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 §2.) 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, §7] 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, §7] 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 §3.)

1 Odd means that if c ∈ 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 r|ΓL 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, §5.1])

{rλ :ΓF → GL56(M λ)}λ, 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, §8]) 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 §1, 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, §4]) 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 {rλ}λ 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 {rλ}λ 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. Lie primitive subgroups of exceptional groups which act irreducibly on a minuscule representation are classified in [LS04]. In ongoing research I’m also producing new results along the lines of Theorem 2.1 and Corollary 2.2 using other Lie primitive subgroups of exceptional groups. In particular, my current research will produce a classification of potentially automorphic Galois representations with exceptional monodromy which can be produced using this method, in terms of the classification of Lie primitive subgroups. A main motivation for [Chi19b] was the paper [BCE+19], which proves a theorem similar to Theorem 2.1 for any CM field F with E7 replaced by E6 (although using different methods – they use a finite group + theory argument which doesn’t generalize well to E7). [BCE 19] also proves that the system {rλ}λ comes from taking λ-adic realizations of a pure motive. I’m also currently working on proving motivic results using Theorem 2.1, and planning to do the same with other similar theorems that I am producing. This would produce proofs of new cases of the Fontiane-Mazur conjecture ([FM95]).

3. A sparsity of automorphic points

Let F be a number field and G a reductive algebraic Z-group scheme. Let ρ :ΓF → G(Fp) denote a 0 Galois representation. Say two lifts ρ, ρ :ΓF → G(Zp) are equivalent if they are conjugate by an element of the kernel of G(Zp) → G(F p). A deformation of ρ is an equivalence class of lifts of ρ. Many of the results mentioned so far are proved using moduli spaces of lifts or deformations of Galois representations ρ (first introduced in [Maz89]).

When ρ :ΓQ → GL2(Fp) is irreducible and odd, not only does ρ have a modular lift (by the proof of Serre’s conjecture), but modular points are dense in quite general deformation spaces associated to ρ ([B01]).¨ + Similar density results are known/expected when Q is replaced by a totally real field F , GL2 is replaced by a higher rank group G, and ρ is suitably “irreducible and odd,” for example in [All19]. When F is a general number field, or when ρ is not odd, automorphic points are expected to become much more sparse, due to an apparent lack of corresponding automorphic representations, e.g. the vanishing of cuspidal cohomology (see e.g. [Har87] or [BW00]) outside of “parallel weight.” Using parallel weight constraints, [CM09, §7] gives examples of ρ :ΓF → GL2(Fp) where F/Q is quadratic imaginary, and 1- dimensional spaces of deformations of ρ for which modular points are discrete.

3.1. Results. Let G be a reductive group, and F/Q a CM extension. Any Galois representations associated to automorphic representations of G(AF ) (which satisfies an expected local-global compatibility) also has parallel Hodge-Tate weights. On the motivic side, any pure motive X over F satisfies Hodge symmetry, which also translates into parallel Hodge-Tate weights for the `-adic realizations of X (see e.g. [Pat17, §2]). Guided by these higher rank analogues of “parallel weight,” I have proven the following generalization to [CM09, Thm. 7.1].

Theorem 3.1 ([Chi19a, Thm. 1.2]). Let G denote a reductive Z-group, p a sufficiently large prime, F a CM number field satisfying mild hypotheses, and ρ :ΓF → G(Fp) a Galois representation satisfying certain local 4 RESEARCH STATEMENT – KEVIN CHILDERS

3 der hypotheses whose image contains G (Fp). Let r equal the product of the degree of F/Q and the semi-simple rank of G. Then there exists a Galois representation lifting ρ

ρ :ΓF → G(Zp X1,...,Xr ) J K such that the set of Qp-points of ρ which satisfy the hypotheses of the Fontaine-Mazur conjecture and have parallel Hodge-Tate weights has positive codimension in the rigid analytic open unit r-ball. In particular, the motivic points of ρ are sparse, as are the points which are known to be automorphic and satisfy local-global compatibility. 3.2. Significance. The main purpose of Theorem 3.1 is to illustrate the stark differences between tradition- ally studied odd Galois representations with the more mysterious non-odd representations. The proof also contributes some new generalizations of the method of Ramakrishna for lifting Galois representations developed in [Ram99] and [Ram02] (see also [Tay03]) and extended to more general groups and fields in [CHT08], [Pat16], and [FKP19]. All of these references apply only to odd Galois representations, and produce lifts which satisfy the hypotheses of the Fontaine-Mazur conjecture. Extending ideas of [CM09], I develop a form of the Ramakrishna method (comparible in generality to that of [Pat16]) for non-odd representations which produces, in particular, the lifts of Theorem 3.1. 3.3. Ongoing work. The details of the proof of Theorem 3.1 were worked out using ideas from [Pat16]. Since then, [FKP19] has vastly improved the methods of [Pat16]. In ongoing research, I plan to apply the methods of [FKP19] to relax the hypotheses of Theorem 3.1, as well as to explore other cases where automorphic points are conjecturally sparse. For example, over general number fields which are not totally real or CM, the Hodge-Tate weights are expected to descend to the maximal CM subfield. The method used in the proof of Theorem 3.1 only proves that the the parallel weight specializations of ρ live in a subspace of positive codimension. However, the proof also suggests that the parallel weight specializations should live in codimension r/2. I’m also investigating the codimension further by exploring when certain hyper-surfaces intersect properly, or can be perturbed to intersect properly using methods similar to [Chi19a].

