Analytic Tate spaces and reciprocity

Analytic Tate spaces and reciprocity Ricardo García López Universitat de Barcelona Kraków, August, 2013 Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 1 / 1

Weil reciprocity Theorem Let k be a perfect field, C a smooth irreducible complete curve over k, f and g two meromorphic functions on C. Denote div(f ) = n p p, div(g) = m q q with n p, m q > 0 their corresponding divisors. If the supports of div(f ) and div(g) are disjoint, then mq f (q) = n p g(p). In short, f (div(g)) = g(div(f )). Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 2 / 1

Weil reciprocity Theorem Let k be a perfect field, C a smooth irreducible complete curve over k, f and g two meromorphic functions on C. Denote div(f ) = n p p, div(g) = m q q with n p, m q > 0 their corresponding divisors. If the supports of div(f ) and div(g) are disjoint, then mq f (q) = n p g(p). In short, f (div(g)) = g(div(f )). There are several proofs, e.g. in Serre s Groups algébriques et corps de classes or in Griffiths-Harris Principles of algebraic geometry (for k = C). Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 2 / 1

Alternative form Weil reciprocity can be reformulated as follows: Define the tame symbol of f, g at p C as [ ] (f, g) p = ( 1) vp(f ) vp(g) f vp(g) g vp(f ) (p) where v p is the valuation at p. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 3 / 1

Alternative form Weil reciprocity can be reformulated as follows: Define the tame symbol of f, g at p C as [ ] (f, g) p = ( 1) vp(f ) vp(g) f vp(g) g vp(f ) (p) where v p is the valuation at p. Then, Theorem (Weil reciprocity, alternative form) For any f, g meromorphic functions on C, (f, g) p = 1. p C The formula above will be referred as a product formula. The label reciprocity is justified e.g. because if k is a finite field, Weil reciprocity is a special case of Artin reciprocity. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 3 / 1

In this talk, my objective is to present a construction which allows to define an extension of the tame symbol. Its features are: Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 4 / 1

In this talk, my objective is to present a construction which allows to define an extension of the tame symbol. Its features are: 1 The construction can be applied both in the complex analytic and in the p-adic cases. 2 The symbols obtained are defined for pairs of invertible analytic functions, possibly with essential singularities. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 4 / 1

In this talk, my objective is to present a construction which allows to define an extension of the tame symbol. Its features are: 1 The construction can be applied both in the complex analytic and in the p-adic cases. 2 The symbols obtained are defined for pairs of invertible analytic functions, possibly with essential singularities. 3 For functions defined on curves, these symbols satisfy a form of Weil reciprocity. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 4 / 1

In this talk, my objective is to present a construction which allows to define an extension of the tame symbol. Its features are: 1 The construction can be applied both in the complex analytic and in the p-adic cases. 2 The symbols obtained are defined for pairs of invertible analytic functions, possibly with essential singularities. 3 For functions defined on curves, these symbols satisfy a form of Weil reciprocity. Several extensions of the tame symbol are already known, among them: Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 4 / 1

Extensions of the tame symbol Contou-Carrère s symbol: Let A be an artinian local ring. Contou-Carrère attaches a symbol to a pair of units f, g A((t)). The construction is algebraic, the Contou-Carrère s symbol coincides with the tame symbol if k is a field. Deligne s symbol: If f, g are two invertible functions defined on a punctured disk D, holomorphic away from the center, Deligne defines a line bundle with connection on D and the symbol of the pair f, g is defined as the monodromy of this bundle. The construction is transcendental. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 5 / 1

Extensions of the tame symbol Contou-Carrère s symbol: Let A be an artinian local ring. Contou-Carrère attaches a symbol to a pair of units f, g A((t)). The construction is algebraic, the Contou-Carrère s symbol coincides with the tame symbol if k is a field. Deligne s symbol: If f, g are two invertible functions defined on a punctured disk D, holomorphic away from the center, Deligne defines a line bundle with connection on D and the symbol of the pair f, g is defined as the monodromy of this bundle. The construction is transcendental. Later I will give a few more details on Deligne s symbol. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 5 / 1

Central extensions In the 60, Tate gives an algebraic proof of the residue theorem which avoids transcendental arguments. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 6 / 1

Central extensions In the 60, Tate gives an algebraic proof of the residue theorem which avoids transcendental arguments. Quoting Tate: as thought one had an abstract Stokes theorem available. He uses in an essential way the notion of commensurability: If V is a vector space and A, B V are subspaces, they are commensurable, denoted A B, if A B has finite codimension in both A and B. Arbarello, de Concini and Kac reprove Weil s reciprocity in the spirit of Tate s proof. If A B, they consider the determinant of A, B, which is the 1-dimensional vector space Det(A, B) := (A/A B) (B/A B) where denotes highest alternate power. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 6 / 1

