Book of abstracts. Recent Developments on the Arithmetic of Special Values of L-functions 11/12/ /12/2017

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1 Book of abstracts Recent Developments on the Arithmetic of Special Values of L-functions 11/12/ /12/2017 EPFL, CIB June 6, 2018

2 Fabrizio Andreatta Università Statale di Milano Triple product p-adic L-functions associated to nite slope p-adic families of modular forms. I will explain how one can extend the methods developed together with Iovita and Pilloni to construct p-adic deformations of nite slope de Rham classes. As an application I will dene p-adic L-functions attached to a triple of p- adic nite slope families of modular forms. This is joint work with Adrian Iovita. Bas Edixhoven University of Leiden Pink's conjecture on unlikely intersections and families of semi-abelian varieties I will report on joint work with Daniel Bertrand, concerning Richard Pink's conjecture on unlikely intersections in families of semi-abelian varieties. For the conjecture, see Pink's preprint `A common generalization of the conjectures of AndréOort, Manin-Mumford, and Mordell-Lang.' The work with Bertrand is about special points and special subvarieties in universal Poincaré torsors and so-called Ribet sections. Michele Fornea McGill University Twisted triple product p-adic L-functions and Hirzebruch-Zagier cycles For L/F a quadratic extension of totally real number elds and p an unramied prime, we construct a p-adic L-function interpolating the central values of the twisted triple product L-functions attached to nearly ordinary families of Hilbert modular forms. When L is a real quadratic eld and p is a split prime, we prove a p-adic Gross-Zagier formula relating the values of the p-adic L-functions outside the range of interpolation to the syntomic Abel-Jacobi image of generalized Hirzebruch-Zagier cycles. The formula, in combination with a recent work of Y. Liu, gives a result in the spirit of the Bloch-Kato conjecture for some p-adic representation attached to elliptic curves A/L and E/Q. This is joint work with Iván Blanco-Chacón. Lennart Gehrmann Universität Duisburg-Essen Derived Hecke algebra and automorphic L-invariants Hilbert modular cusp forms of parallel weight two only show up in the mid- 1

3 dle degree cohomology of the appropiate locally symmetric space, whereas the analogous automorphic forms over number elds of arbitrary sign (e.g. Bianchi modular forms) show up in several degrees. In this talk I dene automorphic L-invariants for each of these degrees and show that Venkatesh's conjecture on derived Hecke algebras implies a precise relationship between these L-invariants. Guido Kings Universität Regensburg Étale and motivic Eisenstein cohomology for Hilbert modular varieties Eisenstein cohomology classes in étale cohomology of a modular curve play a crucial role in the study of p-adic properties of L-functions of modular forms and in the construction of Euler systems, so for example in the work of Kato or in the study of the Euler system for Rankin convolutions of modular forms. The strength of this approach relies on the fact that these Eisenstein classes are in fact images of motivic cohomology classes under the étale regulator map and that these images allow a p-adic interpolation through an Iwasawa cohomology. In the case a Hilbert modular variety, Harder has constructed the Eisenstein cohomology in singular cohomology using transcendental methods, which does not allow to dene integral étale Eisenstein classes. Building upon the thesis of Graf, who uses the topological polylogarithm to give a topological construction of Harder's Eisenstein cohomology, we present in this talk a motivic version of Graf's construction, which can also be carried out in étale cohomology with integral coecients. This allows to dene integral étale Eisenstein cohomology classes and to prove p-adic interpolation results exactly as in the case of modular curves. Daniel Kriz Princeton University A new p-adic Maass-Shimura operator We discuss the construction of a new p-adic Maass-Shimura operator, how it can be used to study the p-adic analysis of modular forms on the supersingular locus of Shimura curves, and the resulting applications to the arithmetic of certain abelian varieties. Chao Li Columbia University 2

4 Arithmetic fundamental lemma in the minuscule case The arithmetic Gan-Gross-Prasad conjecture generalizes the Gross-Zagier formula to Shimura varieties associated to unitary or orthogonal groups. The arithmetic fundamental lemma (AFL), formulated by Wei Zhang in the unitary case, is a key local ingredient in the relative trace formula approach towards arithmetic GGP. In joint work with Yihang Zhu, we prove an explicit formula for the arithmetic intersection numbers in both unitary and orthogonal cases, under a minuscule assumption. In particular, our work gives a new proof of the theorem of Rapoport-Terstiege-Zhang on the AFL. Yifeng Liu Northwestern University Bloch-Kato conjecture for Rankin-Selberg motives, I. In this talk, I will formulate explicitly the Bloch-Kato conjecture for symplectic conjugate self-dual motives of GL(m)*GL(n) over CM elds, which relates Selmer groups and Rankin-Selberg L-functions in the central critical case. When m=n+1, a theorem concerning rank 0 and rank 1 will be stated, which is the main theme of an ongoing joint work with Y. Tian, L. Xiao, W. Zhang, and X. Zhu. Then I will introduce various Shimura varieties of unitary type which we use. If time permits, I will state an arithmetic level raising result for odd dimensional unitary Shimura varieties, which is a key ingredient in our approach. David Loeer University of Warwick Heegner points in Coleman families Using Heegner points, or more general Heegner cycles, one can build Galois cohomology classes for a modular form twisted by an anticyclotomic Grössencharacter of an imaginary quadratic eld. I will report on a joint project with Jetchev and Zerbes in which we show that these classes naturally interpolate into a 2-parameter family, with the modular form varying through a Coleman family, and the ring class character also varying. (This extends earlier results due to many authors including Howard, Disegni, and Kobayashi.) I will also explain a general programme to obtain new Euler systems by specialising such families at critical-slope Eisenstein series; in this Heegnerpoint setting this should give a new construction of the elliptic units, and potentially a generalisation of elliptic units to higher-degree CM elds. 3

