In sections 2 and 3, we define the analog of dirichlet lfunctions for an. The technique of completion, continuation, and functional equation apply more generally to the dedekind zeta function of a number. The new tool is the parity hermitian l function operator. I suggest you to try working out riemanns method yourself. A b and read f is a function from the set a to the set b. We prove consequences of functional equations of padic lfunctions for elliptic curves at supersingular primes p. The following example may illustrate how nontrivial this modularity. By weierstrass theorem, this integral is holomorphic in the entire complex plane. Haruzo hida we prove analytic continuation of hecke l functions of number. In mathematics, an automorphic lfunction is a function ls. Automorphic forms, lfunctions and number theory people.
Shouwu zhang, triple lseries and effective mordell conjecture some history is in. Their fundamental properties, in particular the artin conjecture described below, have turned out to be resistant to easy proof. At the other extreme, in the forefront of research, during the last two to three decades, the celebrated youngbaxter functional equation has been at the heart of many di. A note on the triple laplace transform and its applications to some kind of thirdorder differential equation atangana, abdon, abstract and applied analysis, 20 periods, subconvexity of l functions and representation theory bernstein, joseph and reznikov, andre, journal of differential geometry, 2005. Now it is a theorem that has analytic continuation to and satisfies a functional equation relating and, so the bsd conjecture actually makes sense. We examine the number of vanishings of quadratic twists of the lfunction associated to an elliptic curve. In mathematics, a dirichlet lseries is a function of the form. The functoriality conjecture of langlands, as stated in 33, offers a general organizational schemeforallautomorphicrepresentations. The artin l function, is then the infinite product over all prime ideals of these factors. Functional equation satisfied by certain lfunctions. Functional equations in mathematical olympiads 2017 2018. A simple proof of the functional equation of a certain l. I functional equations in mathematical olympiads problems and solutions vol.
Pdf 100 functional equations problems with solutions. On the archimedean euler factors for spin lfunctions. In x 2 we give a detailed derivation of our conjectures in the case of moments on. In 6, artins aim is the functional equation of his lfunctions.
The proof of this hard theorem is via the famous modularity theorem due to wiles, taylor, breuil, conrad and diamond. Artin must have been interested in class eld theory. The riemann zeta function is based on the following functional equation 1 using the input of 0. This equation is known as the functional equation of the function. Global automorphic lfunction and functional equation lecture and notes by dan dore stanford number theory learning seminar may 2, 2018 1 introduction. The conjecture is that the roots of this complex parity hermitian lfunction operator will match the roots of the zeta function to prove rh. The study of triple integral equations with generalized legendre functions singh, b.
In mathematics, an artin lfunction is a type of dirichlet series associated to a linear representation. By analytic continuation, this function can be extended to a meromorphic function on the whole complex plane, and is then called a dirichlet lfunction and also denoted ls. The reason is that an lfunction is usually thought of as being \attached to something. To obtain smooth functional equations, the partial lfunction z fs has to be completed with an eu. Hankel transform, langlands functoriality and functional equation of automorphic lfunctions. Consequences of functional equations for pairs of p adic l. There is a similar equation for the dirichlet lfunctions, but this time relating them in pairs. Owing to the zeros of the sine function, the functional equation implies that. The reason is that an l function is usually thought of as being \attached to something. Introduced by dirichlet 1837 to prove that there are in nitely many primes in arithmetic progressions. These l functions are associated to tempered balanced unitary cuspidal automorphic representations of gln. Using approximate functional equations to build l functions. As artin reciprocity shows, when g is an abelian group these l functions have a second description as dirichlet l functions when k is the rational number field, and as hecke l functions in general.
For the other lfunctions the analytic continuation is not so easily effected. Some special cases of the symmetric power lfunctions of gl forms are known to. Dirichlet lfunction haruzo hida we extend hurwitzs method of proving analytic continuation and functional equation to dirichlet lfunction. The equation relates values of the riemann zeta function at the points s and 1. The case n 1 is the classical hecke theory, the case n 2 was done by andrianov in an. By analytic continuation, this function can be extended to a meromorphic function on the whole complex plane, and is then called a dirichlet l function and also denoted ls. On the functional equation of the artin lfunctions robert. In the rst problem, i evaluate a twisted average of l values using the approximate functional equation in order. I 2017 2018 by amir hossein parvardi the university of british columbia mathematics department vancouver, canada may 2018 dedicated to my lovely wife, nadia. Here we associate an lfunction to every abelian variety over a number. Then ls, f g h has an analytic continuation and functional equation s 1s. This is very similar to the rankinselberg lfunction. Booker i n march of this year, my student, ce bian,announced the computation of some degree 3 transcendental lfunctions at a workshop1 at the american institute of mathematics aim.
In order to establish the functional equation for the hecke lfunctions, we have intoduce. Trilinear forms and subconvexity of the triple product lfunction. The classical proof of the functional equation via mellin transforms and theta series is then generally valid. It is conjectured that, like other lfunctions, that each motivic lfunction can be analytically continued to a meromorphic function on the entire complex plane and satisfies a functional equation relating the lfunction ls, m of a motive m to l1.
