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Research Statement – Kevin Childers RESEARCH STATEMENT { KEVIN CHILDERS 1. Summary 1.1. L-functions. The first example of an L-function is the Riemann ζ-function. In his famous paper [Rie75], Riemann proves that the ζ-function, defined for complex numbers s with <(s) > 1 as X Y ζ(s) = n−s = (1 − p−s)−1; n≥1 p has meromorphic continuation to the complex plane and satisfies a functional equation. He also states the famous \Riemann hypothesis," and explains connections between ζ(s) and the density and pattern of prime integers. Since then, other L-functions have been attached to many arithmetically interesting objects X { such as algebraic varieties, Galois representations, and automorphic objects { and many deep conjectures have been made which encode arithmetic data of X into its L-functions. Arithmetic conjectures about L-functions often depend on first proving that they are analytic in a right half plane, have meromorphic continuation, and satisfy a desired functional equation. 1.2. Galois and automorphic representations. Proving analytic properties of L-functions associated to Galois representations is, in general, considered quite hard (e.g. Artin's conjecture). In [Chi19b], I construct Galois representations ρ with Zariski dense image in a p-adic Lie group of type E7 and prove that certain L-functions naturally attached to ρ are analytic in a right half plane, have meromorphic continuation to the complex plane, and satisfy the expected functional equation. These are the first examples of their kind in the literature, and in my current research I am extending these methods to produce other new examples of L-functions attached to Galois representations satisfying desired analytic properties. The proof follows by associating ρ to automorphic objects, and deducing analytic properties of the L-function attached to ρ from analytic properties of automorphic L-functions. Due to Riemann, Hecke ([Hec83]), Tate ([Tat67]), Godement and Jaquet ([GJ72]), and many others, L- functions associated to automorphic representations of GLn are known to satisfy all of the above analytic properties. The way that Wiles was able to prove Fermat's last theorem ([Wil95] and [TW95]) was to show that p-adic Galois representations ρ are \modular," in the sense that ρ has the same L-function as a modular form (and the automorphic representation it generates). In [Chi19b], I employ generalizations of Wiles's methods (e.g. [BLGGT14]) to prove that ρ is \potentially automorphic," from which analytic properties of L-functions are deduced. (More details can be found in x2.) 1.3. Deformations of Galois representations. For a perfect field F let F denote the algebraic closure of F and ΓF := Gal(F =F ). Wiles's method studies the reduction of p-adic Galois representations modulo p, moduli spaces of p-adic lifts of such residual representations, and the modular points in such spaces. If ρ :ΓQ ! GL2(Fp) is a 2-dimensional Galois representation, a conjecture of Serre ([Ser87]) now a Theorem ([KW09a] and [KW09b]) asserts that if ρ is absolutely irreducible and odd1, then ρ has a modular lift. In fact, [B01]¨ essentially shows that, under the hypotheses of Serre's conjecture, modular lifts are dense is the moduli space of lifts of ρ. Similar results are expected (and often known) if Q is replaced with any totally real field. In contrast, [CM09, x7] shows that modular lifts are sparse for representations ρ :ΓF ! GL2(Fp) where F=Q is a quadratic imaginary field. In [Chi19a], I have extended the results of [CM09, x7] from quadratic imaginary fields to more general CM number fields, and from 2-dimensional Galois representations to Galois representations valued in any reductive algebraic group G, producing mod p representations with a sparsity of automorphic lifts2. This provides an example of the stark contrast between the totally real case (where many things are known) and the non-totally real case (where very little is known). (More details can be found in x3.) 1 Odd means that if c 2 ΓQ is a choice of complex conjugation, then det ρ(c) = −1. 2Assuming that Galois representations associated to automorphic representations satisfy certain conjectural properties. 1 2 RESEARCH STATEMENT { KEVIN CHILDERS 2. Analytic continuation of an E7 L-function Let F denote a totally real or CM number field and AF is its ring of adeles. Then [Sch15], [HLTT16], and the earlier work of many others construct Galois representations associated to any regular algebraic automorphic representation of GLn(AF ), in the sense that they have the same L-functions. In the other direction, if r :ΓF ! GLn(Q`) is a sufficiently well behaved Galois representation, [BLGGT14] and it's predecessors (and successors) prove that r is \potentially automorphic," in the sense that there is a finite field extension L=F such that rjΓL is automorphic. 