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Introduction to Siegel Modular Forms and Dirichlet Series by Anatoli Andrianov

By Anatoli Andrianov

Advent to Siegel Modular varieties and Dirichlet sequence supplies a concise and self-contained advent to the multiplicative thought of Siegel modular varieties, Hecke operators, and zeta services, together with the classical case of modular varieties in a single variable. It serves to draw younger researchers to this gorgeous box and makes the preliminary steps extra friendly. It treats a couple of questions which are not often pointed out in different books. it's the first and basically e-book thus far on Siegel modular kinds that introduces such vital issues as analytic continuation and the sensible equation of spinor zeta capabilities of Siegel modular kinds of genus two.

Unique positive aspects include:
* New, simplified techniques and a clean outlook on classical problems
* The summary conception of Hecke–Shimura earrings for symplectic and comparable groups
* The motion of Hecke operators on Siegel modular forms
* functions of Hecke operators to a learn of the multiplicative houses of Fourier coefficients of modular forms
* The facts of analytic continuation and the practical equation (under yes assumptions) for Euler items linked to modular varieties of genus two
*Numerous routines

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Extra resources for Introduction to Siegel Modular Forms and Dirichlet Series (Universitext)

Example text

D|n where |f (a)| ≤ cF ak/2 , where ζ (s) is the Riemann zeta function. [Hint: Show first that the function F(z) − f(0)(2ζ (k))−1 Ek (z) is a cusp form. 35. 20. 36. Show that the function ∆ (z) = ((2ζ (4))−1 E4 (z))3 − ((2ζ (6))−1 E6 (z))2 is a nonzero cusp form of the space M12 (Γ1 ) and that ∆ (z) and E12 (z) span the space. ] Petersson Scalar Product. Every space Nk (K, χ ) of cusp forms of an integral weight k and a congruence character χ for a congruence subgroup K of the symplectic group can be endowed with the structure of a Hilbert space by means of the scalar product.

By substituting F(i/qy) = q(iy)k F ∗ (iy) in the first integral, we can rewrite the last relation in the form Φ(s; F) = ik q1−s ∞ q−1/2 F ∗ (iy)yk−s−1 dy + ∞ q−1/2 F(iy)ys−1 dy. 24). 23) both converge absolutely and uniformly on any compact subset of the s-plane. This proves part (1). 21). Finally, if, say, F ∈ N− (q, χ ), then it satisfies (−i)k q−k/2+1 F ∗ = −F, and so the functional equation turns into the equation Φ(k − s; F) = ik qs−k+1 (−1)ik qk/2−1 Φ(s; F) = −(−1)k qs−k/2 Φ(s; F). 2. Let F ∈ Mk (q, χ ) be an arbitrary modular form with Fourier expansion ∞ F(z) = f(0) + ∑ f(m)e2π imz .

Holomorphic on Hn−1 , and is bounded on each subset Hn−1 ε The next lemma gives useful characterizations of cusp forms in terms of Siegel operators. 29. Let K be a congruence subgroup of Sn and χ a congruence character of K and let F ∈ Mk (K, χ ). Then the following three conditions are equivalent: (1) The function F is a cusp form; (2) F satisfies (F|k M)|Φ = 0 for all M ∈ Gn (Q) = Gn where Φ is the Siegel operator; (3) F satisfies (F|k γ )|Φ = 0 Q2n 2n , for all γ ∈ Γn . Proof. 54) that condition 1 implies condition 2.

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