On functional equations of Euler systems

Functional equations for double series of Euler-Hurwitz-Barnes type with coefficients

We first survey the known results on functional equations for the double zeta-function of Euler type and its various generalizations. Then we prove two new functional equations for double series of Euler-Hurwitz-Barnes type with complex coefficients. The first one is of general nature, while the second one is valid when the coefficients are Fourier coefficients of a cusp form.

Functional equations for double series of Euler type with coefficients

We prove two types of functional equations for double series of Euler type with complex coefficients. The first one is a generalization of the functional equation for the Euler double zeta-function, proved in a former work of the second-named author. The second one is more specific, which is proved when the coefficients are Fourier coefficients of cusp forms and the modular relation is essentially used in the course of the proof. As a consequence of functional equation we are able to determine trivial zero divisors.

A relation between the zeros of different two $L$-functions which have the Euler product and functional equation

As automorphic $L$-functions or Artin $L$-functions, several classes of $L$-functions have Euler products and functional equations. In this paper we study the zeros of $L$-functions which have the Euler products and functional equations. We show that there exists some relation between the zeros of the Riemann zeta-function and the zeros of such $L$-functions. As a special case of our results, we find the relations between the zeros of the Riemann zeta-function and the zeros of automorphic $L$-functions attached to elliptic modular forms or the zeros of Rankin-Selberg $L$-functions attached to …

Testing the functional equations of a high-degree Euler product

We study the L-functions associated to Siegel modular forms (equivalently, automorphic representations of ${\rm GSp}(4,\mathbb{A}_{\mathbb{Q}})$) both theoretically and numerically. For the L-functions of degrees 10, 14, and 16 we perform representation theoretic calculations to cast the Langlands L-function in classical terms. We develop a precise notion of what it means to test a conjectured functional equation for an L-function, and we apply this to the degree 10 adjoint L-function associated to a Siegel modular form.

Local epsilon-isomorphisms for rank two p-adic representations of Gal(overline{Q}_p/Q_p) and a functional equation of Kato's Euler system

In this article, we prove (many parts of) the rank two case of the Kato's local epsilon-conjecture using the Colmez's p-adic local Langlands correspondence for GL_2(Q_p). We show that a Colmez's pairing defined in his study of locally algebraic vectors gives us the conjectural epsilon-isomorphisms for (almost) all the families of p-adic representations of Gal(overline{Q}_p/Q_p) of rank two, which satisfy the desired interpolation property for the de Rham and trianguline case. For the de Rham and non trianguline case, we also show this interpolation property for the "critical" range of Hodge-Ta…

Jacob阶梯、Hardy-Littlewood积分(1918)及Euler Gamma函数的新渐近函数方程，连同Fermat-Wiles定理的第十等价式

Jacob's ladders, Hardy-Littlewood integral (1918) and new asymptotic functional equations for Euler's Gamma function together with the tenth equivalent of the Fermat-Wiles theorem

本文基于Hardy-Littlewood积分的几乎线性增量构造了新$Γ$泛函，生成了Fermat-Wiles定理的$Γ$等价式及Euler $Γ$函数的新因式分解公式集。