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HG Duan, VI Prokhorenko, RJ Cogdell, K Ashraf, AL Stevens, M Thorwart, ... Proceedings of the National Academy of Sciences 114 (32), 8493-8498, 2017. 299: 2017:
- ✪ χ(n)bn D2(s, χ) = ns
- Automorphic Forms
- A = R × Y′ Qp = Y′ Qv ⊂ Yv Qv
- 2 Automorphic Forms on GLn
- GLn(A) = Y′ GLn(kv) GLn(Ov)
- v Y′ GLn(kv) ⊃ Kf = Y GLn(Ov). v<∞
- 3 Smooth Automorphic Forms
- A∞ = A∞(GLn(k)\GLn(A)).
- Automorphic Representations
- Hv = lim −→ ξi,v ∗ Hv ∗ ξi,v =
- V ol(Lv)
- Vφ = R(H)φ = φ ∗ H ⊂ A,
- 2 Smooth automorphic representations
- r < ∞ for all X ∈ U(g)} g∈G(A)
- 5 Connections with classical forms
- Fourier Expansions and Multiplicity One Theorems
- 2 Whittaker models
- Definition 4.1
- Eulerian Integral Representations
- ✪ 2 v dhv.
- 3 GLn × GLn
- 0 (GLn) and φ′ ∈ Vπ′ ⊂ A∞
- Z φ(g)φ′(g)E(g, s) dg
- ∈ W(π′ v, ψ−1
- Local L-functions: the Non-Archimedean Case
- 3 The local functional equation
- The Unramified Calculation
- 1 Unramified representations
- J Tr(ρ
- 3 Calculating the integral
- 1 −−−−→ C× −−−−→ WR −−−−→ Gal(C/R) −−−−→ 1
- L(s, τi).
- 4 Is the L-factor correct?
- Ψ(s, Wi, W′ i)
- Global L-functions
- Definition 9.1
- L(s, π × π′) = L(s, πv × π′ v)
- ✪ = WN
- 2 Meromorphic continuation
- LS(s, π × π′)
- L(s, π × π′)
- • L(s, π × π′) = ε(s, π × π′)L(1 − s,
- 5 Boundedness in vertical strips
- I(s, φi, φ′ i) m = n − 1.
- L(s, π × π′) = I(s, φ) with φ ∈ Vπ ⊗Vπ′ ˆ
- 8 Generalized Strong Multiplicity One
- Q(A) (τ1 ⊗ · · · ⊗ τr).
- Converse Theorems
- 1 Converse Theorems for GLn
- Vπ ֒→ A∞ 0 (GLn(k)\GLn(A))?
- v ⊃ GLn−1(OS).
- Vξ(g) = X Wξ(αmqg) where now N′ = α−1
- Functoriality
- 1 The Weil-Deligne group
- 2 The dual group
- 3 The local Langlands conjecture
- 4 Local Functoriality
- Principle of Functoriality: Associated to the L-homomorphism r : LG → LGLn
- XX EE
- 5 Global functoriality
- Principle of Functoriality: Associated to the L-homomorphism r : LG → LGLn
- 6 Functoriality and the Converse Theorem
- Functoriality for the Classical Groups
- 1 The result
- 2 Construction of a candidate lift
- L(s, π × π′) = ε(s, π × π′)L(1 − s, ×
- YY EE
and set Λi(s, χ) = (2π)−sΓ(s)Di(s, χ). Suppose that there exists an N such that for all primitive characters χ of conductor q prime to N we have the Λi(s, χ) extend to entire functions of s, the Λi(s, χ) are bounded in vertical strips, we have the functional equation
In this lecture I want to begin the passage from classical modular forms f to automorphic forms φ and finally to automorphic representations π. This will entail a change of tools from the theory of one complex variable to the use of non-abelian harmonic analysis, that is, representation theory.