ADCK symbols Given A V, they define groups GL(V, A) = {g GL(V ) ga A} ĜL(V, A) = {(g, a) g GL(V, A), a Det(A, ga), a 0} and consider the central extension 1 k ĜL(V, A) GL(V, A) 1 Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 7 / 1

ADCK symbols Given A V, they define groups GL(V, A) = {g GL(V ) ga A} ĜL(V, A) = {(g, a) g GL(V, A), a Det(A, ga), a 0} and consider the central extension 1 k ĜL(V, A) GL(V, A) 1 Definition Given two commuting elements f, g GL(V, A), choose liftings f, ĝ ĜL(V, A), define the ADCK symbol of f, g as (f, g) A := [ˆf, ĝ] k. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 7 / 1

ADCK symbols ADCK prove that this symbol depends only on the commensurability class of A. Take V = k((t)) and A = k[[t]] Given f, g k((t)), they might be regarded as elements of GL(V, A) (namely, as multiplication operators V V ). Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 8 / 1

ADCK symbols ADCK prove that this symbol depends only on the commensurability class of A. Take V = k((t)) and A = k[[t]] Given f, g k((t)), they might be regarded as elements of GL(V, A) (namely, as multiplication operators V V ). Since they obviously commute, the ADCK symbol is defined and one easily checks that it coincides with the tame symbol. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 8 / 1

ADCK symbols Moreover, the ADCK symbol satisfies an abstract reciprocity law: Given A, B V and g, h GL(V, A) GL(V, B) such that g h = h g, one has (g, h) A (g, h) B = (g, h) A B (g, h) A+B Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 9 / 1

ADCK symbols Moreover, the ADCK symbol satisfies an abstract reciprocity law: Given A, B V and g, h GL(V, A) GL(V, B) such that g h = h g, one has (g, h) A (g, h) B = (g, h) A B (g, h) A+B If C is a smooth irreducible projective curve and f, g are meromorphic functions on C, the abstract reciprocity law applied to suitable V gives (f, g) p = 1. that is, Weil s reciprocity. p C Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 9 / 1

Tate spaces Given a vector space V, in order to define the abstract symbols of ADCK one has to fix a commensurability class. A topology can be used to fix such a class. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 10 / 1

Tate spaces Given a vector space V, in order to define the abstract symbols of ADCK one has to fix a commensurability class. A topology can be used to fix such a class. Definition Let k be a field. A topological vector space V is linearly compact if it has the following properties: 1 V is complete and Hausdorff. 2 V has a basis of neighborhoods of 0 consisting of vector subspaces. 3 Each open subspace of V has finite codimension. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 10 / 1

Tate spaces Given a vector space V, in order to define the abstract symbols of ADCK one has to fix a commensurability class. A topology can be used to fix such a class. Definition Let k be a field. A topological vector space V is linearly compact if it has the following properties: 1 V is complete and Hausdorff. 2 V has a basis of neighborhoods of 0 consisting of vector subspaces. 3 Each open subspace of V has finite codimension. Example k[[t]] with the t-adic topology. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 10 / 1

Tate spaces Definition A Tate space is a topological vector space which has an open linearly compact subspace. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 11 / 1

Tate spaces Definition A Tate space is a topological vector space which has an open linearly compact subspace. Example Example: k((t)) with its usual topology (i.e., the subspaces t n k[[t]] are a basis of neighborhoods of 0) is a Tate space, since k[[t]] k((t)) is open and linearly compact. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 11 / 1

Tate spaces Definition A Tate space is a topological vector space which has an open linearly compact subspace. Example Example: k((t)) with its usual topology (i.e., the subspaces t n k[[t]] are a basis of neighborhoods of 0) is a Tate space, since k[[t]] k((t)) is open and linearly compact. Tate spaces were considered by Lefschetz (under the name locally linearly compact space) and more recently by Kapranov, Drinfeld,... Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 11 / 1

Properties of Tate spaces 1 If V is a Tate space, then there exist a discrete subspace D and a linearly compact subspace C such that V = C D. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 12 / 1

Properties of Tate spaces 1 If V is a Tate space, then there exist a discrete subspace D and a linearly compact subspace C such that V = C D. For example, k((t)) = k[[t]] t 1 k[t 1 ]. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 12 / 1

