Dirichlet character twisted nzhang28@illinois.edu University of - PowerPoint PPT Presentation

Dirichlet character twisted nzhang28@illinois.edu University of Illinois at Urbana-Champaign International Workshop on Algebraic Topology 2019 August 19, 2019 Eisenstein series and J -spectra Ningchuan Zhang 张凝川 . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . .. . . . . .

. . . . . . . . . . . . . . . . . . Background Relations with twisted Eisenstein series . . . . . . . . . . . . . . . . . . . . . . Background Twisted J -spectra

. Background . . . . . . . . . . Relations with twisted Eisenstein series . Bernoulli numbers Defjnition Bernoulli numbers are defjned to be the coeffjcients in the Taylor expansion: shows up in number theory: shows up in algebraic topology: is equal to the numerator of . Question . . . . . . . . . . . . . . . Is this a coincidence? . . . . . . . . . . . . . Twisted J -spectra

. . . . . . . . . . . . Background . Relations with twisted Eisenstein series Bernoulli numbers Defjnition Taylor expansion: shows up in number theory: shows up in algebraic topology: is equal to the numerator of . Question . . . . . . . . . . . . . . . . . . . . . . . Is this a coincidence? . . . . Twisted J -spectra Bernoulli numbers B n are defjned to be the coeffjcients in the te t t n ∞ ∑ B n n ! . e t − 1 = n = 0

. . . . . . . . . . . . . . Background Relations with twisted Eisenstein series Bernoulli numbers Defjnition Taylor expansion: shows up in algebraic topology: is equal to the numerator of . Question . . . . . . . . . . . . . . Is this a coincidence? . . . . . . . . . . . . Twisted J -spectra Bernoulli numbers B n are defjned to be the coeffjcients in the te t t n ∞ ∑ B n n ! . e t − 1 = n = 0 B n shows up in number theory: E 2 k ( q ) = 1 − 4 k σ 2 k − 1 ( n ) q n . ∞ ∑ B 2 k n = 1

. . . . . . . . . . . . . . . . Background Relations with twisted Eisenstein series Bernoulli numbers Defjnition Taylor expansion: Question . . . . . . . . . . . . . Is this a coincidence? . . . . . . . . . . . Twisted J -spectra Bernoulli numbers B n are defjned to be the coeffjcients in the te t t n ∞ ∑ B n n ! . e t − 1 = n = 0 B n shows up in number theory: E 2 k ( q ) = 1 − 4 k σ 2 k − 1 ( n ) q n . ∞ ∑ B 2 k n = 1 B n shows up in algebraic topology: ∣ π 4 k − 1 ( J )∣ is equal to the numerator of 4 k / B 2 k .

. . . . . . . . . . . . . . . . Background Relations with twisted Eisenstein series Bernoulli numbers Defjnition Taylor expansion: Question . . . . . . . . . . . . . Is this a coincidence? . . . . . . . . . . . Twisted J -spectra Bernoulli numbers B n are defjned to be the coeffjcients in the te t t n ∞ ∑ B n n ! . e t − 1 = n = 0 B n shows up in number theory: E 2 k ( q ) = 1 − 4 k σ 2 k − 1 ( n ) q n . ∞ ∑ B 2 k n = 1 B n shows up in algebraic topology: ∣ π 4 k − 1 ( J )∣ is equal to the numerator of 4 k / B 2 k .

1 Katz used a Riemann-Hilbert type correspondence to prove 3 Chromatic resolution shows 4 There is a spectral sequence 5 The image of . Background . . . . . . . . . Answer Relations with twisted Eisenstein series Sketch of the answer . This is not a coincidence. acts trivially on 2 measures congruences of a -representation . . completed at each prime is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Twisted J -spectra

1 Katz used a Riemann-Hilbert type correspondence to prove 3 Chromatic resolution shows 4 There is a spectral sequence 5 The image of . Background . . . . . . . . . Answer Relations with twisted Eisenstein series Sketch of the answer . This is not a coincidence. acts trivially on 2 measures congruences of a -representation . . completed at each prime is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Twisted J -spectra

3 Chromatic resolution shows 4 There is a spectral sequence 5 The image of . . . . . . . . . . . Sketch of the answer Background Relations with twisted Eisenstein series . Answer This is not a coincidence. 2 measures congruences of a -representation . . completed at each prime is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Twisted J -spectra 1 Katz used a Riemann-Hilbert type correspondence to prove p acts trivially on ( Z p ) ⊗ k mod p m ⇐ mod p m . ⇒ Z × E k ≡ 1

3 Chromatic resolution shows 4 There is a spectral sequence 5 The image of . . . . . . . . . . . . . . Background Relations with twisted Eisenstein series Sketch of the answer Answer This is not a coincidence. . completed at each prime is . . . . . . . . . . . . . . . . . . . . . . . . . . . Twisted J -spectra 1 Katz used a Riemann-Hilbert type correspondence to prove p acts trivially on ( Z p ) ⊗ k mod p m ⇐ mod p m . ⇒ Z × E k ≡ 1 2 ( M / p ∞ ) Z × p measures congruences of a Z × p -representation M .

