STARK’S CONJECTURES AND HILBERT’S 12TH PROBLEM Samit Dasgupta Duke University Fields Institute, Toronto Online Seminar—8/26/2020
CLASS FIELD THEORY Class field theory describes the Galois group of the maximal abelian extension of a number field . F Gal ( F ab / F ) ≅ A * F / F * F >0 ∞ The right hand side uses information intrinsic to only itself. F Explicit class field theory asks for the construction of the field , again using only information intrinsic to . F ab F 2
̂ KRONECKER-WEBER THEOREM Let . F = Q Class field theory: Z * ≅ ∏ Gal ( Q ab / Q ) ≅ Z * p p Explicit class field theory: (Kronecker-Weber) ∞ Q ab = ⋃ Q ( e 2 π i / n ) n =1 3
COMPLEX MULTIPLICATION Quadratic imaginary fields. F = Q ( d = positive integer . − d ), Theorem . where is an elliptic F n = F ( j ( E ), w ( E [ n ])) E curve with complex multiplication by and 𝒫 F “Weber function.” w = j ( q ) = q − 1 + 744 + 196884 q + 2149360 q 2 + ⋯ Here is the usual modular function. For , modular − d ) F = Q ( functions play the role of the exponential function for . F = Q 4
HILBERT’S 12TH PROBLEM (1900) “The theorem that every abelian number field arises from the realm of rational numbers by the composition of fields of roots of unity is due to Kronecker.” “Since the realm of the imaginary quadratic number fields is the simplest after the realm of rational numbers, the problem arises, to extend Kronecker’s theorem to this case.” “Finally, the extension of Kronecker’s theorem to the case that, in the place of the realm of rational numbers or of the imaginary quadratic field, any algebraic field whatever is laid down as the realm of rationality, seems to me of the greatest importance. I regard this problem as one of the most profound and far-reaching in the theory of numbers and of functions.” 5
APPROACHES USING L-FUNCTIONS ➤ Stark stated a series of conjectures proposing the existence of elements in abelian extensions whose absolute values are related to H / F L -functions (1971-80). ➤ Tate made Stark’s conjectures more precise and stated the Brumer-Stark conjecture. (1981) ➤ Gross refined the Brumer-Stark conjecture using -adic L -functions. p This is called the Gross-Stark conjecture (1981). ➤ Rubin (1996), Burns (2007), and Popescu (2011) made the higher rank version of Stark’s conjectures more precise. ➤ Burns , Popescu , and Greither made partial progress on Brumer-Stark building on work of Wiles . 6
THE BRUMER-STARK AND GROSS-STARK CONJECTURES Let be a totally real number field. Let be a finite CM F H abelian extension . F ➤ The Brumer-Stark conjecture predicts the existence of certain elements called Brumer-Stark units that u ∈ H * are related to L -functions of in a specific way. F ➤ The Gross-Stark conjecture predicts that these units are related to -adic L -functions of in a specific way. p F 7
SOME OF MY PRIOR WORK IN THIS AREA Stated a conjectural exact formula for Brumer-Stark units in several joint works, with: Henri Darmon Pierre Charollois Matthew Greenberg Michael Spiess
SOME OF MY PRIOR WORK IN THIS AREA Proved the Gross-Stark conjecture* Benedict Gross in joint works with: Henri Darmon Robert Pollack Mahesh Kakde Kevin Ventullo
NEW RESULTS* (WITH MAHESH KAKDE) Theorem 1. The Brumer-Stark conjecture holds if we invert 2 (i.e. up to a bounded power of 2). Theorem 2. My conjectural exact formula for Brumer-Stark units holds, up to a bounded root of unity. 10
P-ADIC SOLUTION TO HILBERT’S 12TH PROBLEM Hilbert’s 12th problem is viewed as asking for the construction of the field using analytic functions depending only on . F ab F The Brumer-Stark units, together with other explicit and easy to describe elements, generate the field . F ab Our exact formula expresses the Brumer-Stark units as p -adic integrals of analytic functions depending only on . F Therefore the proof of this conjecture can be viewed as a p -adic solution to Hilbert’s 12th problem. 11
STARK’S CONJECTURE = finite abelian ext of number fields, G = Gal ( H / F ) . H / F = place of that splits completely in v F H . = a set of places of containing the infinite places, ramified places, and . S F v . e = # μ ( H ) Conjecture (Stark 1971-80). There exists such that for every place u ∈ H * | u | w = 1 and for every character of , w ∤ v χ G S ( χ ,0) = − 1 e ∑ . L ′ χ ( σ )log | u | σ − 1 w σ ∈ G Furthermore, is an abelian extension of . H ( u 1/ e ) F 12 Technical remark: For this formulation, must assume at least 3 archimedean or ramified places of . F
