Let g = c(D)q D and f = a n q n be modular forms of half-integral weight k + 1/2 and integral weight 2k respectively that are associated to each other under the Shimura-Kohnen correspondence. For suitable fundamental discriminants D, a theorem of Waldspurger relates the coefficient c(D) to the central critical value L(f, D, k) of the Hecke L-series of f twisted by the quadratic Dirichlet character of conductor D. This paper establishes a similar kind of relationship for central critical derivatives in the special case k = 1, where f is of weight 2. The role of c(D) in our main theorem is played by the first derivative in the weight direction of the Dth Fourier coefficient of a p-adic family of half-integral weight modular forms. This family arises naturally, and is related under the Shimura correspondence to the Hida family interpolating f in weight 2. The proof of our main theorem rests on a variant of the Gross-Kohnen-Zagier formula for Stark-Heegner points attached to real quadratic fields, which may be of some independent interest. We also formulate a more general conjectural formula of Gross-Kohnen-Zagier type for StarkHeegner points, and present numerical evidence for it in settings that seem inaccessible to our methods of proof based on p-adic deformations of modular forms.

Generalizing the method of Faltings-Serre, we rigorously verify that certain abelian surfaces without extra endomorphisms are paramodular. To compute the required Hecke eigenvalues, we develop a method of specialization of Siegel paramodular forms to modular curves.Date: August 10, 2018. 2010 Mathematics Subject Classification. 11F46, 11Y40.1 is made precise by a conjecture of Brumer-Kramer [8, Conjecture 1.1], restricted here for simplicity.Conjecture 1.1.1 (Brumer-Kramer). To every abelian surface A over Q of conductor N with End(A) = Z, there exists a cuspidal, nonlift Siegel paramodular newform f of degree 2, weight 2, and level N with rational Hecke eigenvalues, such thatMoreover, f is unique up to (nonzero) scaling and depends only on the isogeny class of A; and if N is squarefree, then this association is bijective.Conjecture 1.1.1 is often referred to as the paramodular conjecture. As pointed out by Frank Calegari, in general it is necessary to include abelian fourfolds with quaternionic multiplication for the converse assertion: for a precise statement for arbitrary N and further discussion, see Brumer-Kramer [7, Section 8].Extensive experimental evidence [8,47] supports Conjecture 1.1.1. There is also theoretical evidence for this conjecture when the abelian surface A is potentially of GL 2 -type, acquiring extra endomorphisms over a quadratic field: see Johnson-Leung-Roberts [34] for real quadratic fields and Berger-Dembélé-Pacetti-Şengün [5] for imaginary quadratic fields. For a complete treatment of the many possibilities for the association of modular forms to abelian surfaces with potentially extra endomorphisms, see work of Booker-Sijsling-Sutherland-Voight-Yasaki [11]. What remains is the case where End(A Q al ) = Z, which is to say that A has minimal endomorphisms defined over the algebraic closure Q al ; we say then that A is typical. (We do not say generic, since it is not a Zariski open condition on the moduli space.)Recently, there has been dramatic progress in modularity lifting theorems for nonlift Siegel modular forms (i.e., forms not of endoscopic type): see Pilloni [44] for p-adic overconvergent modularity lifting, as well as recent work by Calegari-Geraghty [12, §1.2], Berger-Klosin with Poor-Shurman-Yuen [2] establishing modularity in the reducible case when certain congruences are provided, and a paper in preparation by Boxer-Calegari-Gee-Pilloni [6] establishing potential modularity over totally real fields. 1.2. Main result. For all prime levels N < 277, the paramodular conjecture is known: there are no paramodular forms of the specified type by work of Poor-Yuen [47, Theorem 1.2], and correspondingly there are no abelian surfaces by work of Brumer-Kramer [8, Proposition 1.5]. At level N = 277, there exists a cuspidal, nonlift Siegel paramodular cusp form, unique up to scalar multiple, by work of Poor-Yuen [47, Theorem 1.3]: this form is given explicitly as a rational function in Gritsenko lifts of ten weight 2 theta blocks-see (6.2.2).Our main result is as follows. Theorem 1.2.1. ...

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Given a Hecke eigenform f of weight 2 and square-free level N, by the work of Kohnen, there is a unique weight 3/2 modular form of level 4N mapping to f under the Shimura correspondence. Furthermore, by the work of Waldspurger the Fourier coefficients of such a form are related to the quadratic twists of the form f. Gross gave a construction of the half integral weight form when N is prime, and such construction was later generalized to square-free levels. However, in the non-square free case, the situation is more complicated since the natural construction is vacuous. The problem being that there are too many special points so that there is cancellation while trying to encode the information as a linear combination of theta series. In this paper, we concentrate in the case of level p2, for p > 2 a prime number, and show how the set of special points can be split into subsets (indexed by bilateral ideals for an order of reduced discriminant p2) which gives two weight 3/2 modular forms mapping to f under the Shimura correspondence. Moreover, the splitting has a geometric interpretation which allows to prove that the forms are indeed a linear combination of theta series associated to ternary quadratic forms. Once such interpretation is given, we extend the method of Gross–Zagier to the case where the level and the discriminant are not prime to each other to prove a Gross-type formula in this situation.