Congruence Graphs
Samuele Anni
joint with Vandita Patel
University of Warwick
BMC
Bristol, 23rd March 2016
Samuele Anni
Congruence Graphs
Newforms
A modular form of level n and weight k is an holomorphic function on
the complex upper half-plane satisfying a functional equation and a
growth condition for the the coefficients of its power series expansion
f (z) =
∞
X
an q n ,
where
q = e 2πiz .
0
A newform is a cuspidal modular form (a0 = 0), normalized (a1 = 1),
which is an eigenform for the Hecke operators and arises from level n.
P
Let f be a newform: f = an q n , then Q ({an }) is a number field, called
Hecke eigenvalue field of f .
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Congruence Graphs
Congruence between newforms
Let f and g be two newforms.
X
f =
an q n
g=
X
bn q n .
Definition
We say that f and g are congruent mod p, if there exists an ideal p
above p in the compositum of the Hecke eigenvalue fields of f and g
such that ∀n
an ≡ bn mod p.
Samuele Anni
Congruence Graphs
Congruence Graphs
Vertices: each vertex in the graph corresponds to a Galois orbit of
newforms of level and weight in a given set.
Edges: an edge between two vertices is drawn if there exists a prime
p such that a congruence mod p holds between two forms in the
Galois orbits considered.
Let S be the set of divisors of a positive integer and let W be a finite set
of weights, we will denote the associated graph by GS,W .
These graphs are related to the dual graphs of the spectrum of the Hecke
algebra of given level and weight.
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Congruence Graphs
G[1,2,43,86],[2]
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Congruence Graphs
G[1,2,43,86],[2,4]
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Congruence Graphs
The graph GS,W may not be connected. The computations suggest the
following conjecture:
Conjecture
Given S and W , there exists a finite set W 0 , with W ⊆ W 0 , such that
the graph GS,W 0 is connected.
This conjecture is a theorem if we assume Maeda’s conjecture for level 1
modular forms.
Samuele Anni
Congruence Graphs
Another observation obtained through extensive computations is the
existence of forms which are congruent modulo different primes to each
of the newforms in the set considered.
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Congruence Graphs
Subgraph of G[1,2,43,86],[2,4]
Samuele Anni
Congruence Graphs
A mod 2 phenomena?
Congruences modulo small primes occurr often. Anyway, removing such
primes and increasing the set of weights the resulting graphs are still
connected.
G[1,2,3,4,6,12],[2,4,6]
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Congruence Graphs
G[1,2,3,4,6,12],[2,4,6] no 2, 3
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G[1,2,3,4,6,12],[2,4,6,8] no 2, 3
Congruence Graphs
G[1,2,3,4,6,12],[2,4,6,8,··· ,32] no 2, 3
Samuele Anni
Congruence Graphs
Motivations
Study of chain of congruences techniques developed by Dieulefait;
Generalizations of Maeda’s conjecture;
Factorization of characteristic polynomials of Hecke operators
modulo primes;
Building databases or residual modular Galois representation;
Prove cases of base change in the Langlands program.
Samuele Anni
Congruence Graphs
Congruence Graphs
Samuele Anni
joint with Vandita Patel
University of Warwick
BMC
Bristol, 23rd March 2016
Thanks!
Samuele Anni
Congruence Graphs