The zeta function has an euler product, a functional equation and though very much studied still keeps secret many of its proper ties, the greatest mystery being. Firstly, the existence of a functional equation seems to always be directly related to the lfunction coming from an automorphic representation, and the idea that every lfunction from algebraic geometry aka motivic lfunction also comes from an automorphic representation is in some sense the numbertheoretic incarnation of the global. Our result seems to be a dirichlet series analogue of the well known lagrangeburmann formula for power series. Hankel transform, langlands functoriality and functional. The functional equation shows that there are no more zeroes in the halfplane lfunction z fs has to be completed with an eu. Consequences of functional equations for pairs of p adic. Just the same shape of functional equation holds for the dedekind zeta function of a number field k, with an appropriate gammafactor that depends only on the embeddings of k in algebraic terms, on the tensor product of k with the real field. On lfunctions with poles satisfying maasss functional. The composition of two derivations is no longer a derivation, but we still can. Analytic continuation and functional equation of ls september 27, 2015 as for s, the integral from 1n to 1is nicely convergent for all s2c, and gives an entire function. By considering the case that f is the minimal eisenstein series on gl3, one sees that the sign of the functional equation of l s. The piece that really seems to have piqued his interest was the factorization of zeta functions and l functions as embodied in the theorem of weber and hecke above. Journal of number theory 3, 155158 1971 a simple proof of the functional equation of a certain lfunction yoshiyuki kitaoka department of mathematics, nagoya university, chikusaku, nagoya 464, japan communicated by y. The functional equation of thelfunction associated to.
The results include a relationship between the leading and subleading terms for which we use ideas of wuthrich and bianchi, a parity result of orders of vanishing, and invariance of iwasaswa invariants under conjugate twists of the padic l functions. This is in fact the functional equation for the l function attached to p and 7r which has been conjectured in the context of an arbitrary reductive group by langlands 10. By the functional equation and the euler product, the only other possiblezerosofl. In the rst problem, i evaluate a twisted average of lvalues using the approximate functional equation in order. Uncoveringanew lfunction american mathematical society. Continuations and functional equations the riemann. On lfunctions with poles satisfying maasss functional equation. More generally, suppose that ls x1 n1 a n ns is any lfunction with known analytic continuation and functional equation, and consider the twist ls. The above sets are believed to contain all primitive l functions that are expected to satisfy analogues of the riemann hypothesis, and con. On the functional equation of the artin lfunctions robert p.
Bernsteinzelevinsky derivatives and prove the local functional equation in the case of gl 2mf, for f a nonarchimedean local eld. Kawada received march 15, 1970 by using character sums, stark proved in 1968 the functional equation of the epstein zeta function with a character. The functoriality conjecture will allow us to derive the meromorphic continuation and functional equation of general automorphic lfunctions from the special case of standard lfunctions. It follows from this formula and from formulas 2 and 4 that the functions and are entire functions for all. The sign of the functional equation of the lfunction of an orthogonal motive takeshi saito 1 inventiones mathematicae volume 120, pages 119 142 1995 cite this article. If a function f is defined on the set a to the set b, we write f. We give a rankinselberg integral representation for the spin degree eight lfunction on pgsp6 that applies to the cuspidal automorphic representations associated to siegel modular forms. The above bound can be improved a little further using the full strength of 5, theorem 0. These functions were introduced in 1923 by emil artin, in connection with his research into class field theory. A key is to write down explicitly the hecke lfunction as a linear combination of shintani. This article aims to explain some of the motivation behind the workshop and why bians computations are striking. Applying a conjecture for the full asymptotics of the moments of critical lvalues we obtain a conjecture for the. Booker i n march of this year, my student, ce bian,announced the computation of some degree 3 transcendental lfunctions at a workshop1 at the american institute of mathematics.
Apr 02, 2015 then verify the functional equation 1. We prove consequences of functional equations of padic l functions for elliptic curves at supersingular primes p. The sign of the functional equation of the lfunction of an. By considering the case that f is the minimal eisenstein series on gl3, one sees that the sign of the functional equation of ls. Michael woodbury subconvexity of triple product lfunction. This is in fact the functional equation for the lfunction attached to p and 7r which has been conjectured in the context of an arbitrary reductive group by langlands 10. Yes, this has an analytic continuation and functional equation. The results include a relationship between the leading and subleading terms for which we use ideas of wuthrich and bianchi, a parity result of orders of vanishing, and invariance of iwasaswa invariants under conjugate twists of the padic lfunctions. The piece that really seems to have piqued his interest was the factorization of zeta functions and lfunctions as embodied in the theorem of weber and hecke above. The functional equation shows that there are no more zeroes in the halfplane functional equation and euler product. Pdf functional equations in mathematical olympiads 2017. The goal is to use the functional equation to convert the integral from 0 to 1nto the other sort, and then.
Some fixed point theorems for meirkeeler condensing operators with applications to integral equations alotaibi, a. There is a similar equation for the dirichlet l functions, but this time relating them in pairs. For this reason we restrict to even heckemaass forms in. This result together with a result of langlands 11 concerning the meromorphicity of these.