Modularity of the Abelian Surface of Conductor 277 David S. Yuen - - PowerPoint PPT Presentation

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Modularity of the Abelian Surface of Conductor 277 David S. Yuen - - PowerPoint PPT Presentation

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Modularity of the Abelian Surface of Conductor 277 David S. Yuen Lake Forest College joint work with Armand Brumer Fordham University Cris Poor Fordham University John Voight Dartmouth College Modular Forms and Curves of Low Genus:

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Modularity of the Abelian Surface of Conductor 277

David S. Yuen Lake Forest College joint work with Armand Brumer Fordham University Cris Poor Fordham University John Voight Dartmouth College Modular Forms and Curves of Low Genus: Computational Aspects ICERM, September 2015

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 1 / 25

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Outline

Outline of talk

  • 1. Paramodular Conjecture and evidence.
  • 2. The abelian surface of conductor 277 and the paramodular form f277.
  • 3. Computing eigenvalues by specialization.
  • 4. Making floating point calculations rigorous.
  • 5. Results.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 2 / 25

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Paramodular Conjecture and Evidence Paramodular Conjecture

All abelian surfaces A/Q are paramodular

Paramodular Conjecture (Brumer and Kramer 2009) Let N ∈ N. There is a bijection between

  • 1. isogeny classes of abelian surfaces A/Q with conductor N and

endomorphisms EndQ(A) = Z,

  • 2. lines of Hecke eigenforms f ∈ S2(K(N))new that have rational

eigenvalues and are not Gritsenko lifts from Jcusp

2,N .

In this correspondence we have L(A, s, Hasse-Weil) = L(f , s, spin).

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 3 / 25

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Paramodular Conjecture and Evidence Background evidence

Do the arithmetic and automorphic data match up?

Looks like it.

1997: Brumer makes a (short) list of N < 1000 that could possibly be the conductor of an abelian surface A/Q. Theorem (PY 2009) Let p < 600 be prime. If p ∈ {277, 349, 353, 389, 461, 523, 587} then S2(K(p)) consists entirely of Gritsenko lifts. This list of primes {277, . . . , 587} exactly matches Brumer’s “Yes there is an abelian surface” list for prime levels. This is a lot of evidence for the Paramodular Conjecture because prime levels p < 600 that don’t have abelian surfaces over Q also don’t have paramodular cusp forms beyond the Gritsenko lifts.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 4 / 25

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Paramodular Conjecture and Evidence Background evidence

Proof. We can inject the weight two space into weight four spaces: 1) For g1, g2 ∈ Grit

  • Jcusp

2,p

  • ⊆ S2 (K(p)), we have the injection:

S2(K(p)) ֒ → {(H1, H2) ∈ S4(K(p)) × S4(K(p)) : g2H1 = g1H2} f → (g1f , g2f ) 2) The dimensions of S4(K(p)) are known by Ibukiyama; we still have to span S4(K(p)) by computing products of Gritsenko lifts, traces of theta series and by smearing with Hecke operators. 3) Millions of Fourier coefficients mod 109 later, dim S2(K(p)) ≤ dim{(H1, H2) ∈ S4(K(p)) × S4(K(p)) : g2H1 = g1H2} 4) When the dimension might be bigger, this method also gives a candidate nonlift as a quotient of a known weight 4 form divided by a known weight 2 form.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 5 / 25

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Paramodular Conjecture and Evidence Equality of L-series complete examples

Equality of L-series modularity examples

The lift to a paramodular Hecke eigenform of a certain Hilbert modular form over a real quadratic field by Johnson-Leung and Roberts (2012) is modular with respect to a certain abelian surface. Some work of Demb´ el´ e and Kumar is related to this. An example has conductor 1932.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 6 / 25

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Paramodular Conjecture and Evidence Equality of L-series complete examples

Equality of L-series modularity examples

The lift to a paramodular Hecke eigenform of a certain Hilbert modular form over a real quadratic field by Johnson-Leung and Roberts (2012) is modular with respect to a certain abelian surface. Some work of Demb´ el´ e and Kumar is related to this. An example has conductor 1932. For a similar but different example, constructing a lift from Bianchi modular forms, Berger, Demb´ el´ e, Pacetti, Sengun used an imaginary quadratic number field. An example is conductor N = 2232.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 6 / 25

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Abelian Surface and Modular Form of Conductor 277 The form and the surface

Conductor 277

The form and the surface

(Theorem PY 2009) dim S2(K(277)) = 11 but dim Jcusp

2,277 = 10. There

is a Hecke eigenform f277 ∈ S2(K(277)) that is not a Gritsenko lift. A277 is the Jacobian of the hyperelliptic curve y2 + y = x5 + 5x4 + 8x3 + 6x2 + 2x

