Values of \(p\)-adic \(L\)-functions and a \(p\)-adic Poisson kernel.

*(English)*Zbl 0731.11065Let \(B\) be a definite quaternion algebra over \({\mathbb Q}\) of discriminant \(N\) and let \(p\) be a prime which splits \(B\). Let \({\mathcal O}\) be a maximal \({\mathbb Z}[1/p]\)-order in \(B\) and let \(\Gamma\) be a congruence subgroup of \({\mathcal O}^*\). The author denotes by \(S_ k(\Gamma)\) the space of rigid modular forms on the \(p\)-adic upper half plane \({\mathcal H}_ p\), which is viewed as a rigid analytic subspace of \({\mathbb P}^ 1\). Let \({\mathbb T}\) be the Hecke algebra for \(\Gamma\). As a \({\mathbb T}\)-module, the space \(S_ k(\Gamma)\) is isomorphic to a space of modular forms for the indefinite quaternion algebra with discriminant \(N_ p\). Associated to a modular form \(f\in S_ k(\Gamma)\) is a harmonic cocycle \(c_ f\) which, in fact, determines \(f\). This is a type of harmonic function defined on the tree \({\mathcal T}\) of \(SL_ 2({\mathbb Q}_ p).\)

Let \(P_ k\) be the \(k+1\) dimensional vector space of degree \(k\) polynomials in \(T\) over the field \({\mathbb C}_ p\). The author considers two maps \(\psi\),\(\lambda: S_ k(\Gamma)\to H^ 1(\Gamma,P_{k-2})\), defined from the harmonic cocycles and from the Coleman \(p\)-adic integral, respectively. If \(\psi^{\text{new}}\) and \(\lambda^{\text{new}}\) denote the restriction of \(\psi\) and \(\lambda\) to the new part of \(S_ k(\Gamma)\), then he proves that there exists an element \({\mathcal L}\) in \({\mathbb T}\otimes {\mathbb Q}_ p\) such that \(\lambda^{\text{new}}={\mathcal L}\psi^{\text{new}}\). This invariant \({\mathcal L}\) can be thought of as a “period” associated to \(f\) and it generalizes the \({\mathcal L}\) invariant for weight two modular forms constructed by B. Mazur, J. Tate and the author [(MTT) Invent. Math. 84, 1–48 (1986; Zbl 0699.14028)]. In particular, when \(f\) is a form of weight two, the invariant \({\mathcal L}\) can be derived from the \(p\)-adic period matrix of the Jacobian of \({\mathbb H}_ p/\Gamma.\)

The author introduces a Poisson kernel for the \(p\)-adic upper half plane. By means of this kernel, he constructs an explicit inverse to the map from rigid analytic modular forms to harmonic cocycles. The result is used to obtain a new formula for the Coleman integrals which enter in the definition of the map \(\lambda\).

The \(p\)-adic uniformization theory for Shimura curves, due to Cherednik, makes it possible to formulate a conjecture for the special values of \(p\)- adic \(L\)-functions associated to modular forms of even weight which are lifts, by Jacquet-Langlands, from indefinite quaternion algebras. This Teitelbaum conjecture implies the weight two conjecture, formulated by Mazur, Tate and Teitelbaum in the article quoted above, for those forms which come from quaternion algebras. As in [MTT], it is obtained in this way an “Exceptional Zero Conjecture” in which the invariant \({\mathcal L}\) plays the main role.

This new exceptional zero conjecture is tested, mod \(p^ 5\), for the prime \(p=3\) and the modular form \(f(z)=(\eta (z)\eta (2z)\eta (3z)\eta (6z))^ 2\) which has weight 4 and level 6. The test involves the computation of the modular symbols and the \(p\)-adic \(L\)-function for \(f\) as well as the computation of the invariant \({\mathcal L}\). The hard part in this last step is the computation of \(\lambda\); it is carried out by means of the evaluation of the Coleman integrals.

Let \(P_ k\) be the \(k+1\) dimensional vector space of degree \(k\) polynomials in \(T\) over the field \({\mathbb C}_ p\). The author considers two maps \(\psi\),\(\lambda: S_ k(\Gamma)\to H^ 1(\Gamma,P_{k-2})\), defined from the harmonic cocycles and from the Coleman \(p\)-adic integral, respectively. If \(\psi^{\text{new}}\) and \(\lambda^{\text{new}}\) denote the restriction of \(\psi\) and \(\lambda\) to the new part of \(S_ k(\Gamma)\), then he proves that there exists an element \({\mathcal L}\) in \({\mathbb T}\otimes {\mathbb Q}_ p\) such that \(\lambda^{\text{new}}={\mathcal L}\psi^{\text{new}}\). This invariant \({\mathcal L}\) can be thought of as a “period” associated to \(f\) and it generalizes the \({\mathcal L}\) invariant for weight two modular forms constructed by B. Mazur, J. Tate and the author [(MTT) Invent. Math. 84, 1–48 (1986; Zbl 0699.14028)]. In particular, when \(f\) is a form of weight two, the invariant \({\mathcal L}\) can be derived from the \(p\)-adic period matrix of the Jacobian of \({\mathbb H}_ p/\Gamma.\)

The author introduces a Poisson kernel for the \(p\)-adic upper half plane. By means of this kernel, he constructs an explicit inverse to the map from rigid analytic modular forms to harmonic cocycles. The result is used to obtain a new formula for the Coleman integrals which enter in the definition of the map \(\lambda\).

The \(p\)-adic uniformization theory for Shimura curves, due to Cherednik, makes it possible to formulate a conjecture for the special values of \(p\)- adic \(L\)-functions associated to modular forms of even weight which are lifts, by Jacquet-Langlands, from indefinite quaternion algebras. This Teitelbaum conjecture implies the weight two conjecture, formulated by Mazur, Tate and Teitelbaum in the article quoted above, for those forms which come from quaternion algebras. As in [MTT], it is obtained in this way an “Exceptional Zero Conjecture” in which the invariant \({\mathcal L}\) plays the main role.

This new exceptional zero conjecture is tested, mod \(p^ 5\), for the prime \(p=3\) and the modular form \(f(z)=(\eta (z)\eta (2z)\eta (3z)\eta (6z))^ 2\) which has weight 4 and level 6. The test involves the computation of the modular symbols and the \(p\)-adic \(L\)-function for \(f\) as well as the computation of the invariant \({\mathcal L}\). The hard part in this last step is the computation of \(\lambda\); it is carried out by means of the evaluation of the Coleman integrals.

Reviewer: Pilar Bayer (Barcelona)

##### MSC:

11S40 | Zeta functions and \(L\)-functions |

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

11F85 | \(p\)-adic theory, local fields |

14G20 | Local ground fields in algebraic geometry |

##### Keywords:

quaternion algebra; p-adic L function; periods; rigid analytic modular forms; harmonic cocycles; Coleman integrals; Shimura curves; special values of p-adic L-functions; modular forms of even weight; modular symbols
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\textit{J. T. Teitelbaum}, Invent. Math. 101, No. 2, 395--410 (1990; Zbl 0731.11065)

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