• Corpus ID: 245385727

On Shafarevich-Tate groups and analytic ranks in families of modular forms, II. Coleman families

@inproceedings{Pati2021OnSG,
  title={On Shafarevich-Tate groups and analytic ranks in families of modular forms, II. Coleman families},
  author={Maria Rosaria Pati and Gautier Ponsinet and Stefano Vigni},
  year={2021}
}
This is the second article in a two-part project whose aim is to study algebraic and analytic ranks in p-adic families of modular forms. Let f be a newform of weight 2, square-free level N and trivial character, let Af be the abelian variety attached to f , whose dimension will be denoted by df , and for every prime number p ∤ N at which f has finite slope let f (p) be a p-adic Coleman family through f over a suitable open disc in the p-adic weight space. We prove that, for all but finitely… 

References

SHOWING 1-10 OF 65 REFERENCES
On Shafarevich-Tate groups and analytic ranks in families of modular forms, I. Hida families
Let f be a newform of weight 2, square-free level and trivial character, let Af be the abelian variety attached to f and for every good ordinary prime p for f let f (p) be the p-adic Hida family
P-adic Banach spaces and families of modular forms
Let p be a prime, Cp the completion of an algebraic closure of the p-adic numbers Qp and K a nite extension of Qp contained in Cp. Let v be the valuation on Cp such that v(p) = 1 and let | | be the
Modular p-adic L-functions attached to real quadratic fields and arithmetic applications
Let f ∈ Sk0+2(Γ0(Np)) be a normalized N -new eigenform with p N and such that ap 6= pk0+1 and ordp (ap) < k0 + 1. By Coleman’s theory, there is a p-adic family F of eigenforms whose weight k0 + 2
Motives for modular forms
Let M be a pure motive over a number field F of rank n with coefficients in T ⊂ C; M may be thought of as a direct factor of the cohomology of a smooth projective variety X over F cut out by an
On the semi-simplicity of the $U_p$-operator on modular forms
For and positive integers, let denote the -vector space of cuspidal modular forms of level and weight . This vector space is equipped with the usual Hecke operators , . If we need to consider several
Where the Slopes Are
Fix a prime number p and choose, once and for all, an embedding of the algebraic closure of Q into Qp. Let k and N be integers, and suppose N is not divisible by p. If f is a modular form of weight
On the parity ranks of Selmer groups
Assume that p > 3 and that / is ordinary at p, i.e. that ap(f) 6 Fp is a p-adic unit. According to Hida's theory, there is a p-adic family of ordinary modular forms of varying weights containing /
Central derivatives of L-functions in Hida families
We prove a result of the following type: given a Hida family of modular forms, if there exists a weight two form in the family whose L-function vanishes to exact order one at s = 1, then all but
Kolyvagin's Conjecture and patched Euler systems in anticyclotomic Iwasawa theory
Let E/Q be an elliptic curve of conductor N and let K be an imaginary quadratic field. Under a certain Heegner hypothesis, Kolyvagin constructed cohomology classes for E using K-CM points and
Two variable p-adic L functions attached to eigenfamilies of positive slope
AbstractConsider a classical cusp eigenform f=∑n=1∞an(f)qn of weight k≥2 for Γ0(N) with a Dirichlet character ψ mod N, and let Lf(s,χ)=∑n=1∞χ(n)an(f)n-s denote the L-function of f twisted with an
...
...