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 7 / 25

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Abelian Surface and Modular Form of Conductor 277 The form and the surface

Conductor 277

The form and the surface

(Theorem PY 2009) dim S2(K(277)) = 11 but dim Jcusp

2,277 = 10. There

is a Hecke eigenform f277 ∈ S2(K(277)) that is not a Gritsenko lift. A277 is the Jacobian of the hyperelliptic curve y2 + y = x5 + 5x4 + 8x3 + 6x2 + 2x Magma will compute lots of Euler factors for L(A277, s, H-W)

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 7 / 25

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Abelian Surface and Modular Form of Conductor 277 The form and the surface

Conductor 277

The form and the surface

(Theorem PY 2009) dim S2(K(277)) = 11 but dim Jcusp

2,277 = 10. There

is a Hecke eigenform f277 ∈ S2(K(277)) that is not a Gritsenko lift. A277 is the Jacobian of the hyperelliptic curve y2 + y = x5 + 5x4 + 8x3 + 6x2 + 2x Magma will compute lots of Euler factors for L(A277, s, H-W) By contrast, it takes much more work to compute Euler factors for L(f277, s, spin), and one of the main goals of this talk is to present

  • ne method for computing high eigenvalues of f277.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 7 / 25

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Abelian Surface and Modular Form of Conductor 277 The form and the surface

Conductor 277

The form and the surface

(Theorem PY 2009) dim S2(K(277)) = 11 but dim Jcusp

2,277 = 10. There

is a Hecke eigenform f277 ∈ S2(K(277)) that is not a Gritsenko lift. A277 is the Jacobian of the hyperelliptic curve y2 + y = x5 + 5x4 + 8x3 + 6x2 + 2x Magma will compute lots of Euler factors for L(A277, s, H-W) By contrast, it takes much more work to compute Euler factors for L(f277, s, spin), and one of the main goals of this talk is to present

  • ne method for computing high eigenvalues of f277.

In 2009, we computed the 2, 3 and 5 Euler factors of L(f277, s, spin) and they agree with those of L(A277, s, H-W).

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 7 / 25

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Abelian Surface and Modular Form of Conductor 277 Proving existence of f277

How can we prove a weight two nonlift cusp form exists?

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 8 / 25

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Abelian Surface and Modular Form of Conductor 277 Proving existence of f277

How can we prove a weight two nonlift cusp form exists?

Proof (2009). 1) We have a candidate f = H1/g1 ∈ Mmero

2

(K(p)). 2) Find a weight four cusp form F ∈ S4(K(p)) and prove F g2

1 = H2 1 in S8(K(p)).

Since F = H1 g1 2 is holomorphic, so is f = H1 g1 .

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 8 / 25

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Abelian Surface and Modular Form of Conductor 277 Proving existence of f277

How can we prove a weight two nonlift cusp form exists?

Proof (2009). 1) We have a candidate f = H1/g1 ∈ Mmero

2

(K(p)). 2) Find a weight four cusp form F ∈ S4(K(p)) and prove F g2

1 = H2 1 in S8(K(p)).

Since F = H1 g1 2 is holomorphic, so is f = H1 g1 . Proof (new). We make a Borcherds product in S2(K(p)) and show the Borcherds product is not a Gritsenko lift.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 8 / 25

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Abelian Surface and Modular Form of Conductor 277 f277

Formula for f277

One presentation of f277 is f277 = ( − 14G 2

1 − 20G8G2 + 11G9G2 + 6G 2 2 − 30G7G10 + 15G9G10

+ 15G10G1 − 30G10G2 − 30G10G3 + 5G4G5 + 6G4G6 + 17G4G7 − 3G4G8 − 5G4G9 − 5G5G6 + 20G5G7 − 5G5G8 − 10G5G9 − 3G 2

6

+ 13G6G7 + 3G6G8 − 10G6G9 − 22G 2

7 + G7G8 + 15G7G9 + 6G 2 8

− 4G8G9 − 2G 2

9 + 20G1G2 − 28G3G2 + 23G4G2 + 7G6G2

− 31G7G2 + 15G5G2 + 45G1G3 − 10G1G5 − 2G1G4 − 13G1G6 − 7G1G8 + 39G1G7 − 16G1G9 − 34G 2

3 + 8G3G4 + 20G3G5

+ 22G3G6 + 10G3G8 + 21G3G9 − 56G3G7 − 3G 2

4 )/

( − G4 + G6 + 2G7 + G8 − G9 + 2G3 − 3G2 − G1). for some ten Gritsenko lifts G1, . . . , G10.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 9 / 25

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Abelian Surface and Modular Form of Conductor 277 f277