INSIDE THE ABSOLUTE VALUE Stark’s formula can be manipulated to calculate under | u | each embedding . H ↪ C Can one refine this and propose a formula for itself? u The presence of the absolute value represents a gap between Stark’s Conjecture and Hilbert’s 12th problem—if we had an analytic formula for , this would give a way of constructing u canonical nontrivial elements of . H There are interesting conjectures in this direction by Ren-Sczech and Charollois-Darmon. 13
THE BRUMER-STARK CONJECTURE Fix primes , above . 𝔮 , 𝔯 ⊂ 𝒫 F 𝔔 ⊂ 𝒫 H 𝔮 = {infinite places, ramified places}. S Conjecture (Tate-Brumer-Stark). There exists such that under each u ∈ 𝒫 H [1/ 𝔮 ]* | u | = 1 embedding , H ↪ C L S ( χ ,0)(1 − χ ( σ 𝔯 ) N 𝔯 ) = ∑ χ − 1 ( σ ) ord 𝔔 ( σ ( u )) σ ∈ G for all characters of , and . u ≡ 1 (mod 𝔯𝒫 H ) χ G 14
Ludwig Stickelberger Armand Brumer Harold Stark John Tate 15
RESULTS Theorem (D-Kakde). There exists u ∈ 𝒫 H [1/ 𝔮 ]* ⊗ Z [1/2] satisfying the conditions of the Brumer-Stark conjecture. There is a “higher rank” version of the Brumer-Stark conjecture due to Karl Rubin. We obtain this result as well, after tensoring with Z [1/2] . 16
GROUP RINGS AND STICKELBERGER ELEMENTS Theorem. (Deligne-Ribet, Cassou-Noguès) There is a unique such that Θ ∈ Z [ G ] χ ( Θ ) = L S ( χ − 1 ,0)(1 − χ − 1 ( σ 𝔯 ) N 𝔯 ) for all characters of . χ G 17
CLASS GROUP Define Cl 𝔯 ( H ) = I ( H )/ ⟨ ( u ) : u ≡ 1 (mod 𝔯𝒫 H ) ⟩ . This is a -module. G Brumer-Stark states: annihilates Cl 𝔯 ( H ) . Θ For this, it su ffi ces to prove Θ ∈ Ann Z p [ G ] ( Cl 𝔯 ( H ) ⊗ Z p ) for all primes . p 18
STRONG BRUMER-STARK Theorem. For odd primes , we have p Θ ∈ Fitt Z p [ G ] ( Cl 𝔯 ( H ) ∨ , − ) Fitt Z p [ G ] ( Cl 𝔯 ( H ) ∨ , − ) ⊂ Ann Z p [ G ] ( Cl 𝔯 ( H ) − ) . 19
REFINEMENTS: CONJECTURES OF KURIHARA AND BURNS Theorem. For odd primes , we have p Fitt Z p [ G ] ( Cl 𝔯 ( H ) ∨ , − ) = Θ S ∞ ∏ ( N I v , 1 − σ − 1 v e v ) v ∈ S ram Theorem. For odd primes , we have p Fitt Z p [ G ] ( Sel 𝔯 S ( H ) − p ) = ( Θ S ) 20
(DIAGRAM H/T BARRY MAZUR) RIBET’S METHOD ? L-functions Class Groups Galois Cohomology Eisenstein Series Classes Galois Cusp Forms Representations
GROUP RING VALUED MODULAR FORMS Hilbert modular forms over of weight with M k ( G ) = F k Fourier coe ffi cients in such that for every Z p [ G ] character of , applying yields a form of nebentype . χ G χ χ Example: Eisenstein Series. 2 d Θ + ∑ E 1 ( G ) = 1 ∑ σ 𝔟 q 𝔫 𝔫⊂𝒫 𝔟⊃𝔫 ,( 𝔟 , S )=1 This must be modified in level 1. 22
GROUP RING CUSP FORM f = E 1 ( G ) V k − Θ 2 d H k +1 ( G ) is cuspidal at infinity, where and have constant term 1. H k +1 ( G ) V k Choose , where away from trivial zeroes. V k ≡ 1 (mod p N ) Θ ∣ p N . f ≡ E 1 ( G ) (mod Θ ) The existence of and are non-trivial theorems of V k H k +1 ( G ) Silliman, generalizing results of Hida and Chai. This can be modified to yield a cusp form satisfying . f ≡ E f 23
GALOIS REPRESENTATION We hereafter assume that is an eigenform. f The Galois representation associated to can be chosen as: f ρ f ( σ ) = ( d ( σ ) ) ∈ GL 2 ( Q p [ G ]) a ( σ ) b ( σ ) c ( σ ) where , . a ( σ ) ≡ 1 (mod Θ ) d ( σ ) ≡ [ σ ] (mod Θ ) This is because and f ≡ E 1 ( G ) (mod Θ ) a ℓ ( E 1 ( G )) = 1 + [ σ ℓ ] . Let B = Z p [ G ] ⟨ b ( σ ): σ ∈ G F ⟩ 24
COHOMOLOGY CLASS Then implies b ( στ ) = a ( σ ) b ( τ ) + b ( σ ) d ( τ ) , hence b ( στ ) ≡ b ( τ ) + [ τ ] b ( σ ) (mod Θ ) κ ( σ ) = [ σ ] − 1 b ( σ ) ∈ H 1 ( G F , B / Θ B ) . The class is unramified outside the level and since is. κ ρ f p Problem : In general, is not unramified at . κ p To deal with this in the proof of IMC, Wiles invented “ horizontal Iwasawa theory ,” which led to the Taylor-Wiles method . Issue: In our context, this method meets with obstacles that appear insurmountable. 25
(CASE H/F UNRAMIFIED) SPLITTING FIELD Pretend that is unramified at The splitting field of is an κ p . κ extension of whose Galois group is a quotient of Cl 𝔯 ( H ) : H Cl 𝔯 ( H ) − ↠ B / Θ B Hence Fitt ( Cl 𝔯 ( H ) − ) ⊂ Fitt ( B / Θ B ) ⊂ ( Θ ) since is a faithful -module. B Z p [ G ] An analytic argument shows that this is an . ⊂ =
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