\(ABC\) implies no ``Siegel zeros for \(L\)-functions of characters with negative discriminant

DOI10.1007/s002229900036zbMath0967.11033OpenAlexW2058408537WikidataQ55968421 ScholiaQ55968421MaRDI QIDQ1977004

H. M. Stark, Andrew Granville

Publication date: 12 July 2001

Published in: Inventiones Mathematicae (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s002229900036

zbMATH Keywords

Dirichlet \(L\)-function\(abc\)-conjectureimaginary quadratic fieldreal zeroclass fieldexceptional zero

Mathematics Subject Classification ID

Holomorphic modular forms of integral weight (11F11) Real zeros of (L(s, chi)); results on (L(1, chi)) (11M20) Zeta functions and (L)-functions of number fields (11R42) Cubic and quartic Diophantine equations (11D25)

Related Items (20)

On $abc$ and discriminants ⋮ Lecture on the abc Conjecture and Some of Its Consequences ⋮ Representation by ternary quadratic forms ⋮ Planck-scale mass equidistribution of toral Laplace eigenfunctions ⋮ The \(abc\)-conjecture for algebraic numbers ⋮ Smooth solutions to the \(abc\) equation: the \(xyz\) conjecture ⋮ Problème de Lehmer sur les courbes elliptiques à multiplications complexes ⋮ OMEGA THEOREMS FOR $\frac{L'}{L}(1, \chi_D)$ ⋮ An efficient algorithm for computing DirichletL-functions ⋮ Shifted products that are coprime pure powers ⋮ On the Euler-Kronecker constants of global fields and primes with small norms ⋮ Asymptotic behaviour of the kernel of an integer ⋮ An explicit CM type norm formula and effective nonvanishing of class group \(L\)-functions for CM fields ⋮ Some unlikely intersections beyond André–Oort ⋮ The ABC conjecture, arithmetic progressions of primes and squarefree values of polynomials at prime arguments ⋮ A height inequality for rational points on elliptic curves implied by the abc-conjecture ⋮ Minimal polynomials of singular moduli ⋮ Some examples related to the $abc$-conjecture for algebraic number fields ⋮ Effective bounds on differences of singular moduli that are S-units ⋮ On Landau–Siegel zeros and heights of singular moduli

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