4 There is a spectral sequence 5 The image of . . . . . . . . . . . . . . . Background Relations with twisted Eisenstein series Sketch of the answer Answer This is not a coincidence. completed at each prime is . . . . . . . . . . . . . . . . . . . . . . . . . . Twisted J -spectra 1 Katz used a Riemann-Hilbert type correspondence to prove p acts trivially on ( Z p ) ⊗ k mod p m ⇐ mod p m . ⇒ Z × E k ≡ 1 2 ( M / p ∞ ) Z × p measures congruences of a Z × p -representation M . 3 Chromatic resolution shows ( M / p ∞ ) Z × c ( Z × p ; M ) . p ≃ H 1

5 The image of . . . . . . . . . . . . . . . Background Relations with twisted Eisenstein series Sketch of the answer Answer This is not a coincidence. completed at each prime is . . . . . . . . . . . . . . . . . . . . . . . . . . Twisted J -spectra 1 Katz used a Riemann-Hilbert type correspondence to prove p acts trivially on ( Z p ) ⊗ k mod p m ⇐ mod p m . ⇒ Z × E k ≡ 1 2 ( M / p ∞ ) Z × p measures congruences of a Z × p -representation M . 3 Chromatic resolution shows ( M / p ∞ ) Z × c ( Z × p ; M ) . p ≃ H 1 4 There is a spectral sequence c ( Z × p ; π t ( K ∧ p )) � ⇒ π t − s ( S 0 K ( 1 ) ) . E s,t = H s 2

. . . . . . . . . . . . . . . . Background Relations with twisted Eisenstein series Sketch of the answer Answer This is not a coincidence. . . . . . . . . . . . . . . . . . . . . . . . . Twisted J -spectra 1 Katz used a Riemann-Hilbert type correspondence to prove p acts trivially on ( Z p ) ⊗ k mod p m ⇐ mod p m . ⇒ Z × E k ≡ 1 2 ( M / p ∞ ) Z × p measures congruences of a Z × p -representation M . 3 Chromatic resolution shows ( M / p ∞ ) Z × c ( Z × p ; M ) . p ≃ H 1 4 There is a spectral sequence c ( Z × p ; π t ( K ∧ p )) � ⇒ π t − s ( S 0 K ( 1 ) ) . E s,t = H s 2 5 The image of J completed at each prime is S 0 K ( 1 ) .

. Background . . . . . . . . . . Relations with twisted Eisenstein series . Generalized Bernoulli numbers and Eisenstein series Let be a primitive Dirichlet character. Defjnitions The generalized Bernoulli numbers associated to are defjned by: The Eisenstein series associated to is defjned by: with the . . . . . . . . . . . . . . . -expansion of its normalization given by: . . . . . . . . . . . . . Twisted J -spectra

. . . . . . . . . . . . Background . Relations with twisted Eisenstein series Generalized Bernoulli numbers and Eisenstein series Defjnitions The generalized Bernoulli numbers associated to are defjned by: The Eisenstein series associated to is defjned by: with the . . . . . . . . . . . . . . . . . . . . . . . . . . . -expansion of its normalization given by: Twisted J -spectra Let χ ∶ ( Z / N ) × → C × be a primitive Dirichlet character.

. . . . . . . . . . . . . . . Background Relations with twisted Eisenstein series Generalized Bernoulli numbers and Eisenstein series Defjnitions defjned by: The Eisenstein series associated to is defjned by: with the . . . . . . . . . . . . . . -expansion of its normalization given by: . . . . . . . . . . . Twisted J -spectra Let χ ∶ ( Z / N ) × → C × be a primitive Dirichlet character. The generalized Bernoulli numbers B n,χ associated to χ are χ ( a ) te at F χ ( t ) = N t n ∞ ∑ ∑ B n,χ n ! . e Nt − 1 = a = 1 n = 0

. . . . . . . . . . . . . . . . Background Relations with twisted Eisenstein series Generalized Bernoulli numbers and Eisenstein series Defjnitions defjned by: with the . . . . . . . . . . . . . -expansion of its normalization given by: . . . . . . . . . . . Twisted J -spectra Let χ ∶ ( Z / N ) × → C × be a primitive Dirichlet character. The generalized Bernoulli numbers B n,χ associated to χ are χ ( a ) te at F χ ( t ) = N t n ∞ ∑ ∑ B n,χ n ! . e Nt − 1 = a = 1 n = 0 The Eisenstein series associated to χ is defjned by: χ ( n ) G k ( z ; χ ) ∶ = ∑ ( mNz + n ) k , ( m,n ) ≠ ( 0 , 0 )