Non-commutative Iwasawa theory for modular forms.

*(English)*Zbl 1291.11089Let \(f\) be a primitive cusp form of weight \(k> 2\), of conductor \(N\) and trivial character, with rational Fourier coefficients. Let \(M(f)\) denote the motive attached to \(f\). Let \(p\) be an odd prime number.

“The aim of the present paper is to provide some evidence, largely numerical, for the validity of the non-commutative main conjecture of Iwasawa theory for \(M(f)\) over the \(p\)-adic Lie extension \[ F_\infty= \mathbb Q(\mu_{p^\infty}, m^{1/p^n},\;n= 1,2,\dots), \] which is obtained by adjoint to \(\mathbb Q\) the group \(\mu_{p^\infty}\) of all \(p\)-power roots of unity, and all \(p\)-power roots of some fixed integer \(m> 1\).”

“The predictions of the main conjecture are rather intricate in this case because there is more that one critical point, and also there is no canonical choice of periods. Nevertheless, our numerical data agree perfectly with all aspects of the main conjecture, including Kato’s mysterious congruence between the cyclotomic Manin \(p\)-adic \(L\)-function, and the cyclotomic \(p\)-adic \(L\)-function of a twist of the motive by a certain non-abelian Artin character of the Galois group of this extension.”

“Our numerical computations (see section 6) verify the first congruence (5) for the prime \(p= 3\) and a substantial range of cube-free integers \(m> 1\), for three forms \(f\) of weight 4 and conductors 5, 7, 121, and one form \(f\) of weight \(6\) and conductor \(5\), all of which are ordinary at \(3\).”

“When \(f\) is a complex multiplication form, the congruence (4) has also been studied by D. Delbourgo and T. Ward [Ann. Inst. Fourier 58, No. 3, 1023–1055 (2008; Zbl 1165.11077)] and D. Kim [Math. Proc. Camb. Philos. Soc. 155, No. 3, 483–498 (2013; Zbl 1286.11177)]. However, when \(f\) is not a complex multiplication form, our numerical data seem to provide the first hard evidence in support of the mysterious non-abelian congruence (4) between abelian \(p\)-adic \(L\)-functions.”

“The aim of the present paper is to provide some evidence, largely numerical, for the validity of the non-commutative main conjecture of Iwasawa theory for \(M(f)\) over the \(p\)-adic Lie extension \[ F_\infty= \mathbb Q(\mu_{p^\infty}, m^{1/p^n},\;n= 1,2,\dots), \] which is obtained by adjoint to \(\mathbb Q\) the group \(\mu_{p^\infty}\) of all \(p\)-power roots of unity, and all \(p\)-power roots of some fixed integer \(m> 1\).”

“The predictions of the main conjecture are rather intricate in this case because there is more that one critical point, and also there is no canonical choice of periods. Nevertheless, our numerical data agree perfectly with all aspects of the main conjecture, including Kato’s mysterious congruence between the cyclotomic Manin \(p\)-adic \(L\)-function, and the cyclotomic \(p\)-adic \(L\)-function of a twist of the motive by a certain non-abelian Artin character of the Galois group of this extension.”

“Our numerical computations (see section 6) verify the first congruence (5) for the prime \(p= 3\) and a substantial range of cube-free integers \(m> 1\), for three forms \(f\) of weight 4 and conductors 5, 7, 121, and one form \(f\) of weight \(6\) and conductor \(5\), all of which are ordinary at \(3\).”

“When \(f\) is a complex multiplication form, the congruence (4) has also been studied by D. Delbourgo and T. Ward [Ann. Inst. Fourier 58, No. 3, 1023–1055 (2008; Zbl 1165.11077)] and D. Kim [Math. Proc. Camb. Philos. Soc. 155, No. 3, 483–498 (2013; Zbl 1286.11177)]. However, when \(f\) is not a complex multiplication form, our numerical data seem to provide the first hard evidence in support of the mysterious non-abelian congruence (4) between abelian \(p\)-adic \(L\)-functions.”

Reviewer: Andrzej Dąbrowski (Szczecin)

##### MSC:

11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |

11F33 | Congruences for modular and \(p\)-adic modular forms |

11R23 | Iwasawa theory |