Formula for f277

One presentation of f277 is f277 = ( − 14G 2

1 − 20G8G2 + 11G9G2 + 6G 2 2 − 30G7G10 + 15G9G10

+ 15G10G1 − 30G10G2 − 30G10G3 + 5G4G5 + 6G4G6 + 17G4G7 − 3G4G8 − 5G4G9 − 5G5G6 + 20G5G7 − 5G5G8 − 10G5G9 − 3G 2

6

+ 13G6G7 + 3G6G8 − 10G6G9 − 22G 2

7 + G7G8 + 15G7G9 + 6G 2 8

− 4G8G9 − 2G 2

9 + 20G1G2 − 28G3G2 + 23G4G2 + 7G6G2

− 31G7G2 + 15G5G2 + 45G1G3 − 10G1G5 − 2G1G4 − 13G1G6 − 7G1G8 + 39G1G7 − 16G1G9 − 34G 2

3 + 8G3G4 + 20G3G5

+ 22G3G6 + 10G3G8 + 21G3G9 − 56G3G7 − 3G 2

4 )/

( − G4 + G6 + 2G7 + G8 − G9 + 2G3 − 3G2 − G1). for some ten Gritsenko lifts G1, . . . , G10. This can be used to prove f277 has integer coefficients.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 9 / 25

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Abelian Surface and Modular Form of Conductor 277 f277

And if you must know...

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 10 / 25

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Abelian Surface and Modular Form of Conductor 277 f277

And if you must know... Gi = GritLift(φi), where φi is a theta block given by... φ1 = TB2(2, 4, 4, 4, 5, 6, 8, 9, 10, 14) φ2 = TB2(2, 3, 4, 5, 5, 7, 7, 9, 10, 14) φ3 = TB2(2, 3, 4, 4, 5, 7, 8, 9, 11, 13) φ4 = TB2(2, 3, 3, 5, 6, 6, 8, 9, 11, 13) φ5 = TB2(2, 3, 3, 5, 5, 8, 8, 8, 11, 13) φ6 = TB2(2, 3, 3, 5, 5, 7, 8, 10, 10, 13) φ7 = TB2(2, 3, 3, 4, 5, 6, 7, 9, 10, 15) φ8 = TB2(2, 2, 4, 5, 6, 7, 7, 9, 11, 13) φ9 = TB2(2, 2, 4, 4, 6, 7, 8, 10, 11, 12) φ10 = TB2(2, 2, 3, 5, 6, 7, 9, 9, 11, 12) where a theta block is a product of theta functions and eta functions: TB2(d1, . . . , d10) = η−6 10

i=1 ϑ(τ, diz).

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 10 / 25

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Computing eigenvalues Modularity

What eigenvalues are needed to prove modularity?

  • A. Brumer tells us that to prove A277 is modular, we need to prove that

the eigenvalues λp for it and f277 are equal for p in the following set: B = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 59, 97}

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 11 / 25

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Computing eigenvalues Some Methods

Methods for computing eigenvalues.

The smallest determinant nonzero Fourier coefficient of f277 is a(t0, f277) where t0 =

  • 49

233/2 233/2 277

  • .

Note det(t0) = 3/4. Some methods for computing T(p) are:

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 12 / 25

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Computing eigenvalues Some Methods

Methods for computing eigenvalues.

The smallest determinant nonzero Fourier coefficient of f277 is a(t0, f277) where t0 =

  • 49

233/2 233/2 277

  • .

Note det(t0) = 3/4. Some methods for computing T(p) are: Expand f277 using multiplication and division of the Gritsenko lift (3-variable power series) expanded out to coefficient matrices of determinant 3p2/4.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 12 / 25

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Computing eigenvalues Some Methods

Methods for computing eigenvalues.

The smallest determinant nonzero Fourier coefficient of f277 is a(t0, f277) where t0 =

  • 49

233/2 233/2 277

  • .

Note det(t0) = 3/4. Some methods for computing T(p) are: Expand f277 using multiplication and division of the Gritsenko lift (3-variable power series) expanded out to coefficient matrices of determinant 3p2/4. Expand f277 to p Fourier-Jacobi coefficients using“Jacobi Restriction”.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 12 / 25

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Computing eigenvalues Some Methods

Methods for computing eigenvalues.

The smallest determinant nonzero Fourier coefficient of f277 is a(t0, f277) where t0 =

  • 49

233/2 233/2 277

  • .

Note det(t0) = 3/4. Some methods for computing T(p) are: Expand f277 using multiplication and division of the Gritsenko lift (3-variable power series) expanded out to coefficient matrices of determinant 3p2/4. Expand f277 to p Fourier-Jacobi coefficients using“Jacobi Restriction”. Specialize by plugging something into the argument for f277 and f277|T(p).