2.1. Results. In [Chi19b], I construct an example of a Galois representation ρ :ΓF ! G(Q`) valued in der a group G such that G is an exceptional group of type E7, and such that the composition of ρ with a faithful representation rmin of G r = rmin ◦ ρ :ΓF ! GL56(Q`) satisfies the hypotheses of [BLGGT14]. More precisely, Theorem 2.1 ([Chi19b, Thm. 1.2]). For any prime number p there exists a totally real number field F and a Galois representation ρ :ΓF ! G(Qp) such that: • r = rmin ◦ ρ is potentially automorphic. • r belongs to a strictly pure compatible system of Galois representations (in the sense of [BLGGT14, x5.1]) frλ :ΓF ! GL56(M λ)gλ; where λ ranges over the primes of a number field M. • For every λ, the Zariski closure of the image of rλ is G(M λ), where G is a reductive group with der G = E7. Knowing the Galois representation r is potentially automorphic, doesn't imply that L(ρ, rmin; s) is auto- morphic, but a clever application of Brauer's theorem (see e.g. [Isa06, x8]) and solvable descent allows one to write L(ρ, rmin; s) as a product of integral powers of automorphic L-funcions, from which one can deduce meromorphic continuation and a functional equation for L(ρ, rmin; s) (see e.g. [Tay02]). In particular, as a corollary to Theorem 2.1 I prove the following, which appears to be the first example of such a theorem for E7-valued Galois representations in the literature. Corollary 2.2 ([Chi19b, Thm. 1.1]). Let ρ satisfy the conclusions of Theorem 2.1. Then the L-function associated to ρ with respect to the representation rmin converges in a right half plane, has meromorphic continuation to the complex plane, and satisfies the expected functional equation. 2.2. Significance. As mentioned in x1, proving new cases of meromorphic continuation and functional equations of new L-functions is, in itself, significant in number theory. Besides proving analytic properties of new L-functions, the results of [Chi19b] also provide significant new evidence of other important conjectures. Let G denote a reductive algebraic Z-group scheme. Conjectures of Langlands, Fontaine-Mazur, Serre, and Tate predict connections between automorphic representations of G(AF ) and Galois representations valued in the \L-group" of G, denoted LG (see [Bor79]), for example that they have the same L-functions. For an exceptional group like E7, associating Galois representations and automorphic representations is a totally open problem, but the potential automorphy of r and the monodromy across the compatible system (as proven in Theorem 2.1) strongly suggests that ρ is an example of an automorphic Galois representation valued in an exceptional group of type E7. Cor. 2.2 also has significance to conjectures about automorphic L-functions. If ρ is automorphic, then L(ρ, rmin; s) = L(π; rmin; s) for an automorphic representation π, whose automorphic L-function is only conjecturally defined, and for which meromorphic continuation and a functional equation are totally open (see [Bor79]). Further, ρ is constructed as a lift of a residual representation ρ which is \absolutely irreducible and odd," in a certain sense. Thus connecting ρ to automorphic representations provides new evidence of a generalized Serre conjecture. RESEARCH STATEMENT { KEVIN CHILDERS 3 2.3. Ongoing work. One shortcoming of Theorem 2.1 is that it only proves existence of the field F , where one might like to say that a Galois representation such as ρ exists for any totally real field F . In my current research, I am extending the results of [Chi19b] to general totally real fields F using the following strategy: First, using ideas from [Shi12], prove the existence of certain automorphic representations of (a form of) PGL8(AF ), which produce Galois representations valued in SL8. Then, using a Lie primitive subgroup of E7 which is a form of A7 (as explained in [Chi19b, x4]) prove versions of Theorem 2.1 and Corollary 2.2 which hold over any totally real field F , in particular Q. One interesting consequence of this method is that the compatible system frλgλ has E7-monodromy, but for some prime λ the image of the rλ is degenerate, in the sense that the image is contained in SL8. To motivate the significance of such degenration, consider a case arisng from modular forms. Suppose frλgλ is a compatible system of 2-dimensional Galois representations produced using a cuspidal modular form f. Each rλ is irreducible. Then having the image degenerate to a reducible representation modulo some λ translates into a congruence between f and an Eisenstein series. This phenomenon has been important, e.g. in the field of Iwasawa theory ([SU14]). In the case of an E7-system, degeneration to SL8 should be seen as a congruence between an E7-cuspidal automorphic representation and the (conjectural) functorial transfer of a PGL8-automorphic representation along the map SL8 ! E7, at least if the residual SL8-valued representation has an automorphic lift.
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