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Triple Shifted Sums of Automorphic L-functions

Triple Shifted Sums of Automorphic L-functions

Thomas Hulse Brown University

ICERM Semester Program Automorphic Forms, Combinatorial Representation Theory and Multiple Dirichlet Series Providence, RI

January 29, 2013

Thomas Hulse Triple Shifted Sums of Automorphic L-functions January 29, 2013 1 / 5

Triple Shifted Sums of Automorphic L-functions

Let f(z) and g(z) be even weight k > 0 holomorphic cusp forms on Γ0(N)\H with respective Fourier expansions f(z) =

∞

• m=1

a(m)e2πimz =

∞

• m=1

A(m)m

k−1 2 e2πimz,

g(z) =

∞

• m=1

b(m)e2πimz =

∞

• m=1

B(m)m

k−1 2 e2πimz. Thomas Hulse Triple Shifted Sums of Automorphic L-functions January 29, 2013 2 / 5

Triple Shifted Sums of Automorphic L-functions

Let f(z) and g(z) be even weight k > 0 holomorphic cusp forms on Γ0(N)\H with respective Fourier expansions f(z) =

∞

• m=1

a(m)e2πimz =

∞

• m=1

A(m)m

k−1 2 e2πimz,

g(z) =

∞

• m=1

b(m)e2πimz =

∞

• m=1

B(m)m

k−1 2 e2πimz. Thomas Hulse Triple Shifted Sums of Automorphic L-functions January 29, 2013 2 / 5

Triple Shifted Sums of Automorphic L-functions

In 1965, Selberg[3] constructed and gave meromorphic continuations of shifted convolution sums of the form Selberg:

∞

• m=1

a(m)b(m + h) (2m + h)s , Hoffstein & H:

∞

• m=1

a(m + h)b(m) ms by replacing the real-analytic Eisenstein Series in the Rankin-Selberg Convolution with a real-analytic Poincar´ e Series and then untiling.

Thomas Hulse Triple Shifted Sums of Automorphic L-functions January 29, 2013 3 / 5

Triple Shifted Sums of Automorphic L-functions

In 1965, Selberg[3] constructed and gave meromorphic continuations of shifted convolution sums of the form Selberg:

∞

• m=1

a(m)b(m + h) (2m + h)s , Hoffstein & H:

∞

• m=1

a(m + h)b(m) ms by replacing the real-analytic Eisenstein Series in the Rankin-Selberg Convolution with a real-analytic Poincar´ e Series and then untiling. Such shifted convolution sums have been used to produce subconvexity bounds of moments of L-functions by providing asymptotic estimates of “off-diagonal” terms.

Thomas Hulse Triple Shifted Sums of Automorphic L-functions January 29, 2013 3 / 5

Triple Shifted Sums of Automorphic L-functions

In 1965, Selberg[3] constructed and gave meromorphic continuations of shifted convolution sums of the form Selberg:

∞

• m=1

a(m)b(m + h) (2m + h)s , Hoffstein & H:

∞

• m=1

a(m + h)b(m) ms by replacing the real-analytic Eisenstein Series in the Rankin-Selberg Convolution with a real-analytic Poincar´ e Series and then untiling. Such shifted convolution sums have been used to produce subconvexity bounds of moments of L-functions by providing asymptotic estimates of “off-diagonal” terms. By means of an approximation of a non-square integrable Poincar´ e series devised by Hoffstein, he and I were able to continue a variant of Selberg’s shifted sum with uncoupled denominator[2].

Thomas Hulse Triple Shifted Sums of Automorphic L-functions January 29, 2013 3 / 5

Triple Shifted Sums of Automorphic L-functions

In 1965, Selberg[3] constructed and gave meromorphic continuations of shifted convolution sums of the form Selberg:

∞

• m=1

a(m)b(m + h) (2m + h)s , Hoffstein & H:

∞

• m=1

a(m + h)b(m) ms by replacing the real-analytic Eisenstein Series in the Rankin-Selberg Convolution with a real-analytic Poincar´ e Series and then untiling. Such shifted convolution sums have been used to produce subconvexity bounds of moments of L-functions by providing asymptotic estimates of “off-diagonal” terms. By means of an approximation of a non-square integrable Poincar´ e series devised by Hoffstein, he and I were able to continue a variant of Selberg’s shifted sum with uncoupled denominator[2].

