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Mathematisches Forschungsinstitut Oberwolfach Modular Forms Mathematisches Forschungsinstitut Oberwolfach Report No. 22/2014 DOI: 10.4171/OWR/2014/22 Modular Forms Organised by Jan Hendrik Bruinier, Darmstadt Atsushi Ichino, Kyoto Tamotsu Ikeda, Kyoto Ozlem¨ Imamoglu, Z¨urich 27 April – 3 May 2014 Abstract. The theory of Modular Forms has been central in mathematics with a rich history and connections to many other areas of mathematics. The workshop explored recent developments and future directions with a particular focus on connections to the theory of periods. Mathematics Subject Classification (2010): 11xx. Introduction by the Organisers The workshop Modular Forms, organized by Jan Hendrik Bruinier (Darmstadt), Atsushi Ichino (Kyoto), Tamotsu Ikeda (Kyoto) and Ozlem¨ Imamoglu (Z¨urich) consisted of 19 one-hour long lectures and covered various recent developments in the theory of modular and automorphic forms and related fields. A particular focus was on the connection of modular forms to periods, since there have been important developments in that direction in recent years. In this context, the topics that the workshop addressed include the global Gross- Prasad conjecture and its analogs, which predict a relationship between periods of automorphic forms and central values of L-functions, the theory of liftings and their applications to period relations, as well as explicit aspects of these formulas and relations with a view towards the arithmetic properties of periods. There are two fundamental ways in which automorphic forms are related to periods. First, according to the conjectures of Deligne, Beilinson and Scholl, spe- cial values of motivic automorphic L-functions at integral arguments should be given by periods and encode important arithmetic information, such as ranks of 1222 Oberwolfach Report 22/2014 Chow groups and Selmer groups. Second, the Fourier coefficients of automorphic forms are often given by periods. For instance, by the work of Waldspurger, the coefficients of half integral weight eigenforms are given by period integrals of their Shimura lifts. The majority of the lectures (in particular talks by Wee-Teck Gan, Erez Lapid, Kazuki Morimoto, Anantharam Raghuram, Abhishek Saha and Shun- suke Yamana) discussed (or were motivated by) relations of periods and special values of automorphic L-functions. Periods related to classes in cohomology and Chow groups of Shimura varieties and their connections to automorphic forms were addressed in the talks by Kathrin Bringmann, Yingkun Li, Yifeng Liu, and Tonghai Yang. Other talks discussed the role of automorphic forms in geometry, for instance in context of the Kudla program (Stephan Ehlen, Valery Gritsenko, J¨urg Kramer, Stephen Kudla and Martin Raum). Aspects of the analytic theory of automorphic forms played an important role in the talks by Valentin Blomer, Gautam Chinta, Tomoyoshi Ibukiyama and Ren He Su. In total, 53 researchers participated in the workshop. Out of these, 37 came from 12 countries different from Germany. Beyond the talks, the participants enjoyed ample time for discussions and collaborative research activities. The traditional hike on Wednesday afternoon led us to the Ochsenwirtshof in Schapbach. A further highlight was a piano recital on Thursday evening by Valentin Blomer. The organizers and participants of the workshop thank the Mathematisches Forschungsinstitut Oberfwolfach for hosting the workshop and providing such an ideal working environment. Acknowledgement: The MFO and the workshop organizers would like to thank the National Science Foundation for supporting the participation of junior researchers in the workshop by the grant DMS-1049268, “US Junior Oberwolfach Fellows”. Moreover, the MFO and the workshop organizers would like to thank the Simons Foundation for supporting Stephen S. Kudla in the “Simons Visiting Professors” program at the MFO. Modular Forms 1223 Workshop: Modular Forms Table of Contents Stephen Kudla Product formulas for Borcherds forms ..............................1225 Valentin Blomer (joint with Rizwanur Khan and Matthew Young) Distribution of mass of holomorphic cusp forms .....................1228 Wee-Teck Gan The Shimura-Waldspurger correspondence for Mp2n .................1229 Anantharam Raghuram (joint with Baskar Balasubramanyam) Special values of L-functions and congruences for automorphic forms . .1232 Tomoyoshi Ibukiyama Construction of liftings to vector valued Siegel modular forms .........1234 Kathrin Bringmann (joint with Ben Kane) Meromorphic cycle integrals ......................................1237 Yifeng Liu Central critical L-values and Selmer groups for triple product motives . .1240 Valery Gritsenko (joint with Cris Poor and David Yuen) Borcherds Products Everywhere Theorem ...........................1241 Gautam Chinta Multiple Dirichlet series and prehomogeneous vector spaces ...........1244 Martin