Properties of Tate spaces 1 If V is a Tate space, then there exist a discrete subspace D and a linearly compact subspace C such that V = C D. For example, k((t)) = k[[t]] t 1 k[t 1 ]. One says that C is a compact lattice and D a discrete lattice. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 12 / 1

Properties of Tate spaces 1 If V is a Tate space, then there exist a discrete subspace D and a linearly compact subspace C such that V = C D. For example, k((t)) = k[[t]] t 1 k[t 1 ]. One says that C is a compact lattice and D a discrete lattice. 2 If D 1, D 2 are discrete lattices, then D 1 D 2 and D 1 D 2 is also a discrete lattice. 3 Any Tate space has a distinguished commensurability class, given by the discrete lattices. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 12 / 1

Properties of Tate spaces 1 If V is a Tate space, then there exist a discrete subspace D and a linearly compact subspace C such that V = C D. For example, k((t)) = k[[t]] t 1 k[t 1 ]. One says that C is a compact lattice and D a discrete lattice. 2 If D 1, D 2 are discrete lattices, then D 1 D 2 and D 1 D 2 is also a discrete lattice. 3 Any Tate space has a distinguished commensurability class, given by the discrete lattices. 4 The commensurability class given by the discrete lattices in k((t)) is the one used by ADCK to recover the tame symbol from their abstract symbols. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 12 / 1

What to do in the analytic setting? If one wants to define a symbol for holomorphic functions with arbitrary singularities, it seems instead of k((t)) one should consider the ring of germs of analytic functions in a punctured neighborhood of 0 C, namely { } C{{t}} = a i t i ε > 0 the series converges for 0 < t < ε. i Z This ring can be naturally endowed with a topology: If U C is open, we consider on Γ(U, O C ) the topology determined by the seminorms f K = sup{ f (x) x K } where K U is compact. On C{{t}} = lim 0 U Γ(U {0}, O C ) we consider the direct limit topology. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 13 / 1

Analytic setting Put C{{t}} + = C{{t}} C[[t]] and C{{t}} = C{{t}} C[[t 1 ]] and consider in these subspaces the induced topologies. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 14 / 1

Analytic setting Put C{{t}} + = C{{t}} C[[t]] and C{{t}} = C{{t}} C[[t 1 ]] and consider in these subspaces the induced topologies. We still have a topological decomposition C{{t}} = C{{t}} + t 1 C{{t}}, but it is easy to see that C{{t}} is not a Tate space. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 14 / 1

Analytic setting Put C{{t}} + = C{{t}} C[[t]] and C{{t}} = C{{t}} C[[t 1 ]] and consider in these subspaces the induced topologies. We still have a topological decomposition C{{t}} = C{{t}} + t 1 C{{t}}, but it is easy to see that C{{t}} is not a Tate space. So, if we want to define symbols along the lines of ADCK we need a a diferent notion of Tate space. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 14 / 1

Some definitions from functional analysis... In order to find an analytic version of the notion of Tate space, we need some definitions. From now on, k denotes a local field of characteristic zero. (that is, k = R or C or k is a finite extension of Q p ) Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 15 / 1

Some definitions from functional analysis... In order to find an analytic version of the notion of Tate space, we need some definitions. From now on, k denotes a local field of characteristic zero. (that is, k = R or C or k is a finite extension of Q p ) Let V, W be locally convex topological k-vector spaces (LCTVS). Recall that: 1 For any local field k, a linear map f : V W is compact if the image of any bounded subset of V has compact closure in W. A closely related notion is that of a nuclear map (the definition is a bit technical, nuclear maps are always compact, for the purposes of this talk both notions can be safely identified). Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 15 / 1

Some definitions from functional analysis... In order to find an analytic version of the notion of Tate space, we need some definitions. From now on, k denotes a local field of characteristic zero. (that is, k = R or C or k is a finite extension of Q p ) Let V, W be locally convex topological k-vector spaces (LCTVS). Recall that: 1 For any local field k, a linear map f : V W is compact if the image of any bounded subset of V has compact closure in W. A closely related notion is that of a nuclear map (the definition is a bit technical, nuclear maps are always compact, for the purposes of this talk both notions can be safely identified). 2 V is a nuclear space if for every Banach space B, every linear continuous map V B is nuclear. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 15 / 1

...and some examples Example The space of smooth functions on an open subset of R n or the space of holomorphic functions on an open subset of C n are nuclear. Banach spaces are not nuclear, unless they are finite dimensional. We still need the following: Definition 6 A LCTVS is called Fréchet if it is metrizable and complete. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 16 / 1