p v with respect to the compact open subrings Zp ⊂ Qp. More precisely if we let Sf run over all finite sets of primes then A is the union, or inductive limit,
It should be clear how to define automorphic forms on GLn(k)\GLn(A) for A the ring of adeles for any global field k. For our purposes, we will stick to A being the ring of adeles of a number field k. Let O denote the ring of integers of k. The ring A is then the restricted product of the completions kv of k with respect to the maximal compact subri...
v = lim −→ GLn(kv) × S v∈S Y v/∈S Y is the restricted product with respect to the maximal open compact subgroups Kv = GLn(Ov) ⊂ GLn(kv) for the non-archimedean places. If we agree to now let G = GLn then as before we have
Again, G(k) ֒→ G(A) diagonally as a canonical discrete subgroup with finite co-volume modulo the center Z(A). Let Z = Z(g) denote the center of the universal enveloping algebra U(g) of the complexified Lie algebra g of G∞. Definition 2.2 A smooth function φ : GLn(A) → C is called a (K-finite) automorphic form if it satisfies: [automorphy] φ(γg) = φ...
One would hope to be able to analyze A as a representation of GLn(A) acting by right translation. Unfortunately, this is not possible since condition (ii) in the definition of automorphic forms is not preserved under right translation. More specifically, it is being K∞-finite that is not preserved under right translation by G∞. [To make this more p...
Now GLn(A) does act on A∞ by right translation. Moreover A∞ will carry a limit Fr ́echet topology coming from the uniform moderate growth semi-norms. A∞ is not that far removed from A. By a theorem of Harish-Chandra we know that K∞-finiteness implies uniform moderate growth, so that A ⊂ A∞ and A is precisely the space of φ ∈ A∞ that are K-finite, a...
We have defined our spaces of automorphic forms. Now we turn to our tools. We will analyze A, A∞, or L2(ω) as representation spaces for certain algebras or groups. Throughout we will let G = GLn, although the results remain true for any reductive algebraic group G, let k be a number field, and retain all notations from before.
i Neither H nor any Hv ξi,v ∗ H ∗ ξi,v. i will have an identity, but for each will have ξi as an identity. ξi the subalgebra ξi ∗H∗ξi If v < ∞ is an non-archimedean place of k then Hv = C∞ c (G(kv)) is the algebra of smooth (locally compact) compactly supported functions on Gv = G(kv). It is naturally an algebra under convolution. For each compact ...
where XLv is the characteristic function of Lv. Then ξLv ∗ Hv ∗ ξLv = H(Gv//Lv) is the algebra of Lv-bi-invariant compactly supported functions on Gv. In any representation of Hv the idempotent ξLv will act as a projection onto the Lv-fixed vectors. We will let ξ◦ v denote the fundamental idempotent associated to the maximal compact subgroup Kv. No...
is an admissible H-module. This makes the following definition reasonable. Definition 3.2 An automorphic representation (π, V ) of H is an irreducible (hence admissible) sub-quotient of A(G(k)\G(A)). There is a canonical way to construct admissible representations of H abstractly using the restricted tensor product structure H = ⊗′Hv. Suppose we ha...
Now things are more straight forward on the one hand, since G(A) acts in A∞(G(k)\G(A)) by right translation. However the representation theory is now a bit more complicated. More precisely, for every compact open subgroup L ⊂ Kf the space of L-invariant functions (A∞)L in A∞, namely (A∞)L = {φ ∈ A∞ | φ(gl) = φ(g) for l ∈ L}, is a representation f...
then for any open compact subgroup L ⊂ Kf the space of L-fixed vectors (A∞ r )L in A∞ r is a smooth Frechet representation of moderate growth for G∞ defined by the natural seminorms qX,r(φ) = sup (kgk−r|Xφ(g)|) for X ∈ U(g). g Then as topological representations both (A∞)L = lim −→ (A∞ )L and A∞
Suppose we return to a classical cusp form f for SL2(Z) of weight m. If we follow our passage f 7→φ 7→(πφ, Vφ) then (πφ, Vφ) is an admissible subspace of the space of cuspidal automorphic forms. It need not be irreducible. However, if in addition f is a simultaneous eigen-function for all the classical Hecke operators, then (πφ, Vφ) is irreducible ...
