We present a general existence proof for a wide class of non-linear elliptic equations which can be applied to problems with barrier conditions without specifying any assumptions guaranteeing the uniqueness or local uniqueness of particular solutions. As an application we prove the existence of closed hypersurfaces with curvature prescribed in the tangent bundle of an ambient Riemannian manifold $N$ without supposing any sign condition on the sectional curvatures $K_N$. A curvature flow wouldn't work in this situation, neither the method of successive approximation.

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The aim of this paper is to present a function field analogue of the classical Kronecker limit formula. We first introduce a “non-holomorphic” Eisenstein series on the Drinfeld half plane, and connect its “second term” with Gekeler’s discriminant function. One application is to express the Taguchi height of rank 2 Drinfeld modules with complex multiplication in terms of the logarithmic derivative of the corresponding zeta functions. Moreover, from the integral form of the Rankin-type L-function associated to two “Drinfeld-type” newforms, we then derive a formula for a non-central special derivative of the L-function in question. Adapting the classical approach, we also obtain a Kronecker-type solution for Pell’s equation over function fields.

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We provide a uniform construction of “mixed versions” or “graded lifts” in the sense of Beilinson-Ginzburg–Soergel which works for arbitrary Artin stacks. In particular, we obtain a general construction of graded lifts of many categories arising in geometric representation theory and categorified knot invariants. Our new theory associates to each Artin stack of finite type Y over F_q a symmetric monoidal DG-category Shv_{gr,c}(Y) of constructible graded sheaves on Y along with the six-functor formalism, a perverse t-structure, and a weight (or co-t-)structure in the sense of Bondarko and Pauksztello. This category sits in between the category Shv_{m,c}(Y_n) of constructible mixed ℓ-adic sheaves in the sense of Beilinson–Bernstein–Deligne--Gabber for any F_{q^n} -form Y_n of Y and the category Shv_c(Y) of constructible ℓ-adic sheaves on Y, compatible with the six-functor formalism, perverse t-structures, and Frobenius weights. Classically, mixed versions were only constructed in very special cases. However, the category Shv_{gr,c}(Y) agrees with those previously constructed when they are available. For example, for any reductive group G with a fixed pair T ⊂ B of a maximal torus and a Borel subgroup, we have an equivalence of monoidal DG weight categories Shv_{gr,c}(B\G/B) ≃ Ch^b(SBim_W), where Ch^b(SBim_W) is the monoidal DG-category of bounded chain complexes of Soergel bimodules and W is the Weyl group of G.