...and some examples Example The space of smooth functions on an open subset of R n or the space of holomorphic functions on an open subset of C n are nuclear. Banach spaces are not nuclear, unless they are finite dimensional. We still need the following: Definition 6 A LCTVS is called Fréchet if it is metrizable and complete. 7 A (FN) space is a space which is simultaneously Fréchet and nuclear. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 16 / 1

...and some examples Example The space of smooth functions on an open subset of R n or the space of holomorphic functions on an open subset of C n are nuclear. Banach spaces are not nuclear, unless they are finite dimensional. We still need the following: Definition 6 A LCTVS is called Fréchet if it is metrizable and complete. 7 A (FN) space is a space which is simultaneously Fréchet and nuclear. 8 A (DFN) space is the (strong) dual of a (FN)-space. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 16 / 1

Analytic Tate spaces One can check that C{{t}} is Fréchet and nuclear (so, it is FN). Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 17 / 1

Analytic Tate spaces One can check that C{{t}} is Fréchet and nuclear (so, it is FN). The residue pairing C{{t}} C{{t}} C (f, g) Res(f dg) induces a topological isomorphism between C{{t}} + and the (strong) dual of t 1 C{{t}}, so C{{t}} + is a DFN space. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 17 / 1

Analytic Tate spaces One can check that C{{t}} is Fréchet and nuclear (so, it is FN). The residue pairing C{{t}} C{{t}} C (f, g) Res(f dg) induces a topological isomorphism between C{{t}} + and the (strong) dual of t 1 C{{t}}, so C{{t}} + is a DFN space. We have a topological decomposition C{{t}} = C{{t}} + t 1 C{{t}} where the first summand is (DFN) and the second is (FN). Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 17 / 1

Analytic Tate spaces The example suggest the following definition: Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 18 / 1

Analytic Tate spaces The example suggest the following definition: Definition A LCTVS V will be said to be an analytic Tate space if there is a (topological) isomorphism V = G H where G is (DFN) and H is (FN). Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 18 / 1

Analytic Tate spaces The example suggest the following definition: Definition A LCTVS V will be said to be an analytic Tate space if there is a (topological) isomorphism V = G H where G is (DFN) and H is (FN). We will say that G is a (DFN)-lattice and H is a (FN)-lattice in V. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 18 / 1

Examples of analytic Tate spaces Examples For k = R or C, the ring of function germs analytic on a punctured neighborhood of 0 k. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 19 / 1

Examples of analytic Tate spaces Examples For k = R or C, the ring of function germs analytic on a punctured neighborhood of 0 k. The ring k((t)) = k[[t]] t 1 k[t 1 ] (endowed with the topology given by k[[t]] = N k, k[t 1 ] = N k). More generally, if V is a finite dimensional k vector space, then ( N V ) ( N V ) is an analytic Tate space. In particular, if g is a finite dimensional topological k-lie algebra, the loop algebra g((t)) = g k k((t)) is what might be called an analytic Tate Lie algebra, with the bracket [g p, h q] = [g, h] p q Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 19 / 1

One more example: Robba rings Example Let k be non-archimedean. On studying p-adic cohomology and p-adic differential equations, a ring which plays the role of k((t)) is the so-called Robba ring, defined as { } R = a i t i λ < 1 a a i k, i λ i 0 for i ε < 1 a i ε i 0 for i + i Z Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 20 / 1

1 λ Elements of the Robba ring converge on an annulus of outer radious 1 Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 21 / 1

1 λ Elements of the Robba ring converge on an annulus of outer radious 1 Put R + = R k[[t]], R = R k[[t 1 ]]. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 21 / 1

1 λ Elements of the Robba ring converge on an annulus of outer radious 1 Put R + = R k[[t]], R = R k[[t 1 ]]. The direct sum decomposition R = R + t 1 R is topological, the k-vector space R + is (FN) and t 1 R is (DFN) (notice that the role of positive and negative powers is reversed with respect to the complex analytic case). Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 21 / 1

1 λ Elements of the Robba ring converge on an annulus of outer radious 1 Put R + = R k[[t]], R = R k[[t 1 ]]. The direct sum decomposition R = R + t 1 R is topological, the k-vector space R + is (FN) and t 1 R is (DFN) (notice that the role of positive and negative powers is reversed with respect to the complex analytic case). Thus R is an analytic Tate space. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 21 / 1

Problems of this definition Analytic Tate spaces keep some of the properties of Tate spaces, but if L 1, L 2 are both (FN) or (DFN) lattices, it is not true in general that L 1 L 2, so there is no distinguished commensurability class and the definition of determinant given by ADCK does not work. Example Let f be a unit of C{{t}}, then C{{t}} = C{{t}} + t 1 C{{t}} = f C{{t}} + t 1 f C{{t}}, so C{{t}} + and f C{{t}} + are (DFN)-lattices. But if f has an essential singularity at zero, then and so C{{t}} + f C{{t}} +. C{{t}} + (f C{{t}} + ) = {0} Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 22 / 1