Raum (joint with Jan Hendrik Bruinier) Symmetric Formal Fourier Jacobi Series and Kudla’s Conjecture ......1247 Shunsuke Yamana (joint with Eyal Kaplan) Symmetric square L-functions of GL(n) ............................1249 Erez M. Lapid (joint with Zhengyu Mao) Whittaker coefficients of cuspidal representations of the metaplectic group ................................................... .......1250 Yingkun Li Real-dihedral harmonic Maass forms and CM-values of Hilbert modular functions ................................................... ....1254 Abhishek Saha (joint with Ameya Pitale, Ralf Schmidt) Structure and arithmeticity for nearly holomorphic Siegel cusp forms of degree 2 ................................................... .....1257 1224 Oberwolfach Report 22/2014 Kazuki Morimoto On special values of L-functions for quaternion unitary groups of degree 2 and GL(2) .............................................. ......1259 Tonghai Yang CM values of automorphic Green functions and L-functions ..........1262 Ren He Su Eisenstein series in Kohnen plus space for Hilbert modular forms ......1265 Stephan Ehlen (joint with Jan Hendrik Bruinier, Eberhard Freitag) Lattices with many Borcherds products .............................1267 J¨urg Kramer (joint with Jos´eBurgos Gil) A geometrical approach to Jacobi forms, revisited ....................1270 Modular Forms 1225 Abstracts Product formulas for Borcherds forms Stephen Kudla In a now classic pair of Inventiones papers in 1995 and 1998, Borcherds constructed meromorphic modular forms on the arithmetic quotient of a bounded domain D associated to a rational quadratic space V , ( , ) of signature (n, 2). These forms have various remarkable properties, for example, their divisor is explicitly given. But perhaps most striking is that, in a suitable neighborhood of each 0-dimensional boundary component, they are given by a product formula reminiscent of that for the Dedekind η function. In this talk, I will describe analogous product formulas for Borcherds forms, now valid in a suitable neighborhood of each 1-dimensional boundary component, assuming that V admits 2-dimensional isotropic subspaces. Let D = w V (C) (w, w)=0, (w, w¯) < 0 /C× P(V (C)) { ∈ | } ⊂ be the ‘quadric’ model of the symmetric space associated to V . Fix an even integral lattice M M ∨ in V , let ⊂ Γ Γ = γ SO(V ) γM = M, γ =1 ⊂ M { ∈ | |M ∨/M } be a subgroup of finite index, and let XΓ =Γ D be the corresponding arithmetic ∨ \ quotient. Let SM = C[M /M] be the group algebra of the discriminant group of M, which we view as a subspace of S(V (Af )), the space of locally constant functions of compact support on the finite ad`ele points of V . The group SL2(Z), for n even, or a central extension of it, for n odd, acts on the space SM via the Weil representation ρM . Recall that in [2], Borcherds takes as input a weakly n holomorphic modular form F : H SM of weight 1 2 and type ρM . In particular F has a Fourier expansion → − F (τ)= c(m) qm, c(m) S ∈ M m X with only a finite number of nonvanishing coefficients c(m) for m< 0. Assuming that for m 0, c(m) Z[M ∨/M], Borcherds associates to F a meromorphic mod- ular form Ψ(≤F ) on D∈of weight c(0)(0)/2 with respect to a finite index subgroup of ΓM . Now suppose that ′ V = U + V0 + U is a Witt decomposition of V , where U is an isotropic 2-plane, U ′ is an isotropic ′ ⊥ complement and V0 = (U + U ) . The complex curve (U)= w U(C) span w, w¯ = U(C) /C× P(U(C)) P(U(R)) C { ∈ | { } } ≃ − lies in the closure of D in P(V (C)) and defines a 1-dimensional rational boundary BB component in the Bailey-Borel compactification XΓ of XΓ. 1226 Oberwolfach Report 22/2014 ′ ′ Choose a Z-basis e1, e2 for the lattice M U and let e1 and e2 be a dual basis for U ′. The Witt decomposition then determines∩ an isomorphism ∼ D (τ , w , τ ′ ) C V (C) C 4v v′ Q(w w¯ ) > 0 , −→ { 1 0 2 ∈ × 0 × | 1 2 − 0 − 0 } ′ ′ 1 where v1 = (τ1), v2 = (τ2), Q(x)= 2 (x, x), and the inverse map is obtained by taking ℑ ℑ ′ ′ ′ ′ ′ w(τ1, w0, τ2)= τ2e1 + (τ1τ2 Q(w0))e2 + w0 + τ1e1 + e2. ′ − − Note that, as v2 for τ1 and w0 in bounded sets, the isotropic line Cw in D goes to the isotropic→ ∞ line C( e + τ e ) in (U). − 1 1 2 C Theorem. In a region of the form w(τ , w , τ ′ ) v′ > Av + (Q( (w )) + B)v−1 , { 1 0 2 | 2 1 ℑ 0 1 } for suitable positive constants A and B, the Borcherds form Ψ(F )(w) is given as the product of the following factors: (a) c(−Q(x))(x) 1 e( (x, w)) , − − x∈M ∨ (x,eY2)=0 (x,e1)>0 mod M∩Qe2 (b) c(−Q(x))(x)/2 ϑ( (x, w), τ1) 1 (x,e′ ) − e((x, w) (xU , w)) 1 , η(τ1) − 2 x∈M ∨∩U ⊥ modYM∩U Q(x)6=0 (c) c(0)(x)/2 ϑ( (x, w), τ1) 1 (x,e′ ) − e( (x, w)) 1 , η(τ1) 2 x∈M ∨∩U/M∩U xY6=0 (d) c(0)(0) I0 κ η(τ1) q2 , where ϑ(z, τ) is the Jacobi theta function and I = c( m)(x) σ (m Q(x)). 0 − − 1 − m x∈M ∨∩U ⊥ X modXM∩U ′ Here q2 = e(τ2) and σ1(n) is the sum of the positive divisors of n if n > 0, σ (0) = 1/24, and σ (n)=0 if n< 0.
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