4. The inverse Galois problem The inverse Galois problem (IGP) asks which finite groups are Galois groups of extensions of Q (or any other field F ). There are many known cases, for instance IGP is known for all solvable groups, Sn and An for any n, and all but one sporadic group (see e.g. [Ser08] and [MM18]). However, many cases of IGP remain open for groups of Lie type. A (vague) strategy for proving IGP for a group G of Lie type is the following: • Construct automorphic representations π with a certain set of properties P. • Construct (compatible systems of) Galois representations associated to π and satisfying some set of conditions S, implied by π satisfying the conditions P. • Show that residually, some Galois representation satisfying S must have image equal to G. For example, cuspidal modular forms are used to prove cases of IGP for PSL2(Fp) ([Zyw15]) and some symplectic cases are proven using this strategy in [KLS08] and [Shi11]. My ongoing improvements to Theorem 2.1 require an argument along these lines, proving new cases of IGP along the way for groups of type A7. Also, new cases of IGP for groups of type E7 will result as corollaries to new and improved versions of Theorem 2.1.

4.1. Other ongoing work. IGP for SL2(Fp) is surprisingly difficult, but the following provides some insight on how new cases might be proven. Suppose that IGP is known for SL2(Fp) (this is true for some small primes p), manifested by some (surjective) Galois representation ρ :ΓQ → SL2(Fp). Then for some quadratic imaginary F/Q, ρ :ΓF → SL2(Fp) is surjective, and one can use arguments from [BLGGT14] to show that 2 Sym ρ :ΓF → SO3(Fp) is a residual representation in a compatible system of potentially automorphic 3-dimensional Galois representations. I am collecting known examples of SL2(Fp)-extensions K/Q, and computationally finding explicit auto- morphic representations of GL3(AF ) which give rise to K/Q in this manner.

3Most notably that if v is a prime of F dividing p, then the restriction of ρ to a decomposition group of v lands in a fixed Borel subgroup of G in a “non-split” way. RESEARCH STATEMENT – KEVIN CHILDERS 5

4.2. Significance. The ultimate goal is to glean which properties P and S could be used in the strategy outlined above, and to prove new cases of IGP for SL2(Fp).

References [All19] Patrick B. Allen. On automorphic points in polarized deformation rings. Amer. J. Math., 141(1):119–167, 2019. [B01]¨ Gebhard B¨ockle. On the density of modular points in universal deformation spaces. Amer. J. Math., 123(5):985–1007, 2001. [BCE+19] George Boxer, Frank Calegari, Matthew Emerton, Brandon Levin, Keerthi Madapusi Pera, and Stefan Patrikis. Compatible systems of Galois representations associated to the exceptional group E6. Forum Math. Sigma, 7:e4, 29, 2019. [BLGGT14] Thomas Barnet-Lamb, Toby Gee, David Geraghty, and Richard Taylor. Potential automorphy and change of weight. Ann. of Math. (2), 179(2):501–609, 2014. [Bor79] A. Borel. Automorphic L-functions. In Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, pages 27–61. Amer. Math. Soc., Providence, R.I., 1979. [BW00] A. Borel and N. Wallach. Continuous cohomology, discrete subgroups, and representations of reductive groups, volume 67 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, second edition, 2000. [Chi19a] Kevin Childers. Galois deformation spaces with a sparsity of automorphic points. 2019. preprint. [Chi19b] Kevin Childers. Potential automorphy and E7. 2019. preprint. [CHT08] Laurent Clozel, Michael Harris, and Richard Taylor. Automorphy for some l-adic lifts of au- tomorphic mod l Galois representations. Publ. Math. Inst. Hautes Etudes´ Sci., (108):1–181, 2008. With Appendix A, summarizing unpublished work of Russ Mann, and Appendix B by Marie-France Vign´eras. [CM09] Frank Calegari and Barry Mazur. Nearly ordinary Galois deformations over arbitrary number fields. J. Inst. Math. Jussieu, 8(1):99–177, 2009. [FKP19] Najmuddin Fakhruddin, Chandrashekhar Khare, and Stefan Patrikis. Relative deformation theory and lifting irreducible Galois representations. arXiv e-prints, page arXiv:1904.02374, Apr 2019. [FM95] Jean-Marc Fontaine and Barry Mazur. Geometric Galois representations. In Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993), Ser. Number Theory, I, pages 41–78. Int. Press, Cambridge, MA, 1995. [GJ72] Roger Godement and Herv´eJacquet. Zeta functions of simple algebras. Lecture Notes in Mathematics, Vol. 260. Springer-Verlag, Berlin-New York, 1972. [Har87] G. Harder. Eisenstein cohomology of arithmetic groups. The case GL2. Invent. Math., 89(1):37– 118, 1987. [Hec83] Erich Hecke. Mathematische Werke. Vandenhoeck & Ruprecht, G¨ottingen,third edition, 1983. With introductory material by B. Schoeneberg, C. L. Siegel and J. Nielsen. [HLTT16] Michael Harris, Kai-Wen Lan, Richard Taylor, and Jack Thorne. On the rigid cohomology of certain Shimura varieties. Res. Math. Sci., 3:Paper No. 37, 308, 2016. [Isa06] I. Martin Isaacs. Character theory of finite groups. AMS Chelsea Publishing, Providence, RI, 2006. Corrected reprint of the 1976 original [Academic Press, New York; MR0460423]. [KLS08] Chandrashekhar Khare, Michael Larsen, and Gordan Savin. Functoriality and the inverse Galois problem. Compos. Math., 144(3):541–564, 2008. [KW09a] Chandrashekhar Khare and Jean-Pierre Wintenberger. Serre’s modularity conjecture. I. Invent. Math., 178(3):485–504, 2009. [KW09b] Chandrashekhar Khare and Jean-Pierre Wintenberger. Serre’s modularity conjecture. II. Invent. Math., 178(3):505–586, 2009. [LS04] Martin W. Liebeck and Gary M. Seitz. Subgroups of exceptional algebraic groups which are irreducible on an adjoint or minimal module. J. Group Theory, 7(3):347–372, 2004. [Maz89] B. Mazur. Deforming Galois representations. In Galois groups over Q (Berkeley, CA, 1987), volume 16 of Math. Sci. Res. Inst. Publ., pages 385–437. Springer, New York, 1989. 6 RESEARCH STATEMENT – KEVIN CHILDERS