Polarizations (after Segal) Let V be a LCTVS. On the set of topological direct sum decompositions V = V + V we consider the following equivalence relation: V + V W + W if and only if the compositions V i V W j and W i V V j ( for i, j {+, }, i j). are nuclear, where in both cases the arrows are the projections attached to the given decompositions. Definition A polarization of V is an equivalence class of decompositions. If we fix a decomposition V = V + V, the decompositions in the same equivalence class are called allowable, as well as the projections onto its summands. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 23 / 1

Polarizations Let V be a polarized LCTVS, choose an allowable decomposition V = V + V, denote GL(V ) the group of bicontinuous automorphisms of V. Given g GL(V ), set 1 ( ) g++ g g = + : V + V V + V g + g 1 I.e., g i,j : V i V j the composition of the restriction g Vi with the allowable projection V V j, where i, j {+, }. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 24 / 1

Polarizations Let V be a polarized LCTVS, choose an allowable decomposition V = V + V, denote GL(V ) the group of bicontinuous automorphisms of V. Given g GL(V ), set 1 ( ) g++ g g = + : V + V V + V g + g Definition The restricted linear group of V is the group GL res (V ) = {g GL(V ) g +, and g,+ are nuclear }. 1 I.e., g i,j : V i V j the composition of the restriction g Vi with the allowable projection V V j, where i, j {+, }. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 24 / 1

Polarizations The essential step to obtain the reciprocity laws in this setting is: Proposition Let V + be a (FN)-space and V a (DFN)-space. Then: 1 Every continuous linear map f : V + V is nuclear. 2 Every continuous linear map g : V V + is nuclear. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 25 / 1

Polarizations The essential step to obtain the reciprocity laws in this setting is: Proposition Let V + be a (FN)-space and V a (DFN)-space. Then: 1 Every continuous linear map f : V + V is nuclear. 2 Every continuous linear map g : V V + is nuclear. Proof (sketch): 1: One can assume that V = i N W i is the union of a nested sequence of Banach spaces. By a theorem of Grothendieck, there is a i N such that f factors as V + W i V. Then since V + is nuclear, the first map is nuclear, and then so is the composition. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 25 / 1

Polarizations The essential step to obtain the reciprocity laws in this setting is: Proposition Let V + be a (FN)-space and V a (DFN)-space. Then: 1 Every continuous linear map f : V + V is nuclear. 2 Every continuous linear map g : V V + is nuclear. Proof (sketch): 1: One can assume that V = i N W i is the union of a nested sequence of Banach spaces. By a theorem of Grothendieck, there is a i N such that f factors as V + W i V. Then since V + is nuclear, the first map is nuclear, and then so is the composition. 2. Follows from a theorem proved in the 60 by Ed Dubinsky. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 25 / 1

Polarizations Corollary 1 An analytic Tate space has a essentially unique polarization, that is, if we have a topological isomorphism V + V = W+ W where V +, W + are (FN)-spaces and V, W are (DFN)-spaces, then these decompositions are equivalent. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 26 / 1

Polarizations Corollary 1 An analytic Tate space has a essentially unique polarization, that is, if we have a topological isomorphism V + V = W+ W where V +, W + are (FN)-spaces and V, W are (DFN)-spaces, then these decompositions are equivalent. 2 If V is an analytic Tate space, then GL(V ) = GL res (V ). Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 26 / 1

Determinants (after Quillen) We need an analogue of the determinant lines defined by ADCK. This will be provided by the notion of determinant line. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 27 / 1

Determinants (after Quillen) We need an analogue of the determinant lines defined by ADCK. This will be provided by the notion of determinant line. Let V, W be topological vector spaces, f : V W a Fredholm map (a continuous map such that Ker(f ) and Coker(f ) are finite dimensional). Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 27 / 1

Determinants (after Quillen) We need an analogue of the determinant lines defined by ADCK. This will be provided by the notion of determinant line. Let V, W be topological vector spaces, f : V W a Fredholm map (a continuous map such that Ker(f ) and Coker(f ) are finite dimensional). Definition The Quillen determinant of f is: Det Q (V, W, f ) = ( Ker(f ) ) Coker(f ). Here, = highest exterior power. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 27 / 1

Determinants (after Segal) Let f : V W be Fredholm, for simplicity assume its index is zero. An operator g : V W is a nuclear perturbation of f if f g is nuclear. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 28 / 1