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 12 / 25

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Computing eigenvalues Some Methods

Methods for computing eigenvalues.

The smallest determinant nonzero Fourier coefficient of f277 is a(t0, f277) where t0 =

  • 49

233/2 233/2 277

  • .

Note det(t0) = 3/4. Some methods for computing T(p) are: Expand f277 using multiplication and division of the Gritsenko lift (3-variable power series) expanded out to coefficient matrices of determinant 3p2/4. Expand f277 to p Fourier-Jacobi coefficients using“Jacobi Restriction”. Specialize by plugging something into the argument for f277 and f277|T(p).

Plug in a specific numerical point and evaluate numerically. Get numbers.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 12 / 25

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Computing eigenvalues Some Methods

Methods for computing eigenvalues.

The smallest determinant nonzero Fourier coefficient of f277 is a(t0, f277) where t0 =

  • 49

233/2 233/2 277

  • .

Note det(t0) = 3/4. Some methods for computing T(p) are: Expand f277 using multiplication and division of the Gritsenko lift (3-variable power series) expanded out to coefficient matrices of determinant 3p2/4. Expand f277 to p Fourier-Jacobi coefficients using“Jacobi Restriction”. Specialize by plugging something into the argument for f277 and f277|T(p).

Plug in a specific numerical point and evaluate numerically. Get numbers. Specialize to a modular curve. Get elliptic modular forms.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 12 / 25

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Computing eigenvalues Some Methods

Methods for computing eigenvalues.

The smallest determinant nonzero Fourier coefficient of f277 is a(t0, f277) where t0 =

  • 49

233/2 233/2 277

  • .

Note det(t0) = 3/4. Some methods for computing T(p) are: Expand f277 using multiplication and division of the Gritsenko lift (3-variable power series) expanded out to coefficient matrices of determinant 3p2/4. Expand f277 to p Fourier-Jacobi coefficients using“Jacobi Restriction”. Specialize by plugging something into the argument for f277 and f277|T(p).

Plug in a specific numerical point and evaluate numerically. Get numbers. Specialize to a modular curve. Get elliptic modular forms. Specialize to a Humbert surface. Get Hilbert modular forms.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 12 / 25

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Computing eigenvalues Specialization to modular curves

Specialization to modular curves

Fix any positive definite 2 × 2 symmetric matrix s0. Let f (Ω) =

  • t

a(t; f ) exp(2πit, Ω). Then f (s0τ) =

  • n

 

  • t:s0,t=n

a(t; f )   qn

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 13 / 25

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Computing eigenvalues Specialization to modular curves

Specialization to modular curves

Fix any positive definite 2 × 2 symmetric matrix s0. Let f (Ω) =

  • t

a(t; f ) exp(2πit, Ω). Then f (s0τ) =

  • n

 

  • t:s0,t=n

a(t; f )   qn For example, for s0 = 554 233 233 98

  • ,

f277(s0τ) = −3q3 + 6q6 + 6q9 + O(q10) (f277|T(2))(s0τ) = 6q3 − 12q6 − 12q9 + O(q10) and hence λ2(f277) = −2.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 13 / 25

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Computing eigenvalues Specialization to modular curves

Specialization to modular curves

Fix any positive definite 2 × 2 symmetric matrix s0. Let f (Ω) =

  • t

a(t; f ) exp(2πit, Ω). Then f (s0τ) =

  • n

 

  • t:s0,t=n

a(t; f )   qn For example, for s0 = 554 233 233 98

  • ,

f277(s0τ) = −3q3 + 6q6 + 6q9 + O(q10) (f277|T(2))(s0τ) = 6q3 − 12q6 − 12q9 + O(q10) and hence λ2(f277) = −2. How is the second of above two series computed?

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 13 / 25

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Computing eigenvalues Coset representatives for T(p)

Coset representatives for T(p)

When p | N, we can use the following coset representatives in expressing T(p) as a sum of cosets. T(p) = K(N) 1 0 0 0

0 1 0 0 0 0 p 0 0 0 0 p

  • K(N) =

p3+p2+p+1

  • ℓ=1

K(N) Aℓ Bℓ Dℓ

  • = K(N)

p 0 0 0

0 p 0 0 0 0 1 0 0 0 0 1

  • +
  • a mod p

K(N) 1 0 a 0

0 p 0 0 0 0 p 0 0 0 0 1

  • +
  • a,b mod p

K(N) p 0 0 0

a 1 0 b 0 0 1 −a 0 0 0 p

  • +
  • a,b,c mod p

K(N)

  • 1 0 a b

0 1 b c 0 0 p 0 0 0 0 p

  • David Yuen et al

Modularity of Abelian Surface A277 ICERM September 2015 14 / 25

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Computing eigenvalues Coset representatives for T(p)

