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We use Borcherds products to give a new construction of the weight $3$ paramodular nonlift eigenform $f_N$ for levels $N = 61, 73, 79$. We classify the congruences of $f_N$ to Gritsenko lifts. We provide techniques that compute eigenvalues to support future modularity applications. Our method does not compute Hecke eigenvalues from Fourier coefficients but instead uses elliptic modular forms, specifically the restrictions of Gritsenko lifts and their images under the slash operator to modular curves.

The subsequent series of responses to big events may exhibit a synchronicity of event number, frequency and energy release in different fault zones. This synchronicity is a reliable source for probing non-intuitive geological structures, assessing regional seismicity hazard map and even predicting the next big events. The synchronicity of main faults in the eastern margin of the Qinghai-Tibetan Plateau is still unknown to us. We propose to examine the correlation of earthquake occurrence among different fault zones to indicate this synchronicity, and to obtain a preliminary understanding of geodynamics processes and the unrecognized characteristics of deep evolution in the eastern margin of the Qinghai-Tibetan Plateau. We estimate temporal changes of completeness level, frequency seismicity, and intensity seismicity, referring respectively to Mc, Z, and E values, of 21 main fault zones, using a seismic catalogue from 1970 to 2015. Our results reveal that six fault zone pairs of fault zones exhibit relative high correlation (>0.6) by all three indicators, while four fault zone pairs are non-adjacent with close internal affinity offsetting the limit of spatial distance, such as the pair of Rongjing-mabian fault and Minjiang-huya fault. Most strikingly, some fault zone pairs showing typical high correlation (>0.8) of seismicity frequency or seismicity intensity, the faults surprisingly belong to neither the same seismic belt nor the same geological block, exhibiting a regional scale remote triggering pattern of earthquakes or structures. An embryonic pattern to predict the next possible events will also be presented. This correlation analysis discovers a previously unrecognized strong coupling relationship among main faults with high earthquake risk in the eastern margin of the Qinghai-Tibetan Plateau.

Several of NASA missions (TESS, JWST, WFIRST, etc.) and mission concepts (LUVOIR, HabEx, and OST) emphasize the exploration and characterization of exoplanets, and the study of the interstellar medium. We anticipate that a much broader set of chemical environments exists on exoplanets, necessitating data from a correspondingly broader set of chemical reactions. Similarly, the conditions that exist in astrophysical environments are very different from those traditionally probed in laboratory chemical kinetics studies. These are areas where quantum mechanical theory, applied to important reactions via well-validated chemical kinetics models, can fill a critical knowledge gap. Quantum chemical calculations are also introduced to study interior of planets, photochemical escape, and many critical chemical pathways (e.g. prebiotic environments, contaminations, etc.) After years of development of the relevant quantum chemical theories and significant advances in computational power, quantum chemical simulations have currently matured enough to describe real systems with an accuracy that competes with experiments. These approaches, therefore, have become the best possible alternative in many circumstances where performing experiments is too difficult, too expensive, or too dangerous, or simply not possible. In this white paper, several existing quantum chemical studies supporting exoplanetary science, planetary astronomy, and astrophysics are described, and the potential positive impacts of improved models associated with scientific goals of missions are addressed. In the end, a few recommendations from the scientific community to strengthen related research efforts at NASA are provided.

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. In the appendix, Serre proves a result extending his work on the reduction of G-invariant bilinear forms modulo primes to the case of G-covariant forms., Comment: 49 pages; proof of Lemma 4.3.6 simplified, still with the new appendix by Serre

We complete the construction of the nonlift weight two cusp paramodular Hecke eigenforms for prime levels N<600, which arise in conformance with the paramodular conjecture of Brumer and Kramer.

We present an algorithm to compute all Borcherds product paramodular cusp forms of a specified weight and level, describing its implementation in some detail and giving examples of its use., Comment: 42 pages, a few typos corrected

We compute the space $S_2(K(N))$ of weight $2$ Siegel paramodular cusp forms of squarefree level $N<300$. In conformance with the paramodular conjecture of A. Brumer and K. Kramer, the space is only the additive (Gritsenko) lift space of the Jacobi cusp form space $J_{2,N}^{\text{cusp}}$ except for $N=249,295$, when it further contains one nonlift newform. For these two values of $N$, the Hasse-Weil $p$-Euler factors of a relevant abelian surface match the spin $p$-Euler factors of the nonlift newform for the first two primes $p\nmid N$.