Determinants (after Segal) Let f : V W be Fredholm, for simplicity assume its index is zero. An operator g : V W is a nuclear perturbation of f if f g is nuclear. Denote Seg(f ) = {(g, λ) f g is nuclear, λ k}. On Seg(f ) consider the equivalence relation generated by (g θ, λ) (g, det(θ) λ) where id θ : V V is nuclear. Here, det(θ) denotes the Fredholm determinant of θ (well defined since θ is a nuclear perturbation of id : V V ). Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 28 / 1

Definition The Segal determinant of f is defined as Det S (V, W, f ) = Seg(f )/. By its very definition, the Segal determinant is invariant under nuclear perturbations. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 29 / 1

Definition The Segal determinant of f is defined as Det S (V, W, f ) = Seg(f )/. By its very definition, the Segal determinant is invariant under nuclear perturbations. For our purposes, the interest of this notion relies in the following Proposition There is a canonical isomorphism Det S (V, W, f ) = Det Q (V, W, f ). In fact, let γ 1,..., γ n be a basis of Coker f and α 1,..., α n a basis of (Ker f ). Choose extensions α i : V k of α i (for k a local field, this is always possible). Then, the isomorphism is given by ( f + ) γ i α i, 1 α 1 α n γ 1 γ n. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 29 / 1

The use of Quillen-Segal determinants in our context Let V be a polarized vector space V +, W + two allowable plus-summands p : V W + be an allowable projection. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 30 / 1

The use of Quillen-Segal determinants in our context Let V be a polarized vector space V +, W + two allowable plus-summands p : V W + be an allowable projection. If q : V W + is another allowable projection, the difference p V+ q V+ is nuclear, so we have a canonical isomorphism Det(V +, W +, p V+ ) = Det(V +, W +, q V+ ) Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 30 / 1

The use of Quillen-Segal determinants in our context Let V be a polarized vector space V +, W + two allowable plus-summands p : V W + be an allowable projection. If q : V W + is another allowable projection, the difference p V+ q V+ is nuclear, so we have a canonical isomorphism Det(V +, W +, p V+ ) = Det(V +, W +, q V+ ) and we put Det(V + : W + ) = lim p Det(V +, W +, p V+ ), Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 30 / 1

Choose allowable subspaces V +, W + V. For g GL res (V ), the subspaces g(v + ), g(w + ) are also allowable and there is a canonical isomorphism Det(V + : g(v + )) Det(W + : g(w + )), Definition P g = lim V+ Det(V + : g(v + )). Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 31 / 1

Choose allowable subspaces V +, W + V. For g GL res (V ), the subspaces g(v + ), g(w + ) are also allowable and there is a canonical isomorphism Det(V + : g(v + )) Det(W + : g(w + )), Definition P g = lim V+ Det(V + : g(v + )). These k-lines P g have several functorial properties, among them Proposition There are canonical isomorphisms ρ f,g : P f P g P fg. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 31 / 1

Group extensions Definition GL + res(v ) = {(f, α) f GL res (V ), α P f α 0}. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 32 / 1

Group extensions Definition GL + res(v ) = {(f, α) f GL res (V ), α P f α 0}. GL + res(v ) is a group with the operation defined by (f, α) (g, β) = (f g, ρ f,g (α β)). Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 32 / 1

Group extensions Definition GL + res(v ) = {(f, α) f GL res (V ), α P f α 0}. GL + res(v ) is a group with the operation defined by (f, α) (g, β) = (f g, ρ f,g (α β)). and we proceed as ADCK. We have a central extension 1 k GL + res(v ) GL res (V ) 1. Definition Given commuting elements f, g GL res (V ), choose liftings f, g GL + res(v ) and define the symbol f, g by f, g = f g f 1 g 1 k. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 32 / 1

Examples 1.- We take V = C{{t}} and f, g two units of C{{t}} Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 33 / 1

Examples 1.- We take V = C{{t}} and f, g two units of C{{t}} Any unit f C{{t}} can be written uniquely as a product f = c t n g(t) h(t 1 ), where c C, n Z, g is holomorphic at zero, g(0) = 1, h(x) = e ϕ(x) and ϕ is an entire function with ϕ(0) = 0 (Weierstrass-Birkhoff decomposition). Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 33 / 1