Coset representatives for T(p)

When p | N, we can use the following coset representatives in expressing T(p) as a sum of cosets. T(p) = K(N) 1 0 0 0

0 1 0 0 0 0 p 0 0 0 0 p

  • K(N) =

p3+p2+p+1

  • ℓ=1

K(N) Aℓ Bℓ Dℓ

  • = K(N)

p 0 0 0

0 p 0 0 0 0 1 0 0 0 0 1

  • +
  • a mod p

K(N) 1 0 a 0

0 p 0 0 0 0 p 0 0 0 0 1

  • +
  • a,b mod p

K(N) p 0 0 0

a 1 0 b 0 0 1 −a 0 0 0 p

  • +
  • a,b,c mod p

K(N)

  • 1 0 a b

0 1 b c 0 0 p 0 0 0 0 p

  • So

f |T(p) =

p3+p2+p+1

  • ℓ=1

f | Aℓ Bℓ Dℓ

  • .

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 14 / 25

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Computing eigenvalues Specializing after slashing by upper triangular

Specializing after slashing by upper triangular

  • f |

A B

0 D

  • (s0τ) =(det D)−k det(AD)2k−3f (As0D−1τ + BD−1)

=(det D)−k det(AD)2k−3 ·

  • n∈Q

 

  • t:As0D−1,t=n

a(t; f ) exp(2πiBD−1, t)   qn.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 15 / 25

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Computing eigenvalues Specializing after slashing by upper triangular

Specializing after slashing by upper triangular

  • f |

A B

0 D

  • (s0τ) =(det D)−k det(AD)2k−3f (As0D−1τ + BD−1)

=(det D)−k det(AD)2k−3 ·

  • n∈Q

 

  • t:As0D−1,t=n

a(t; f ) exp(2πiBD−1, t)   qn. Note that exp(2πiBD−1, t) could yield a pth root of unity.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 15 / 25

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Computing eigenvalues Specialization of f277|T(p)

Specialization of f277|T(p)

Using the formula f277 =  

i,j

αijGiGj   /  

j

βjGj   ,

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 16 / 25

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Computing eigenvalues Specialization of f277|T(p)

Specialization of f277|T(p)

Using the formula f277 =  

i,j

αijGiGj   /  

j

βjGj   , we have (f277|T(p))(s0τ) =

p3+p2+p+1

  • ℓ=1

   

i,j

αijgiℓgjℓ   /  

j

βjgjℓ     , where giℓ =

  • Gi|
  • Aℓ Bℓ

0 Dℓ

  • (s0τ) is a power series in one variable.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 16 / 25

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Computing eigenvalues Specialization of f277|T(p)

Specialization of f277|T(p)

Using the formula f277 =  

i,j

αijGiGj   /  

j

βjGj   , we have (f277|T(p))(s0τ) =

p3+p2+p+1

  • ℓ=1

   

i,j

αijgiℓgjℓ   /  

j

βjgjℓ     , where giℓ =

  • Gi|
  • Aℓ Bℓ

0 Dℓ

  • (s0τ) is a power series in one variable.

Of course, for computations, we compute with truncated power series.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 16 / 25

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Making Floating Point Calculations Rigorous

The issue.

The individual slashes in the sum for T(p) might have pth roots of unity. Solutions:

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 17 / 25

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Making Floating Point Calculations Rigorous

The issue.

The individual slashes in the sum for T(p) might have pth roots of unity. Solutions: Keep track of the primitive pth root of unity ζp symbolically and perform operations in the field Q[ζp], storing elements as polymomials with rational coefficients.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 17 / 25

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SLIDE 39

Making Floating Point Calculations Rigorous

The issue.

The individual slashes in the sum for T(p) might have pth roots of unity. Solutions: Keep track of the primitive pth root of unity ζp symbolically and perform operations in the field Q[ζp], storing elements as polymomials with rational coefficients. Use floating point numbers with a specified precision and keep track

  • f their radii of error.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 17 / 25

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SLIDE 40

Making Floating Point Calculations Rigorous

The issue.

The individual slashes in the sum for T(p) might have pth roots of unity. Solutions: Keep track of the primitive pth root of unity ζp symbolically and perform operations in the field Q[ζp], storing elements as polymomials with rational coefficients. Use floating point numbers with a specified precision and keep track

  • f their radii of error.

