor of the Dirichlet character. We obtain these expressions by computing the spectral decomposition of automorphic functions closely related to these Eisenstein series. As an application, we then evaluate certain sums of modular symbols in a way which parallels past work of Goldfeld, O’Sullivan, Petridis, and Risager. In one case we find less cancelation in this sum than would be predicted by the common phenomenon of “square root cancelation”, while in another case we find more cancelation.

scholarly journals Distribution of Modular Symbols in ℍ3

2020 ◽  
Author(s):  
Petru Constantinescu
Keyword(s):  
Method Of Moments ◽  
Cusp Form ◽  
General Setting ◽  
Kleinian Groups ◽  
Congruence Subgroups ◽  
Modular Symbols ◽  
Ring Of Integers ◽  
Second Moments ◽  
A New Technique ◽  
Imaginary Number

Abstract We introduce a new technique for the study of the distribution of modular symbols, which we apply to the congruence subgroups of Bianchi groups. We prove that if $K$ is a quadratic imaginary number field of class number one and $\mathcal{O}_K$ is its ring of integers, then for certain congruence subgroups of $\textrm{PSL}_2(\mathcal{O}_K)$, the periods of a cusp form of weight two obey asymptotically a normal distribution. These results are specialisations from the more general setting of quotient surfaces of cofinite Kleinian groups where our methods apply. We avoid the method of moments. Our new insight is to use the behaviour of the smallest eigenvalue of the Laplacian for spaces twisted by modular symbols. Our approach also recovers the first and second moments of the distribution.

Lifting modular symbols of non-critical slope

2007 ◽  
Vol 161 (1) ◽  
pp. 141-155 ◽  
Author(s):  
Matthew Greenberg
Keyword(s):  
Modular Symbols ◽  
Critical Slope
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