Examples 1.- We take V = C{{t}} and f, g two units of C{{t}} Any unit f C{{t}} can be written uniquely as a product f = c t n g(t) h(t 1 ), where c C, n Z, g is holomorphic at zero, g(0) = 1, h(x) = e ϕ(x) and ϕ is an entire function with ϕ(0) = 0 (Weierstrass-Birkhoff decomposition). Proposition Given units f 1, f 2 C{{t}}, with decompositions f i = c i t n i g i (t) h i (t 1 ) as above (i = 1, 2), we have f 1, f 2 = c n 2 1 c n 1 2. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 33 / 1

Corollary If f 1, f 2 C{t}[t 1 ] are meromorphic at zero, then f, g 1 equals the tame symbol. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 34 / 1

Corollary If f 1, f 2 C{t}[t 1 ] are meromorphic at zero, then f, g 1 equals the tame symbol. 2.- In the p-adic case, there is an analogue of the Weierstrass-Birkhoff decomposition (due to E. Motzkin). Precisely, if f R is a unit of the Robba ring, there exist unique n Z, c k, g 1 + tr + invertible in R + and h 1 + t 1 R invertible in R such that f = c t n g h. It follows that the previous proposition holds also in this case. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 34 / 1

Reciprocity, complex case Theorem Let C/C be a complete irreducible non-singular curve, S C a finite subset. For s S, let O C,ŝ denote the ring of function germs which are holomorphic in a punctured neighborhood of s in C. If f, g are invertible elements of the ring of analytic functions on C S, then f s, g s s = 1. s S where, s denotes the symbol in O C,ŝ. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 35 / 1

Sketch of proof 1 One proves first that V = s S O C,ŝ is an analytic Tate space and H1 := Image [ Γ(C S, O C ) s S O C,ŝ ] H2 = s S O C,ŝ are (FN)-lattices in it. They determine the same polarization, so the value of f, g is the same if computed with H 1 or with H 2. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 36 / 1

Sketch of proof 1 One proves first that V = s S O C,ŝ is an analytic Tate space and H1 := Image [ Γ(C S, O C ) s S O C,ŝ ] H2 = s S O C,ŝ are (FN)-lattices in it. They determine the same polarization, so the value of f, g is the same if computed with H 1 or with H 2. 2 If one takes H 1, one checks that f, g = 1. 3 If one takes H 2, then f, g = s S f s, g s s. The result follows. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 36 / 1

Sketch of proof 1 One proves first that V = s S O C,ŝ is an analytic Tate space and H1 := Image [ Γ(C S, O C ) s S O C,ŝ ] H2 = s S O C,ŝ are (FN)-lattices in it. They determine the same polarization, so the value of f, g is the same if computed with H 1 or with H 2. 2 If one takes H 1, one checks that f, g = 1. 3 If one takes H 2, then f, g = s S f s, g s s. The result follows. Remark The theorem can also be proved along the lines of the complex analytic proof of Weil reciprocity, using just Stokes theorem. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 36 / 1

On Deligne s symbol (here, k = C) Let C be a Riemann surface, x C, let f, g be two holomorphic functions defined an invertible on a punctured neighborhood of x in C. Let S x C be a small circle around x, fix x 0 S x. Deligne defines a symbol attached to f, g and x by (f, g) Del = exp ( 1 ( x0 log f d g 2πi x 0 g log(g(x 0)) d f f (integral over S x, log f is a branch of the logarithm defined on S x {x 0 }). )). Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 37 / 1

On Deligne s symbol (here, k = C) Let C be a Riemann surface, x C, let f, g be two holomorphic functions defined an invertible on a punctured neighborhood of x in C. Let S x C be a small circle around x, fix x 0 S x. Deligne defines a symbol attached to f, g and x by (f, g) Del = exp ( 1 ( x0 log f d g 2πi x 0 g log(g(x 0)) d f f (integral over S x, log f is a branch of the logarithm defined on S x {x 0 }). In fact, Deligne attaches to f, g a line bundle over S x, and (f, g) Del is its monodromy. )) In case f, g are meromorphic at x, (f, g) Del equals Tate s tame symbol. Deligne s symbol verifies Weil reciprocity.. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 37 / 1

Comparison with Deligne s symbol. If f, g C{{t}} are meromorphic at zero, then f, g 1 = (f, g) Del, since both equal the tame symbol. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 38 / 1

Comparison with Deligne s symbol. If f, g C{{t}} are meromorphic at zero, then f, g 1 = (f, g) Del, since both equal the tame symbol. In general, assume f = f m f e and g = g m g e where f m, g m are meromorphic and f e = e ϕ f (t 1), g e = e ϕg (t 1) as in the Weierstrass-Birkhoff decomposition. An explicit computation gives that if f m = g m = 1, then f, g = (f, g) Del = 1. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 38 / 1

