zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On the order of vanishing of modular L-functions at the critical point. (English) Zbl 0719.11029

Let f(z)= n=1 a n e 2πinz be a Hecke eigenform, newform of weight 2 for Γ 0 (N). Its L-series L(s)= n=1 a n n -s is also the L-function of an elliptic curve E over . A condition for the finiteness of the group of rational points of E is L ' (1,χ d )0 for a certain quadratic character χ d =(-d y)·

In § 2 of [D. Bump, S. Friedberg and J. Hoffstein, Bull. Am. Math. Soc., New Ser. 21, 89-93 (1989; Zbl 0699.10038)] the theorem is announced that L ' (1,χ d )0 holds for infinitely many χ d associated to imaginary quadratic number fields. Their proof goes along the same lines as in their earlier paper [Ann. Mat., II. Ser. 131, 53-127 (1990; Zbl 0699.10039)]. The same result follows from the main theorem in V. K. Murty [Proc. Conf. on Automorphic Forms and Analytic Number Theory, Montréal, June 1989, 89-113 (1990)].

In the paper under review a more quantitative statement is proved: L ' (1,χ d )0 for at least Y 2/3-ϵ primitive quadratic characters with d<Y, for Y large enough. This follows from the estimates

dY |L ' (1,χ d )| 4 Y 2+ϵ , d L ' (1,χ d )F(d/Y)=α F YlogY+β F Y+O(Y 13/14+ϵ )

with α F 0; the test function F is smooth with compact support, and d runs over a set of squarefree numbers.

The proof uses an integral representation for the L-series, the symmetric square L-series associated to f, the large sieve inequality, and other techniques of analytic number theory. It is quite different from the proof of Bump, Friedberg and Hoffstein, which uses automorphic forms more heavily.

11F67Special values of automorphic L-series, etc
11F11Holomorphic modular forms of integral weight
11F66Langlands L-functions; one variable Dirichlet series, etc.
11G05Elliptic curves over global fields
11M41Other Dirichlet series and zeta functions
11N36Applications of sieve methods