Nonvanishing ofL-functions for GL (2)

Rohrlich, David E.

doi:10.1007/bf01389047

Cited by 70 publications

(62 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This was proved by Rohrlich [115] using several non-trivial results as input, not the least of which was a version of the Bombieri-Vinogradov theorem over number fields due to Murty and Murty [101]. Very roughly, one takes m to be a product of prime ideals p such that N(p) − 1 has no large prime factor.…”

Section: Techniques Over Number Fieldsmentioning

confidence: 99%

The role of the Ramanujan conjecture in analytic number theory

Blomer¹,

Brumley²

2013

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

Section: Techniques Over Number Fieldsmentioning

confidence: 99%

The role of the Ramanujan conjecture in analytic number theory

Blomer¹,

Brumley²

2013

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. [1,3,7,9,10,11,13,15,17] and the references given there. This is of interest in various aspects such as the Birch-Swinnerton-Dyer conjecture, the Siegel zero (see [7]) and the theory of modular forms of half-integral weight (see [16,18] [12].…”

mentioning

confidence: 99%

Non-vanishing of class group $L$-functions at the central point

Blomer¹

2004

Annales De L’institut Fourier

View full text Add to dashboard Cite

“…With such motivation, Shimura [14] showed that given a weight k modular form ƒ, there is some character of finite order x such that L(l, ƒ,#) is not zero. This was refined by Rohrlich [13], who proved that for ƒ an arbitrary automorphic form on PGL(2) and for arbitrary s, there exists a character x of finite order such that L(s,f,x) ¥" 0.…”

mentioning

confidence: 98%

A nonvanishing theorem for derivatives of automorphic 𝐿-functions with applications to elliptic curves

Bump¹,

Friedberg²,

Hoffstein³

1989

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

1. A brief history of nonvanishing theorems. The nonvanishing of a Dirichlet series 2 a(n)n~\ or the existence of a pole, at a particular value of s often has applications to arithmetic. Euler gave the first example of this, showing that the infinitude of the primes follows from the pole of Ç(s) at s = 1. A deep refinement was given by Dirichlet, whose theorem on primes in an arithmetic progression depends in a fundamental way upon the nonvanishing of Dirichlet L-functions at s = 1.Among the many examples of arithmetically significant nonvanishing results in this century, one of the most important is still mostly conjectural. Let E be an elliptic curve defined over Q: the set of all solutions to an equationy 2 = x 3 -ax-b where a, b are rational numbers with 4a 3 -27b 2 ^ 0. Mordell showed that E(Q) may be given the structure of a finitely generated abelian group. The Birch-Swinnerton-Dyer Conjecture asserts that the rank of this group is equal to the order of vanishing of a certain Dirichlet series L(s,E) as s = 1-the center of the critical strip-and that the leading Taylor coefficient of this L-function at s = 1 is determined in an explicit way by the arithmetic of the elliptic curve. We refer to the excellent survey article of Goldfeld [5] for details.In 1977, Coates and Wiles [3] proved the first result towards the BirchSwinnerton-Dyer conjecture. The conjecture implies that if the L-series of E does not vanish at 1, then the group of rational points is finite. Coates and Wiles proved this last claim in the special case that E has complex multiplication (nontrivial endomorphisms). In this note, we announce a nonvanishing theorem which, together with work of Kolyvagin and Gross-Zagier, implies that E(Q) is finite when L(l,E) ^ 0 for any modular elliptic curve E. (A modular elliptic curve is one which may be parametrized by automorphic functions. Deuring proved that all elliptic curves with complex multiplication are modular; Taniyama and Weil have conjectured that indeed all elliptic curves defined over Q are modular.)Before giving details of our theorem, we mention several other nonvanishing theorems and arithmetic applications. The following discussion is necessarily not a complete survey Shimura showed that there is a correspondence between modular forms ƒ of even weight k and modular forms ƒ of half-integral weight (k + l)/2.

show abstract

Nonvanishing ofL-functions for GL (2)

Cited by 70 publications

References 15 publications

The role of the Ramanujan conjecture in analytic number theory

The role of the Ramanujan conjecture in analytic number theory

Non-vanishing of class group $L$-functions at the central point

A nonvanishing theorem for derivatives of automorphic 𝐿-functions with applications to elliptic curves

Contact Info

Product

Resources

About