[MM18] Gunter Malle and B. Heinrich Matzat. Inverse Galois theory. Springer Monographs in Mathe- matics. Springer, Berlin, 2018. Second edition [ MR1711577]. [Pat16] Stefan Patrikis. Deformations of Galois representations and exceptional monodromy. Invent. Math., 205(2):269–336, 2016. [Pat17] Stefan Patrikis. Generalized Kuga-Satake theory and good reduction properties of Galois rep- resentations. Algebra Number Theory, 11(10):2397–2423, 2017. [Ram99] Ravi Ramakrishna. Lifting Galois representations. Invent. Math., 138(3):537–562, 1999. [Ram02] Ravi Ramakrishna. Deforming Galois representations and the conjectures of Serre and Fontaine- Mazur. Ann. of Math. (2), 156(1):115–154, 2002. [Rie75] B. Riemann. On the number of prime numbers below a given quantity. Sci. Rep. Kagoshima Univ., (24j):1–8, 1975. Translated from the German with commentary by Yˆoichi Koshiba. [Sch15] Peter Scholze. On torsion in the cohomology of locally symmetric varieties. Ann. of Math. (2), 182(3):945–1066, 2015. [Ser87] Jean-Pierre Serre. Sur les repr´esentations modulaires de degr´e2 de Gal(Q/Q). Duke Math. J., 54(1):179–230, 1987. [Ser08] Jean-Pierre Serre. Topics in Galois theory, volume 1 of Research Notes in Mathematics.AK Peters, Ltd., Wellesley, MA, second edition, 2008. With notes by Henri Darmon. [Shi11] Sug Woo Shin. Galois representations arising from some compact Shimura varieties. Ann. of Math. (2), 173(3):1645–1741, 2011. [Shi12] Sug Woo Shin. Automorphic Plancherel density theorem. Israel J. Math., 192(1):83–120, 2012. [SU14] Christopher Skinner and Eric Urban. The Iwasawa main conjectures for GL2. Invent. Math., 195(1):1–277, 2014. [Tat67] J. T. Tate. Fourier analysis in number fields, and Hecke’s zeta-functions. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pages 305–347. Thompson, Washington, D.C., 1967. [Tay02] Richard Taylor. Remarks on a conjecture of Fontaine and Mazur. J. Inst. Math. Jussieu, 1(1):125–143, 2002. [Tay03] Richard Taylor. On icosahedral Artin representations. II. Amer. J. Math., 125(3):549–566, 2003. [TW95] Richard Taylor and Andrew Wiles. Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2), 141(3):553–572, 1995. [Wil95] Andrew Wiles. Modular elliptic curves and Fermat’s last theorem. Ann. of Math. (2), 141(3):443–551, 1995. [Zyw15] David Zywina. The inverse Galois problem for PSL2(Fp). Duke Math. J., 164(12):2253–2292, 2015.