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By Irving Kaplansky

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97 (1927), no. 1, p. 210–242. [26] J. S. H SIA – “Representations by spinor genera”, Pacific J. Math. 63 (1976), no. 1, p. 147– 152. [27] H. I WANIEC & P. S ARNAK – “Perspectives on the analytic theory of L-functions”, Geom. Funct. Anal. (2000), no. Special Volume, Part II, p. 705–741, GAFA 2000 (Tel Aviv, 1999). Valentin Blomer & Gergely Harcos: L-functions, automorphic forms, and arithmetic 25 [28] M. J UTILA – “The additive divisor problem and its analogs for Fourier coefficients of cusp forms.

Note by the way, that the trivial bound would recover the convexity estimate. Now that we have an explicit description of L(s, π ⊗ χ) as a finite sum, let us try to exhibit cancellation in such sums. Let us first look at a simple example(7) . Suppose you want to prove that | sin x + cos x| ≤ 2. There are certainly many ways of proving this. Here is one: Square the left hand side and add a “spectrally useful" nonnegative quantity: | sin x + cos x|2 + | sin x − cos x|2 = 2. Now drop the second term, and the proof is complete.

Valentin Blomer & Gergely Harcos: L-functions, automorphic forms, and arithmetic 15 Then G acts on L 2 (Γ\G) by the right regular representation, ρ(g )(φ)(x) := φ(xg ) for φ ∈ L 2 (Γ\G), and we have a G-equivariant decomposition L 2 (Γ\G) = C · 1 ⊕ (5) π Vπ ⊕ a R Ha (t ) d t into the constant functions, cuspidal irreducible representations (π,Vπ ) and Eisenstein series for the cusps a (that enter the picture because Γ\G is not compact). Each Vπ decomposes further according to the characters of K : Vπ = q∈2Z Vπ,q (in the Hilbert space sense), and it is known that dimVπ,q ≤ 1.

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