We choose the floating point number method.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 17 / 25

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Making Floating Point Calculations Rigorous Floating point numbers

IEEE standards for floating point calculations

We use a programming package (such as GMP, MPFR, MPC) that follows IEEE standards for floating point calculations.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 18 / 25

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SLIDE 42

Making Floating Point Calculations Rigorous Floating point numbers

IEEE standards for floating point calculations

We use a programming package (such as GMP, MPFR, MPC) that follows IEEE standards for floating point calculations. Choose a precision B. Denote the set of numbers representable by floating numbers of precision B by CB. So CB ⊂ ±({0, . . . , 2B − 1} + i{0, . . . , 2B − 1}) ∗ 2Z.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 18 / 25

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SLIDE 43

Making Floating Point Calculations Rigorous Floating point numbers

IEEE standards for floating point calculations

We use a programming package (such as GMP, MPFR, MPC) that follows IEEE standards for floating point calculations. Choose a precision B. Denote the set of numbers representable by floating numbers of precision B by CB. So CB ⊂ ±({0, . . . , 2B − 1} + i{0, . . . , 2B − 1}) ∗ 2Z. Every standard operation (arithmetic, trigonometric, etc.) has user-specified rounding modes for the real and imaginary parts.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 18 / 25

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SLIDE 44

Making Floating Point Calculations Rigorous Floating point numbers

IEEE standards for floating point calculations

We use a programming package (such as GMP, MPFR, MPC) that follows IEEE standards for floating point calculations. Choose a precision B. Denote the set of numbers representable by floating numbers of precision B by CB. So CB ⊂ ±({0, . . . , 2B − 1} + i{0, . . . , 2B − 1}) ∗ 2Z. Every standard operation (arithmetic, trigonometric, etc.) has user-specified rounding modes for the real and imaginary parts.

RNDN: Round to nearest

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 18 / 25

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SLIDE 45

Making Floating Point Calculations Rigorous Floating point numbers

IEEE standards for floating point calculations

We use a programming package (such as GMP, MPFR, MPC) that follows IEEE standards for floating point calculations. Choose a precision B. Denote the set of numbers representable by floating numbers of precision B by CB. So CB ⊂ ±({0, . . . , 2B − 1} + i{0, . . . , 2B − 1}) ∗ 2Z. Every standard operation (arithmetic, trigonometric, etc.) has user-specified rounding modes for the real and imaginary parts.

RNDN: Round to nearest RNDU: Round up (towards +∞).

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 18 / 25

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SLIDE 46

Making Floating Point Calculations Rigorous Floating point numbers

IEEE standards for floating point calculations

We use a programming package (such as GMP, MPFR, MPC) that follows IEEE standards for floating point calculations. Choose a precision B. Denote the set of numbers representable by floating numbers of precision B by CB. So CB ⊂ ±({0, . . . , 2B − 1} + i{0, . . . , 2B − 1}) ∗ 2Z. Every standard operation (arithmetic, trigonometric, etc.) has user-specified rounding modes for the real and imaginary parts.

RNDN: Round to nearest RNDU: Round up (towards +∞). RNDD: Round down (towards −∞).

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 18 / 25

slide-47
SLIDE 47

Making Floating Point Calculations Rigorous Floating point numbers

IEEE standards for floating point calculations

We use a programming package (such as GMP, MPFR, MPC) that follows IEEE standards for floating point calculations. Choose a precision B. Denote the set of numbers representable by floating numbers of precision B by CB. So CB ⊂ ±({0, . . . , 2B − 1} + i{0, . . . , 2B − 1}) ∗ 2Z. Every standard operation (arithmetic, trigonometric, etc.) has user-specified rounding modes for the real and imaginary parts.

RNDN: Round to nearest RNDU: Round up (towards +∞). RNDD: Round down (towards −∞).

Example usage: add(z, w, RNDNN). The result of an operation should be as if the calculation is done exactly (with infinite precision) and then rounded as requested to an element of CB.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 18 / 25

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SLIDE 48

Making Floating Point Calculations Rigorous A new class

Write a new class in an object oriented language

Call our class MyComplex. For speed, the precision B should be fixed at compile time. Also, set M = ⌊B/4⌋. (Other choices could work.)

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 19 / 25

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SLIDE 49

Making Floating Point Calculations Rigorous A new class

Write a new class in an object oriented language

Call our class MyComplex. For speed, the precision B should be fixed at compile time. Also, set M = ⌊B/4⌋. (Other choices could work.) An object of this class has at least two fields:

z, a complex floating point number of precision B. r, a real floating point number of precision B.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 19 / 25

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SLIDE 50

Making Floating Point Calculations Rigorous A new class

Write a new class in an object oriented language

Call our class MyComplex. For speed, the precision B should be fixed at compile time. Also, set M = ⌊B/4⌋. (Other choices could work.) An object of this class has at least two fields:

z, a complex floating point number of precision B. r, a real floating point number of precision B.

