$L$-functions at $s=1$. IV: First derivatives at $s=0$

DOI10.1016/0001-8708(80)90049-3zbMath0475.12018OpenAlexW1983440497MaRDI QIDQ1159715

Publication date: 1980

Published in: Advances in Mathematics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/0001-8708(80)90049-3

zbMATH Keywords

units complex multiplication reciprocity class field evaluation of derivative of L-functions number of roots of unity totally real extension

Mathematics Subject Classification ID

Other Dirichlet series and zeta functions (11M41) Zeta functions and (L)-functions of number fields (11R42)

Related Items (68)

Elliptic units and genus theory ⋮ $p$-adic Abelian Stark conjectures at $s=1$ ⋮ SIC-POVMs from Stark units: Prime dimensions n2 + 3 ⋮ Relative norms of Gauss sums ⋮ Normal basis and Greenberg's conjecture ⋮ Problèmes de construction en multiplication complexe ⋮ Class numbers of ray class fields of imaginary quadratic fields ⋮ Algebraic number fields ⋮ Stark's conjecture and Abelian $L$-functions with higher order zeros at $s=0$ ⋮ Congruences between derivatives of abelian $L$-functions at $s =0$ ⋮ Stark's conjecture over complex cubic number fields ⋮ Counting points on $\text{CM}$ elliptic curves ⋮ Fourier series and the special values of $L$-functions ⋮ A Stark conjecture ``over ${\mathbb{Z}}$ for abelian $L$-functions with multiple zeros ⋮ Unnamed Item ⋮ On Shintani's ray class invariant for totally real number fields ⋮ On the group of modular units and the ideal class group ⋮ Explicit elliptic units. I ⋮ $p$-adic interpolation of square roots of central values of Hecke $L$-functions ⋮ Continued fractions, special values of the double sine function, and Stark units over real quadratic fields ⋮ Unnamed Item ⋮ Construction of ray class fields by elliptic units ⋮ The equivariant Tamagawa number conjecture and the extended abelian Stark conjecture ⋮ The special value 𝑢=1 of Artin-Ihara 𝐿-functions ⋮ A Kronecker limit formula for indefinite zeta functions ⋮ Unnamed Item ⋮ Computing Stark units for totally real cubic fields ⋮ On a generalization of the rank one Rubin-Stark conjecture ⋮ Theta function identities ⋮ On derivatives of Artin $L$-series ⋮ Stark's conjecture in multi-quadratic extensions, revisited. ⋮ Non-cyclic class groups and the Brumer-Stark conjecture ⋮ The Brauer-Manin-Scharaschkin obstruction for subvarieties of a semi-abelian variety and its dynamical analog ⋮ Functorial Properties of Stark Units in Multiquadratic Extensions ⋮ On the ratios of Barnes' multiple gamma functions to the $p$-adic analogues ⋮ Galoismodulstruktur und elliptische Funktionen. (Galois module structure and elliptic functions) ⋮ A $p$-adic Stark conjecture in the rank one setting ⋮ On the square root of special values of certain $L$-series ⋮ Stark's Conjectures and Hilbert's Twelfth Problem ⋮ Numerical Verification of the Stark-Chinburg Conjecture for Some Icosahedral Representations ⋮ THE EQUIVALENCE OF RUBIN'S CONJECTURE AND THE ETNC/LRNC FOR CERTAIN BIQUADRATIC EXTENSIONS ⋮ Unnamed Item ⋮ A formula for the central value of certain Hecke $L$-functions ⋮ Sign functions of imaginary quadratic fields and applications. ⋮ Extensions of zeta functions -- examples and a study of the double sine functions ⋮ Numerical evidence for higher order Stark-type conjectures ⋮ Remark on elliptic units in a $\mathbb {Z}_p$-extension of an imaginary quadratic field ⋮ A two-variable generalization of the Kummer-Malmstén formula for the logarithm of the double gamma and double sine functions ⋮ Computing the Hilbert class field of real quadratic fields ⋮ On Stark elements of arbitrary weight and their $p$-adic families ⋮ On higher order Stickelberger-type theorems ⋮ Generalized Stark formulae over function fields ⋮ A refinement of Stark's conjecture over complex cubic number fields ⋮ THE FRACTIONAL GALOIS IDEAL FOR ARBITRARY ORDER OF VANISHING ⋮ Elliptic units, indices and $\mathbb Z_p$-extensions ⋮ Noncyclotomic Z_p-Extensions of Imaginary Quadratic Fields ⋮ Special values of Abelian $L$-functions at $s=0$ ⋮ An extension of the first-order Stark conjecture ⋮ An enhancement of Zagier’s polylogarithm conjecture ⋮ Two cases of Stark's conjecture ⋮ Verifying a $p$-adic abelian Stark conjecture at $s=1$. ⋮ A result on the Siegel-Ramachandra class invariant over imaginary quadratic fields ⋮ On the Brumer-Stark conjecture ⋮ On the Stark-Shintani conjecture and cyclotomic ${\mathbb{Z}}_ p$-extensions of class fields over real quadratic fields. II ⋮ Concernant la relation de distribution satisfaite par la fonction $\phi$ associée à un réseau complexe. (On the distribution relation satisfied by the function $\phi$ associated to a complex lattice) ⋮ Lower powers of elliptic units ⋮ Arithmetic on elliptic curves with complex multiplication. II ⋮ On the units generated by Weierstrass forms

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\(L\)-functions at \(s=1\). IV: First derivatives at \(s=0\)