5 Maria Rosaria Pati Università di Padova Exceptional zero formulae for anti-cyclotomic p-adic L-functions of elliptic curves in the ramied case Iwasawa theory of modular forms over anti-cyclotomic Z_p-extensions of imaginary quadratic elds has been studied by several authors under the crucial assumption that the prime p is unramied in K. In this talk we start the systematic study of anti-cyclotomic p-adic L-functions when p is ramied in K. In particular, when f is a weight 2 modular form attached to an elliptic curve E/Q having multiplicative reduction at p, and p is ramied in K, we show an analogue of the exceptional zeroes phenomenon investigated by Bertolini-Darmon in the setting when p is inert in K. More precisely, we consider situations in which the p-adic L-function L_p(E/K) of E over the anti-cyclotomic Z_p-extension of K does not vanish identically but has a zero at certain characters chi of the Hilbert class eld of K. In this case we show that the value at chi of the rst derivative of L_p(E/K) is equal to the formal group logarithm of the specialization at p of a global point on the elliptic curve (actually, this global point is a twisted sum of Heegner points). This generalizes similar results of Bertolini-Darmon, available when p is inert in K and chi is the trivial character. Victor Rotger Universitat Politècnica de Catalunya Stark-Heegner points and the Euler system of diagonal cycles Stark-Heegner points are conjectural substitutes for Heegner points when the imaginary quadratic eld of the theory of complex multiplication is replaced by a real quadratic eld K. They are constructed analytically as local points on elliptic curves with multiplicative reduction at a prime p that remains inert in K, but are conjectured to be rational over ring class elds of K and to satisfy a Shimura reciprocity law describing the action of G_K on them. This amounts to claiming that any linear combination of Stark-Heegner points weighted by the values of a ring class character of K should belong to the corresponding piece of the Mordell-Weil group over the associated ring class eld. In this lecture I will report on joint work with H. Darmon, where we show that such linear combinations arise from the localisation at primes above p of global classes in the idoneous pro-p Selmer group. The proof rests on a direct comparison between Stark-Heegner points and the generalised Kato classes arising from the Euler system of diagonal cycles. 4

6 Shrenik Shah Columbia University Geometry and representation theory of Archimedean regulator pairings We describe some connections between Archimedean regulator pairings and ideas in geometry and representation theory, with an emphasis on Picard modular surfaces and the Shimura fourfolds associated to GU(2, 2). For instance, the relative Fourier-Mukai transform will arise as well as distinction and multiplicity one for representations of p-adic groups. This is joint work with Aaron Pollack. Florian Sprung Princeton University The Birch and Swinnerton-Dyer formula in the non-ordinary case We present a proof of the main conjectures in the non-ordinary case, for elliptic curves, and for weight two modular forms. Ye Tian Academy of Mathematics and System Science A converse theorem of Gross-Zagier and Kolyvagin: CM case Let E be a CM elliptic curve over rationals and p an odd ordinary prime. Assume that p-selmer gorup of E has corank one, we show that the analytic rank of E is also one. This is joint work with Ashay Burungale. Yichao Tian Universität Bonn Bloch-Kato conjecture for Rankin-Selberg motives II This is a continuation of Yifeng Liu's talk on the progress of our on-going project with Liang Xiao, Wei Zhang and Xinwen Zhu. In this talk, I will focus on the case of U(2)*U(3) motives. I will start with explaining the proof of the level raising result in the case of U(2) Shimura varieties, perhaps already stated in Yifeng's talk. Then I will explain how to deduce one side implication of the Bloch-Kato conjecture for U(2)*U(3) motives in the rank 0 case under some additional assumptions. A key ingredient is the generic Tate conjecture for U(3) Shimura varieties over nite elds proved recently by Liang Xiao and Xinwen Zhu. 5

7 Rodolfo Venerucci Universität Duisburg-Essen Diagonal classes and derivatives of Garrett-Rankin p-adic L-functions I discuss exceptional zero formulae relating special values of p-adic Garrett- Ranking L-functions to the Abel-Jacobi images of certain diagonal classes. This is joint work with M. Bertolini and M. A. Seveso. Wei Zhang Massachusetts Institute of Technology Periods and cycles related to central values of L-functions and their derivatives We will recall conjectures and theorems on automorphic period integrals, special cycles on Shimura varieties, and their connection to central values of L-functions and their derivatives. We will mainly focus on the global Gan GrossPrasad conjecture, which will appear in the talks by Yifeng Liu and Yichao Tian respectively. 6