Thomas Hulse Triple Shifted Sums of Automorphic L-functions January 29, 2013 3 / 5

Triple Shifted Sums of Automorphic L-functions

Similarly, in the case where N = 1, this modified Poincar´ e series can be used to give a meromorphic continuation of the shifted convolution sum

∞

• m=1

a(m + h)λℓ(m) ms+ k−1

2

, where the λℓs are the Fourier coefficients of a weight-zero Maass form with eigenvalue 1

2 + itℓ.

Thomas Hulse Triple Shifted Sums of Automorphic L-functions January 29, 2013 4 / 5

Triple Shifted Sums of Automorphic L-functions

Similarly, in the case where N = 1, this modified Poincar´ e series can be used to give a meromorphic continuation of the shifted convolution sum

∞

• m=1

a(m + h)λℓ(m) ms+ k−1

2

, where the λℓs are the Fourier coefficients of a weight-zero Maass form with eigenvalue 1

2 + itℓ.

Combining this construction with the spectral expansion of Selberg’s shifted convolution sum and employing Bochner’s Theorem on the analytic continuation of functions in several variables,[1] we are able to construct meromorphic continuations of the multivariable functions T±(s1, s2, s3) =

∞

• m,h,n≥1

a(m − h)b(m)c(h ± n) ms1hs2ns3 to all (s1, s2, s3) ∈ C3.

Thomas Hulse Triple Shifted Sums of Automorphic L-functions January 29, 2013 4 / 5

Triple Shifted Sums of Automorphic L-functions

By taking inverse Mellin Transforms of T±, we are able to derive the non-trivial estimates

∞

• m,h,n≥1

a(m − h)b(m)c(h ± n) mkh

k 2

e−( m+h+n

X

) =

∞

• m,h,n≥1

A(m − h)B(m)C(h ± n)(1 − h

m)

k 2 (1 ± n

h)

k 2

• (m − h)(m)(h ± n)

e−( m+h+n

X

) = Of(1).

Thomas Hulse Triple Shifted Sums of Automorphic L-functions January 29, 2013 5 / 5

Triple Shifted Sums of Automorphic L-functions

By taking inverse Mellin Transforms of T±, we are able to derive the non-trivial estimates

∞

• m,h,n≥1

a(m − h)b(m)c(h ± n) mkh

k 2

e−( m+h+n

X

) =

∞

• m,h,n≥1

A(m − h)B(m)C(h ± n)(1 − h

m)

k 2 (1 ± n

h)

k 2

• (m − h)(m)(h ± n)

e−( m+h+n

X

) = Of(1). It is expected that certain binomial and integral expansions can remove the unwanted coupling terms.

Thomas Hulse Triple Shifted Sums of Automorphic L-functions January 29, 2013 5 / 5

Triple Shifted Sums of Automorphic L-functions

By taking inverse Mellin Transforms of T±, we are able to derive the non-trivial estimates

∞

• m,h,n≥1

a(m − h)b(m)c(h ± n) mkh

k 2

e−( m+h+n

X

) =

∞

• m,h,n≥1

A(m − h)B(m)C(h ± n)(1 − h

m)

k 2 (1 ± n

h)

k 2

• (m − h)(m)(h ± n)

e−( m+h+n

X

) = Of(1). It is expected that certain binomial and integral expansions can remove the unwanted coupling terms. Since these objects correspond to the “off-diagonal” terms of third moments of L-functions, the ultimate goal of my research is to get a formula for the asymptotics of higher moments and use these to produce subconvexity estimates.

Thomas Hulse Triple Shifted Sums of Automorphic L-functions January 29, 2013 5 / 5

Triple Shifted Sums of Automorphic L-functions

• S. Bochner.

A theorem on analytic continuation of functions in several variables.

• Ann. of Math. (2), 39(1):14–19, 1938.
• J. Hoffstein and T. Hulse.

Multiple dirichlet series and shifted convolutions (in preparation).

• Oct. 2012.
• A. Selberg.

On the estimation of Fourier coefficients of modular forms. In Proc. Sympos. Pure Math., Vol. VIII, pages 1–15. Amer. Math. Soc., Providence, R.I., 1965.

Thomas Hulse Triple Shifted Sums of Automorphic L-functions January 29, 2013 5 / 5