The idea is that a MyComplex object represents every value that is within r of z, namely D(z,r) = {w ∈ C : |w − z| ≤ r}.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 19 / 25

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SLIDE 51

Making Floating Point Calculations Rigorous A new class

Addition in this class

We want the result of MyComplex add((z1, r1), (z2, r2)) to be (z3, r3) such that D(z1,r1) + D(z2,r2) ⊆ D(z3,r3).

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 20 / 25

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SLIDE 52

Making Floating Point Calculations Rigorous A new class

Addition in this class

We want the result of MyComplex add((z1, r1), (z2, r2)) to be (z3, r3) such that D(z1,r1) + D(z2,r2) ⊆ D(z3,r3). We achieve this by:

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 20 / 25

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SLIDE 53

Making Floating Point Calculations Rigorous A new class

Addition in this class

We want the result of MyComplex add((z1, r1), (z2, r2)) to be (z3, r3) such that D(z1,r1) + D(z2,r2) ⊆ D(z3,r3). We achieve this by: z3 = add(z1, z2, RNDNN)

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 20 / 25

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SLIDE 54

Making Floating Point Calculations Rigorous A new class

Addition in this class

We want the result of MyComplex add((z1, r1), (z2, r2)) to be (z3, r3) such that D(z1,r1) + D(z2,r2) ⊆ D(z3,r3). We achieve this by: z3 = add(z1, z2, RNDNN) r3 = add(add(r1, r2, RNDU), const adderr) where const adderr = 21+M−B is a constant. Abort with an error if any input |zi| > 2M. Be sure to check this condition via norm(zi, RNDU) ≤ 2M.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 20 / 25

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SLIDE 55

Making Floating Point Calculations Rigorous A new class

Addition in this class

We want the result of MyComplex add((z1, r1), (z2, r2)) to be (z3, r3) such that D(z1,r1) + D(z2,r2) ⊆ D(z3,r3). We achieve this by: z3 = add(z1, z2, RNDNN) r3 = add(add(r1, r2, RNDU), const adderr) where const adderr = 21+M−B is a constant. Abort with an error if any input |zi| > 2M. Be sure to check this condition via norm(zi, RNDU) ≤ 2M. It can be proven that the above guarantees D(z1,r1) + D(z2,r2) ⊆ D(z3,r3).

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 20 / 25

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SLIDE 56

Making Floating Point Calculations Rigorous A new class

Multiplication in this class

To perform the method MyComplex multiply((z1, r1), (z2, r2))...

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 21 / 25

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SLIDE 57

Making Floating Point Calculations Rigorous A new class

Multiplication in this class

To perform the method MyComplex multiply((z1, r1), (z2, r2))... Calculate ai = norm(zi, RNDU). Abort if either ai > 2M. Calculate z3 = mul(z1, z2, RNDNN) Calculate r3 =add(add(mul(r1, a2, RNDU), mul(r2, myadd(a1, r1, RNDU), RNDU), RNDU), const mulerr, RNDU) where const mulerr = 22M−B is a constant. It can be proven that the above guarantees D(z1,r1) · D(z2,r2) ⊆ D(z3,r3).

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 21 / 25

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SLIDE 58

Making Floating Point Calculations Rigorous A new class

Division in this class

To perform the method MyComplex reciprocal(z, r))...

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 22 / 25

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SLIDE 59

Making Floating Point Calculations Rigorous A new class

Division in this class

To perform the method MyComplex reciprocal(z, r))... Calculate b = norm(z, RNDD). Abort if b < 2−M. (Thus guaranteeing |z| > 2−M.) Abort if r ≥ b. (Thus guaranteeing r < |z| and 0 ∈ D(z, r).) Calculate z3 = div(1.0, z, RNDNN) Calculate r3 = add(div(div(r, b, RNDU), subtract(b, r, RNDD), RNDU), const diverr, RNDU) where const diverr = 2B−M is a constant. It can be proven that the above guarantees 1/D(z,r) ⊆ D(z3,r3).

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 22 / 25

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SLIDE 60

Making Floating Point Calculations Rigorous f277|T(p)

Back to calculating (f277|T(p))(s0τ)

Check that each coefficient, a MyComplex object, has only one integer in its disk.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 23 / 25

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SLIDE 61

Making Floating Point Calculations Rigorous f277|T(p)

Back to calculating (f277|T(p))(s0τ)

Check that each coefficient, a MyComplex object, has only one integer in its disk. Example using precision B = 512 (binary) digits, we calculate: (f277|T(17))(s0τ) = (12 + 9.22337 × 10−82, 1.9 × 10−69)q3 + O(q3+1/17).

