# Manual A p-adic Property of Fourier Coefficients of Modular Forms of Half Integral Weight

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## Number theory seminar: Abstracts

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### Online Theses in Number Theory

Purchase Subscription prices and ordering Short-term Access To purchase short term access, please sign in to your Oxford Academic account above. This article is also available for rental through DeepDyve. View Metrics. Email alerts New issue alert. Thanks to the work of Lafforgue on the Langlands conjecture for GL N over function fields, one obtains a proof of the Ramanujan conjecture by studying the image of functoriality.

Let f be a newform of weight 2k-2 and level N with N odd and square-free. In joint work with Mahesh Agarwal we show roughly half of the Bloch-Kato conjecture in this setting, namely, the size of the Shafarevich-Tate group of the Galois representation associated to f is bounded below by an appropriately normalized special value of the L-function associated to f. We accomplish this by studying congruences among automorphic forms on GSp 4.

We present the theorem and discuss the necessary hypotheses. We will also present a conjecture about twisting special values to ensure they are p-adic units as well as numerical evidence for this conjecture. We prove a Voronoi type summation formula for divisor functions associated with Gaussian integers. We use a method of Beineke and Bump to obtain the two sides of the summation formula as values of a certain Eisenstein series.

This is joint work with D. Part of the work is joint with O. Bet Aharon. The majority of the talk will be on the utility of computations, including how the computer often asks us questions. The remainder of the talk will be on the cusp structure of the Paramodular Group in degree 2 for arbitrary levels. We give the 0-dimensional cusps and 1-dimensional cusps and how they cross. Applications to paramodular forms are given. Let k be an odd integer and N be a positive integer divisibe by 4.

### Duke Mathematical Journal

Applying Waldspurger's theorem to this space allows us to express the critical values of the L-functions of twists of g in terms of the coefficients of modular forms of half-integral weight. Following Tunnell, this often allows us to give a criterion for the n-th twist of an elliptic curve to have positive rank in terms of the number of representations of certain integers by certain ternary quadratic forms. I describe a strategy for finding "tetrahedral elliptic curves" over quadratic fields, that is, elliptic curves whose associated mod p representation, for some prime p, has projective image A4.

We describe a method that uses modular curves. I will discuss some computational and theoretical aspects of dimension formulas for vector-valued Hilbert modular forms of integral and half-integral weight. Friday 10th Back to overview. I will describe the distribution of holonomy angles attached to closed geodesics on a Hilbert modular surface. To study their distribution I will use a trace formula on a space of hybrid Hilbert Maass-Modular forms.

We consider random walks in the plane which consist of n steps of fixed length each taken into a uniformly random direction. Our interest lies in the probability distribution of the distance travelled by such a walk. While excellent asymptotic expressions are known for the density functions when n is moderately large, we focus on the arithmetic properties of short random walks.

In the case of three and four steps, the density functions satisfy differential equations of modular origin. This intertwines with the combinatorics of the corresponding even moments and leads to hypergeometric evaluations of the density functions. Much less is known for the density in case of five random steps, but we use the modularity of the four-step case and the Chowla-Selberg formula to deduce its exact behaviour near zero.

The talk will be based on joint work with Jonathan M. Borwein, James Wan, and Wadim Zudilin. Pippich: On elliptic analogues of the Takhtajan-Zograf metric on moduli spaces. In this talk, we report on a joint project with G. Freixas on an arithmetic Riemann-Roch isometry for singular metrics. The related local index theorem involves an analogue of the Takhtajan-Zograf metric employing so-called elliptic Eisenstein series.

In this lecture, properties of four quintic theta functions are developed in parallel with those of the classical Jacobi null theta functions. The quintic theta functions are shown to satisfy analogues of Jacobi's quartic theta-function identity and counterparts of Jacobi's Principles of Duplication and Dimidiation.

The resulting library of quintic transformation formulas is used to describe the action of Hecke operators of level five and more general quintic dissection operators. Central to the analysis is a new nonlinear coupled system of differential equations satisfied by the quintic theta functions. Our main result is the construction of an infinite family of cuspidal Drinfeld eigenforms with A-expansions, which generalizes results of Bartolome Lopez.

In joint work with T. Browning, S.

Baier and L. Zhao, we investigate the average and mean square of the error term in the conjectural asymptotic for D x , h , as h varies. The main tool in this work is the Hardy-Littlewood circle method. Over C , this is a well-studied, but extremely rich class of dynamical systems.

The special case of postcritically finite polynomials is now better understood.

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The arguments are elementary, from basic properties of valuations and heights. I will be talking about joint work with Fabien Trihan on the parity conjectures in positive characteristic. These conjectures are a weak version of the Birch and Swinnerton-Dyer conjecture modulo 2. Namely it asserts that the analytic rank of an elliptic curve has the same parity as the corank of the l -primary Selmer group. Recently, a lot of work has been done proving them for number fields.

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## Warwick number theory seminar: Abstracts

Recent methods to compute the zeta function of a variety over a finite field exploit the Frobenius structure on the relative cohomology of a family of varieties. When using these methods, one needs an effective bound on the rate of convergence of the Frobenius structure at the bad fibers of the family.

Using the theory of p-adic differential equations, we obtain a bound which is much better than all previous bounds and probably even sharp. This is joint work with Kiran Kedlaya. I will explain some applications of the theory of elementary divisors in p -adic Hodge theory.