We now start with results which are often GLn specific. So we let G = GLn (however one should also keep in mind G = GLn × GLm) and still take k to be a number field.
Consider now the functions W = Wφ which appear in the Fourier expansion of our cusp forms φ ∈ Vπ. These are smooth functions on G(A) satisfying W(ng) = ψ(n)W(g) for all n ∈ N(A). Let
A representation (πv, Vπv) having a Whittaker model is called generic. Of course, the same definition applies in the global situation. Note that for G = GLn this notion is independent of the choice of (non-trivial) ψv or ψ. Now return to our smooth cuspidal representation (π, Vπ), or in fact any ir-reducible admissible smooth representation of G(A)...
We now turn to the integral representations for the L-functions for GLn. We will be interested not only in the integral representations for L(s, π), with π a cuspidal representation of GLn(A), but also the twisted L-functions L(s, π × π′) with π′ a cuspidal representation of GLm(A). We have seen these in Weil’s Converse Theorem and in the lectures ...
Hence our family is once again Eulerian. For this family of integrals the two sides of the functional equation involve slightly different integrals. With a little (?) more work, one shows that the integrals occurring in the right hand side of the functional equation are also Eulerian, with
In the m = n case the paradigm comes not from Hecke but from the classical work of Rankin and Selberg, again with a bit of Tate’s thesis mixed in. One might be first tempted to try
0 (GLn), but this, convergence issues aside, would give an invariant pairing and would be zero unless eπ ≃ π′ ⊗| det |s. (In fact, such integrals will arise as residues of poles of our family of Eulerian integrals.) Instead, we will consider integrals of the form
where E(g, s) is an appropriate Eisenstein series. Murty wrote down the classical version of this in his lectures. To construct our Eisenstein series we begin with a Schwartz-Bruhat function Φ ∈ S(An). Recall that S(An) is a restricted tensor product (topological at the archimedean places), S(An) = ⊗′S(kn ), where for v|∞ we have S(kn ) = S(Rn) or ...
), and if necessary Φv ∈ S(kn ). Next we will first sift the local L-functions L(s, πv×π′ v) from our families of local integrals. We then put these together to form the global L-function L(s, π × π′). Then we relate this global L-function back to the family of global integrals and deduce its analytic properties from theirs. We will do this over th...
For this lecture and the next k will be a non-archimedean local field, O its ring of integers, p its maximal ideal. We will let ̟ denote a uniformizer, so p = (̟). We normalize the absolute value by
From the functional equation of the global integrals we would expect a re-lation between Ψ(s, W, W′) and − This will follow from interpreting these integrals as giving quasi-invariant functionals on Vπ × Vπ′ or W(π, ψ) × W(π′, ψ−1) and then invoking a uniqueness principle. s, R(wn,m)fW , W′). f As for the quasi-invariance, note that the integrals Ψ...
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
πv ✪ φv Φv ✪ Πv. W′ kv Philosophically, through functoriality one hopes to pull back structural results from GLN to the classical groups Gn. In this lecture I would like to outline some of what we know in these cases.
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Apr 23, 2024 · The Cogdell system is located near the Morven and Summerstide systems and consists of a class G5V primary orbited by at least four planets. System History . The Cogdell system was settled prior to the Third Succession War but didn't become significant enough to record on maps until later.
Cogdell is an unincorporated community and census-designated place (CDP) in the northeast corner of Clinch County, Georgia, United States. It is on Georgia State Route 122, 22 miles (35 km) west of Waycross and 24 miles (39 km) east of Lakeland. Homerville, the Clinch county seat, is 10 miles (16 km) to the south.