The iron spin crossover in ferropericlase introduces anomalies in its thermodynamics and thermoelastic properties. Here we investigate how these anomalies can affect the lower mantle geotherm. The effect is examined in mantle aggregates consisting of mixtures of bridgmanite, ferropericlase, and CaSiO$_3$ perovskite, with different Mg/Si ratios varying from harzburgitic to perovskitic (Mg/Si$\sim$1.5 to 0.8). We find that the anomalies introduced by the spin crossover increase the isentropic gradient and thus the geotherm proportionally to the amount of ferropericlase. The geotherms can be as much as $\sim$ 200K hotter than the conventional adiabatic geotherm at deep lower mantle conditions. Aggregate elastic moduli and seismic velocities are also sensitive to the spin crossover and the geotherm, which impacts analyses of lower mantle velocities and composition.

We define an algebraic set in $23$~dimensional projective space whose $\mathbb Q$-rational points correspond to meromorphic, antisymmetric, paramodular Borcherds products. We know two lines inside this algebraic set. Some rational points on these lines give holomorphic Borcherds products and thus construct examples of Siegel modular forms on degree two paramodular groups. Weight $3$ examples provide antisymmetric canonical differential forms on Siegel modular threefolds. Weight $2$ is the minimal weight and these examples, via the Paramodular Conjecture, give evidence for the modularity of some rank one abelian surfaces defined over $\mathbb Q$.

We compute Hecke eigenform bases of spaces of level one, degree~three Siegel modular forms and 2-Euler factors of the eigenforms through weight 22. Our method uses the Fourier coefficients of Siegel Eisenstein series, which are fully known and computationally tractable by the work of H. Katsurada; we also use P. Garrett's decomposition of the pullback of the Eisenstein series through the Witt map. Our results support I. Miyawaki's conjectural lift, and they give examples of eigenforms that are congruence neighbors.

We prove the Borcherds Products Everywhere Theorem, Theorem 6.6, that constructs holomorphic Borcherds Products from certain Jacobi forms that are theta blocks without theta denominator. The proof uses generalized valuations from formal series to partially ordered abelian semigroups of closed convex sets. We present nine infinite families of paramodular Borcherds Products that are simultaneously Gritsenko lifts. This is the first appearance of infinite families with this property in the literature., Comment: 35 pages

The Fourier Jacobi expansions of paramodular forms are characterized from among all formal series of Jacobi forms by two conditions on the Fourier coeffcients of the Jacobi forms: a growth condition and a set of linear relations. Examples, both theoretical and computational, indicate that the growth condition may be superfluous.

We classify Siegel modular cusp forms of weight two for the paramodular group K(p) for primes p< 600. We find that weight two Hecke eigenforms beyond the Gritsenko lifts correspond to certain abelian varieties defined over the rationals of conductor p. The arithmetic classification is in a companion article by A. Brumer and K. Kramer. The Paramodular Conjecture, supported by these computations and consistent with the Langlands philosophy and the work of H. Yoshida, is a partial extension to degree 2 of the Shimura-Taniyama Conjecture. These nonlift Hecke eigenforms share Euler factors with the corresponding abelian variety $A$ and satisfy congruences modulo \ell with Gritsenko lifts, whenever $A$ has rational \ell-torsion.

We give a hyperelliptic formulation of the Ansatz of D'Hoker and Phong. We give an explicit family of binary invariants, one for each genus, that satisfies this hyperelliptic Ansatz. We also prove that this is the unique family of weight eight binary forms over the theta group on the hyperelliptic locus that satisfies this Ansatz. Futhermore, we prove that this solution may also be obtained by applying Thomae's map to multivalued Siegel modular forms of Grushevsky and making certain choices of roots.

Let n >= 2 be an integer and consider the set T_n of n by n permutation matrices pi for which pi_{ij}=0 for j>=i+2. In this paper we study the convex hull of T_n, which we denote by P_n. P_n is a polytope of dimension binom{n}{2}. Our main purpose is to provide evidence for the following conjecture concerning its volume. Let v_n denote the minimum volume of a simplex with vertices in the affine lattice spanned by T_n. Then the volume of P_n is v_n times the product for i varying from 0 to n-2 of frac{1}{i+1} binom{2i}{i}. That is, P_n is the product of v_n and the first n-1 Catalan numbers. We also give a related result on the Ehrhart polynomial of P_n., Comment: this version corrects a small number of typographical errors in the original