Selberg’s conjectures and Artin 𝐿-functions

Abstract

In its comprehensive form, an identity between an automorphic L-function and a "motivic" L-function is called a reciprocity law.The celebrated Artin reciprocity law is perhaps the fundamental example.The conjecture of Shimura-Taniyama that every elliptic curve over Q is "modular" is certainly the most intriguing reciprocity conjecture of our time.The "Himalayan peaks" that hold the secrets of these nonabelian reciprocity laws challenge humanity, and, with the visionary Langlands program, we have mapped out before us one means of ascent to those lofty peaks.The recent work of Wiles suggests that an important case (the semistable case) of the Shimura-Taniyama conjecture is on the horizon and perhaps this is another means of ascent.In either case, a long journey is predicted.To paraphrase the cartographer, it is not a journey for the fainthearted.Indeed, there is a forest to traverse, "whose trees will not fall with a few timid blows.We have to take up the double-bitted axe and the cross-cut saw and hope that our muscles are equal to them."At the 1989 Amalfi meeting, Seiberg [S] announced a series of conjectures which looks like another approach to the summit.Alas, neither path seems the easier climb.Selberg's conjectures concern Dirichlet series, which admit analytic continuations, Euler products, and functional equations.The Riemann zeta function is the simplest example of a function in the family 5? of functions F(s) of a complex variable 5 satisfying the following properties:(i) (Dirichlet series) For Re(s) > 1, F(s) = J2T=\ anlnS > where a\ = 1, and we will write an(F) = an for the coefficients of the Dirichlet series.(ii) (Analytic continuation) F(s) extends to a meromorphic function so that for some integer m > 0, (s -l)mF(s) is an entire function of finite order.(iii) (Functional equation) There are numbers Q > 0, a, > 0, Re(r,) > 0, so that d Q>(s) = QsHr(ais + n)F(s) 1=1 satisfies O(s) = wQ>(l -s) for some complex number w with \w\ = 1 .(iv) (Euler product) F(s) = FTp-W where Fp(s) = expÇZkLi bP"/Phs) where bpk = 0(pke) for some 6 < 1/2, where p denotes a prime number (here and throughout this paper).(v) (Ramanujan hypothesis) an = 0(ne) for any fixed e > 0.

Locations

• arXiv (Cornell University) - PDF
• Bulletin of the American Mathematical Society - View - PDF