Abstract. We study the motion of an interface between two irrotational, incompressible fluids, with elastic bending forces present; this is the hydroelastic wave problem. We prove a global bifurcation theorem for the existence of families of spatially periodic traveling waves on infinite depth. Our traveling wave formulation uses a parameterized curve, in which the waves are able to have multi-valued height. This formulation and the presence of the elastic bending terms allows for the application of an abstract global bifurcation theorem of "identity plus compact" type. We furthermore perform numerical computations of these families of traveling waves, finding that, depending on the choice of parameters, the curves of traveling waves can either be unbounded, reconnect to trivial solutions, or end with a wave which has a self-intersection. Our analytical and computational methods are able to treat in a unified way the cases of positive or zero mass density along the sheet, the cases of single-valued or multi-valued height, and the cases of single-fluid or interfacial waves.

In a prior work, the authors proved a global bifurcation theorem for spatially periodic interfacial hydroelastic traveling waves on infinite depth, and computed such traveling waves. The formulation of the traveling wave problem used both analytically and numerically allows for waves with multivalued height. The global bifurcation theorem required a one-dimensional kernel in the linearization of the relevant mapping, but for some parameter values, the kernel is instead two-dimensional. In the present work, we study these cases with two-dimensional kernels, which occur in resonant and nonresonant variants. We apply an implicit function theorem argument to prove existence of traveling waves in both of these situations. We compute the waves numerically as well, in both the resonant and non-resonant cases.

We carry out some computations of vector valued Siegel modular forms of degree two, weight (k, 2) and level one. Our approach is based on Satoh's description of the module of vector-valued Siegel modular forms of weight (k, 2) and an explicit description of the Hecke action on Fourier expansions. We highlight three experimental results:(1) we identify a rational eigenform in a three dimensional space of cusp forms, (2) we observe that non-cuspidal eigenforms of level one are not always rational and (3) we verify a number of cases of conjectures about congruences between classical modular forms and Siegel modular forms. Harder for clarifying the case δ = 2 of Conjecture 4.2 for us. We thank Neil Dummigan for many insightful comments regarding Conjecture 4.5. We thank Martin Raum for making available code for computing values of symmetric square L-functions.

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