Comparison with Deligne s symbol. If f, g C{{t}} are meromorphic at zero, then f, g 1 = (f, g) Del, since both equal the tame symbol. In general, assume f = f m f e and g = g m g e where f m, g m are meromorphic and f e = e ϕ f (t 1), g e = e ϕg (t 1) as in the Weierstrass-Birkhoff decomposition. An explicit computation gives that if f m = g m = 1, then f, g = (f, g) Del = 1. In general, one has f, g (f, g) Del = exp ( ( d fm Res ϕ g + d g )) m ϕ f f m g m Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 38 / 1

Comparison with Deligne s symbol. If f, g C{{t}} are meromorphic at zero, then f, g 1 = (f, g) Del, since both equal the tame symbol. In general, assume f = f m f e and g = g m g e where f m, g m are meromorphic and f e = e ϕ f (t 1), g e = e ϕg (t 1) as in the Weierstrass-Birkhoff decomposition. An explicit computation gives that if f m = g m = 1, then f, g = (f, g) Del = 1. In general, one has f, g (f, g) Del = exp Meaning of the RHS? ( ( d fm Res ϕ g + d g )) m ϕ f f m g m Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 38 / 1

Reciprocity, p-adic case Let F be a finite field, X /F a projective non-singular curve, Y X an affine curve, S = X Y. From a p-adic point of view, a ring which plays a role similar to that of the ring of regular functions on Y is the ring A Y of overconvergent functions, the rough idea is X/F p F Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 39 / 1

Examples For example, if F = F p and Y = A 1 F, then A Y = i 0 a i t i a i Q p, ε > 1 lim i a i ε i = 0 1 ε Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 40 / 1

Examples Or, if Y = A 1 F {0}, then A Y = { i Z a i t i a i Q p, ε > 1 lim i a i ε i = 0 lim a i ε i = 0 i } Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 41 / 1

Examples Or, if Y = A 1 F {0}, then } A Y = { i Z a i t i a i Q p, ε > 1 lim i a i ε i = 0 lim a i ε i = 0 i To each closed point of s of X one can attach a Robba ring R s so that there is a map A Y R s f f s thus, any f A Y can be regarded as an element of R s. We have: Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 41 / 1

Reciprocity, p-adic case Theorem Under the previous assumptions, if f, g are invertible elements of A Y, then f s, g s s = 1. s S where, s denotes the symbol calculated in the Robba ring at s S. Proof. As in the complex case, considering now the analytic Tate space V = s S R s and computing f, g with the (DFN)-lattices G 1 = s S R s and G 2 = Image [ A Y V ]. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 42 / 1

Reciprocity, p-adic case Theorem Under the previous assumptions, if f, g are invertible elements of A Y, then f s, g s s = 1. s S where, s denotes the symbol calculated in the Robba ring at s S. Proof. As in the complex case, considering now the analytic Tate space V = s S R s and computing f, g with the (DFN)-lattices G 1 = s S R s and G 2 = Image [ A Y V ]. Here, no Stokes theorem is available (to my knowledge). Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 42 / 1

Semiinfinite...? Although it is not true that lattices are in an analytic Tate space belong to the same commensurability class, the following still holds: Proposition Let V be an analytic Tate space. If L 1, L 2 are two lattices of the same kind (either both (FN) or both (DFN)) and L 1 L 2, then the quotient L 1 /L 2 is finite dimensional. This property is the one used by Kapranov ( Semiinfinite symmetric powers ) to construct measures and Fourier transforms in the locally linearly compact case. Kapranov s construction can be mimicked in this context. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 43 / 1

Semiinfinite...? Although it is not true that lattices are in an analytic Tate space belong to the same commensurability class, the following still holds: Proposition Let V be an analytic Tate space. If L 1, L 2 are two lattices of the same kind (either both (FN) or both (DFN)) and L 1 L 2, then the quotient L 1 /L 2 is finite dimensional. This property is the one used by Kapranov ( Semiinfinite symmetric powers ) to construct measures and Fourier transforms in the locally linearly compact case. Kapranov s construction can be mimicked in this context. Similarly, one can adapt to the analytic Tate spaces Kapranov s definition of the semi-infinite de Rham complex associated to a locally linearly compact R-vector space. Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 43 / 1

Dziękuję bardzo za uwagę Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 44 / 1

Dziękuję bardzo za uwagę (According to Google s translator, this is Polish for Thank you very much for your attention ) Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 44 / 1

Dziękuję bardzo za uwagę (According to Google s translator, this is Polish for Thank you very much for your attention ) Koniec Ricardo García López (U.B.) Analytic Tate spaces and reciprocity Kraków, August, 2013 44 / 1