Research Interest: I am interested in number theory and algebraic geometry. I mostly work in arithmetic algebraic geometry.
Some more specific themes in my research have been the integral models of Shimura varieties.
Research Interest: I am interested in number theory and algebraic geometry. I mostly work in arithmetic algebraic geometry.
Some more specific themes in my research have been the integral models of Shimura varieties.
Publications and pre-prints:
On the geometry of splitting models, (with S. Bijakowski & Z. Zhao), 2025, arXiv:2501.05950.
Abstract: We consider Shimura varieties associated to a unitary group of signature (n−s,s) where n is even. For these varieties, by using the spin splitting models from Zachos-Zhao, we construct flat, Cohen-Macaulay, and normal p-adic integral models with reduced special fiber and with an explicit moduli-theoretic description over odd primes p which ramify in the imaginary quadratic field with level subgroup at p given by the stabilizer of a π-modular lattice in the hermitian space. We prove that the special fiber of the corresponding splitting model is stratified by an explicit poset with a combinatorial description, similar to Bijakowski-Hernandez, and we describe its irreducible components. Additionally, we prove the closure relations for this stratification.
2. Semi-stable and splitting models for unitary Shimura varieties over ramified places. II, (with Z. Zhao), International Mathematics Research Notices, Volume 2025, Issue 11, June 2025, https://doi.org/10.1093/imrn/rnaf145.
Abstract: We consider Shimura varieties associated to a unitary group of signature (n-1,1). For these varieties, we construct p-adic integral models over odd primes p which ramify in the imaginary quadratic field with level subgroup at p given by the stabilizer of a vertex lattice in the hermitian space. Our models are given by a variation of the construction of the splitting models of Pappas-Rapoport and they have a simple moduli theoretic description. By an explicit calculation, we show that these splitting models are normal, flat, Cohen-Macaulay and with reduced special fiber. In fact, they have relatively simple singularities: we show that a single blow-up along a smooth codimension one subvariety of the special fiber produces a semi-stable model. This also implies the existence of semi-stable models of the corresponding Shimura varieties.
3. Semi-stable and splitting models for unitary Shimura varieties over ramified places. I, (with Z. Zhao), Forum of Mathematics, Sigma 13 (2025): e119, https://doi.org/10.1017/fms.2025.10079.
Abstract: We consider Shimura varieties associated to a unitary group of signature (n-s,s) where n is even. For these varieties, we construct smooth p-adic integral models for s=1 and regular p-adic integral models for s=2 and s=3 over odd primes p which ramify in the imaginary quadratic field with level subgroup at p given by the stabilizer of a π-modular lattice in the hermitian space. Our construction, which has an explicit moduli-theoretic description, is given by an explicit resolution of a corresponding local model.
4. Semi-stable models for some unitary Shimura varieties over ramified primes, preprint (2024), Algebra & Number Theory, Vol. 18 (2024), No. 9, 1715–1736.
Abstract: We consider Shimura varieties associated to a unitary group of signature (n-2,2). We give regular p-adic integral models for these varieties over odd primes p which ramify in the imaginary quadratic field with level subgroup at p given by the stabilizer of a selfdual lattice in the hermitian space. Our construction is given by an explicit resolution of a corresponding local model.
5. Regular integral models for Shimura varieties of orthogonal type, (with G. Pappas), Compos. Math. 158(4) (2022), 831-867.
Abstract: We consider Shimura varieties for orthogonal or spin groups acting on hermitian symmetric domains of type IV. We give regular p-adic integral models for these varieties over odd primes p at which the level subgroup is the connected stabilizer of a vertex lattice in the orthogonal space. Our construction is obtained by combining results of Kisin and the first author with an explicit presentation and resolution of a corresponding local model.
6. On orthogonal local models of Hodge type, International Mathematics Research Notices, Volume 2023, Issue 13, (2023), 10799–10836.
Abstract: We study local models that describe the singularities of Shimura varieties of non-PEL type for orthogonal groups at primes where the level subgroup is given by the stabilizer of a single lattice. In particular, we use the Pappas-Zhu construction and we give explicit equations that describe an open subset around the ``worst" point of orthogonal local models given by a single lattice. These equations display the affine chart of the local model as a hypersurface in a determinantal scheme. Using this we prove that the special fiber of the local model is reduced and Cohen-Macaulay.
Ph.D Thesis: "On Orthogonal Local Models of Shimura Varieties". The results of my thesis have been superseded by those in the papers (5) & (6). If you would still like to see it, click here.
Co-organizer (with J. Lourenço and E. Viehmann) of the Oberseminar at the University of Münster on "Affine Deligne-Lusztig theory", Spring 2025.