The Gross. Xxxxx conjecture As the Dedekind zeta function extends the definition of the Riemann zeta function to arbitrary number fields, we now want to extend the definition of the Dedekind zeta function to work ∶ ∗→ / / = ( / ) ( ) = ( ) ( ) = with extensions of arbitrary number fields. We are interested in finite abelian extensions of number fields H F . The Galois group of H F , which we denote G Gal H F , is the group of automorphisms of H that fix the base field F . Let R denote a finite set of places of F containing the infinite places of F and those that are ramified in H. Let χ G C be any character of G. As usual, we view χ also as a multiplicative map on the semigroup of integral fractional ideals of F by defining χ q χ σq if q is unramified in H and χ q 0 if q is ramified in H. Here σq is the image of the ideal q under the Artin map of class field theory. We can thus associate to any such χ the Artin L-function (a,R L (χ, s) = ∑ χ(a) = ∏ 1 , s ∈ C, Re(s) > 1. = F /Q ( ) ∶ ( / ) → )=1 Nas q∉R 1 − χ(q)Nq−s Here, the sum is over all non-zero integral ideals of uF that are coprime R, i.e., the ideals that are coprime to each prime ideal in R. The product is over all prime ideals of uF that are not contained in R. Here and from now on we write N N . Similar to the Dedekind zeta function, if χ is non-trivial, we can analytically continue LR χ, s to a holomorphic function on all of C. Write F for the algebraic closure of F . We now let χ Gal F F Q∗ ⊂ ⊂ = ∪ ∪ be a character of the absolute Galois group of F . Fix a rational prime p. We fix embeddings Q C and Q Cp, so χ may be viewed as taking values in C or Cp. We let H denote the fixedfield of the kernel of χ. We now give the construction of the p-adic L-function. Write P for the set of primes of F lying above p and let RP R P . Partition P as Sp R1, where Sp denotes the subset of primes that split completely in H and R1 the set of remaining primes of P . Let ∶ ( ( )/ ) → ( / ) → ω Gal F µ2p F Z 2pZ ∗ µ2(p−1) ∈ > denote the Teichmüller character. For n Z 0, we have let µn denote the cyclic group of nth roots of unity. There is a p-adic meromorphic function

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