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 23 / 25

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SLIDE 62

Making Floating Point Calculations Rigorous f277|T(p)

Back to calculating (f277|T(p))(s0τ)

Check that each coefficient, a MyComplex object, has only one integer in its disk. Example using precision B = 512 (binary) digits, we calculate: (f277|T(17))(s0τ) = (12 + 9.22337 × 10−82, 1.9 × 10−69)q3 + O(q3+1/17). We note that there is only one integer 12 that is inside the coefficient disk

  • f q3. Because we know the coefficients of f277 are integers, we conclude

(f277|T(17))(s0τ) = 12q3 + O(q4).

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 23 / 25

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SLIDE 63

Making Floating Point Calculations Rigorous f277|T(p)

Back to calculating (f277|T(p))(s0τ)

Check that each coefficient, a MyComplex object, has only one integer in its disk. Example using precision B = 512 (binary) digits, we calculate: (f277|T(17))(s0τ) = (12 + 9.22337 × 10−82, 1.9 × 10−69)q3 + O(q3+1/17). We note that there is only one integer 12 that is inside the coefficient disk

  • f q3. Because we know the coefficients of f277 are integers, we conclude

(f277|T(17))(s0τ) = 12q3 + O(q4). Hence λ17(f277) = −4.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 23 / 25

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SLIDE 64

Making Floating Point Calculations Rigorous f277|T(p)

Back to calculating (f277|T(p))(s0τ)

Check that each coefficient, a MyComplex object, has only one integer in its disk. Example using precision B = 512 (binary) digits, we calculate: (f277|T(17))(s0τ) = (12 + 9.22337 × 10−82, 1.9 × 10−69)q3 + O(q3+1/17). We note that there is only one integer 12 that is inside the coefficient disk

  • f q3. Because we know the coefficients of f277 are integers, we conclude

(f277|T(17))(s0τ) = 12q3 + O(q4). Hence λ17(f277) = −4. The above calculation took 140 seconds on a laptop.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 23 / 25

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SLIDE 65

Results

Results: some eigenvalues of f277

p 2 3 5 7 11 13 17 19 23 λp

  • 2
  • 1
  • 1

1

  • 2

3

  • 4
  • 1

3 p 29 31 37 41 43 47 53 59 97 λp

  • 1
  • 10

4 7 4

  • 8

14 1

  • 6

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 24 / 25

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SLIDE 66

Results

Results: some eigenvalues of f277

p 2 3 5 7 11 13 17 19 23 λp

  • 2
  • 1
  • 1

1

  • 2

3

  • 4
  • 1

3 p 29 31 37 41 43 47 53 59 97 λp

  • 1
  • 10

4 7 4

  • 8

14 1

  • 6

Note: the calculation of λ97 took 28 days on one CPU (on a Fordham HPC machine, done in July 2015).

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 24 / 25

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SLIDE 67

Results

Results: some eigenvalues of f277

p 2 3 5 7 11 13 17 19 23 λp

  • 2
  • 1
  • 1

1

  • 2

3

  • 4
  • 1

3 p 29 31 37 41 43 47 53 59 97 λp

  • 1
  • 10

4 7 4

  • 8

14 1

  • 6

Note: the calculation of λ97 took 28 days on one CPU (on a Fordham HPC machine, done in July 2015). This algorithm is parallelizable as the slash by each coset representative can be computed separately.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 24 / 25

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SLIDE 68

Results

Results: some eigenvalues of f277

p 2 3 5 7 11 13 17 19 23 λp

  • 2
  • 1
  • 1

1

  • 2

3

  • 4
  • 1

3 p 29 31 37 41 43 47 53 59 97 λp

  • 1
  • 10

4 7 4

  • 8

14 1

  • 6

Note: the calculation of λ97 took 28 days on one CPU (on a Fordham HPC machine, done in July 2015). This algorithm is parallelizable as the slash by each coset representative can be computed separately. These eigenvalues match those of the abelian surface A277.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 24 / 25

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SLIDE 69

Results

Results: some eigenvalues of f277

p 2 3 5 7 11 13 17 19 23 λp

  • 2
  • 1
  • 1

1

  • 2

3

  • 4
  • 1

3 p 29 31 37 41 43 47 53 59 97 λp

  • 1
  • 10

4 7 4

  • 8

14 1

  • 6

Note: the calculation of λ97 took 28 days on one CPU (on a Fordham HPC machine, done in July 2015). This algorithm is parallelizable as the slash by each coset representative can be computed separately. These eigenvalues match those of the abelian surface A277. Together with the results in Brumer’s talk and and assuming that all the details can be written down for associating a Galois representation to a weight 2 paramodular form, this proves the modularity of the abelian surface of conductor 277.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 24 / 25

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SLIDE 70

Results Further work

Further work

Work in progress on conductors 349, 353, 389, . . . . Thank you.

David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 25 / 25