Algebra Number Theory

Elliptic Curves and Rational Points

Investigation of rational point distribution on elliptic curves and connections to the Birch and Swinnerton-Dyer conjecture.

RESEARCH GAP FRONTIERS

Modular Forms and L-functions

Study of modular forms, their Fourier expansions, and associated L-function properties with applications to number theory.

RESEARCH GAP FRONTIERS

Galois Theory and Field Extensions

Exploration of Galois groups, inverse Galois problems, and absolute Galois group structures in arithmetic geometry.

RESEARCH GAP FRONTIERS

Diophantine Equations and Integer Solutions

Analysis of solvability and solution sets for polynomial equations over integers using algebraic and analytic techniques.

RESEARCH GAP FRONTIERS

Arithmetic Algebraic Geometry

Study of algebraic varieties over number fields combining algebraic geometry with arithmetic methods and height theory.

RESEARCH GAP FRONTIERS

p-adic Analysis and Hensel Lifting

Investigation of p-adic valuations, p-adic numbers, and Hensel lifting techniques for solving congruences and equations.

RESEARCH GAP FRONTIERS

Homological Algebra and Derived Categories

Development of derived functors, spectral sequences, and derived category theory with applications to ring theory.

RESEARCH GAP FRONTIERS

Representation Theory of Finite Groups

Study of irreducible representations, character theory, and modular representations of finite and algebraic groups.

RESEARCH GAP FRONTIERS

Iwasawa Theory and Cyclotomic Fields

Investigation of power series invariants, class groups, and unit groups in towers of cyclotomic number fields.

Computational Number Theory Algorithms

Development and analysis of efficient algorithms for factorization, discrete logarithm, and solving Diophantine equations.

Class Field Theory Extensions

Study of abelian extensions of number fields, class groups, and reciprocity laws in higher-dimensional settings.

Transcendental Number Theory

Investigation of transcendence and algebraic independence of special constants using Diophantine approximation methods.

Lattices and Integer Programming

Study of lattice reduction algorithms, shortest vector problems, and applications to cryptography and optimization.

Commutative Ring Theory

Analysis of prime ideals, valuations, completions, and regular local rings in commutative algebra foundations.

Algebraic K-theory and Cohomology

Study of K-groups, motivic cohomology, and their applications to algebraic geometry and number theory problems.

Abelian Varieties and Jacobians

Investigation of Jacobian varieties, endomorphism rings, and isogeny structures on abelian varieties over number fields.

Automorphic Forms and Langlands Program

Study of automorphic representations, L-functions, and connections between Galois representations and automorphic forms.

Quadratic Forms and Integer Matrices

Analysis of representability by quadratic forms, genus theory, and class numbers of quadratic forms.

Cryptographic Number Theory Applications

Development of elliptic curve cryptography, pairing-based systems, and post-quantum algebraic cryptographic schemes.

Zeta Functions and Analytic Methods

Study of Riemann zeta function, Dedekind zeta functions, and analytic continuation techniques in number theory.

Local Fields and Ramification Theory

Investigation of extensions of p-adic fields, ramification groups, and inertia filtrations in local number theory.

Noncommutative Algebra and Division Rings

Study of central simple algebras, Brauer groups, and noncommutative generalizations of algebraic number theory.

Primes in Arithmetic Progressions

Analysis of prime distribution using Dirichlet characters, Dirichlet L-functions, and sieve theory methods.

Algebraic Integers and Norms

Study of rings of integers, unit groups, discriminants, and norms in algebraic number field extensions.

Polynomial Factorization Over Finite Fields

Development of algorithms and theory for factoring polynomials over finite fields with cryptographic applications.

Arithmetic Progressions and Additive Combinatorics

Investigation of long arithmetic progressions, Szemerédi theorem generalizations, and density-based algebraic results.

Shimura Varieties and Hecke Correspondences

Study of complex multiplication, Hecke algebras, and the arithmetic geometry of Shimura varieties.

Valuations and Absolute Values

Analysis of discrete and non-discrete valuations, completion of fields, and valuation theory in algebraic extensions.

Frobenius Elements and Chebotarev Density

Study of Frobenius conjugacy classes in Galois groups and distribution of primes via Chebotarev density theorem.

Gorenstein Rings and Cohen-Macaulay Properties

Investigation of Gorenstein and Cohen-Macaulay ring structures with applications to algebraic geometry and ring theory.

Biquadratic and Cyclic Field Extensions

Analysis of specific field extension types, their Galois groups, class numbers, and unit structures.

Baker Theory and Linear Forms in Logarithms

Application of lower bounds for linear forms in logarithms to solve exponential Diophantine equations effectively.

Cyclotomic Units and Stickelberger Elements

Study of special units in cyclotomic fields and Stickelberger''s theorem relating class groups to L-function values.

Intersection Theory on Algebraic Varieties

Development of Chow rings, intersection multiplicities, and Fulton-MacPherson intersection theory for varieties.

Arithmetic of Hyperelliptic Curves

Investigation of Jacobians of hyperelliptic curves, rational points, and computational methods for genus computations.

Perfectoid Spaces and Tilting

Study of perfectoid rings, Scholze''s tilting theory, and applications to p-adic geometry and arithmetic.

Ring Extensions and Integral Closures

Analysis of integral dependence, integrally closed rings, and conductor ideals in ring extension theory.

Dirichlet Characters and Gauss Sums

Study of character sums, Gauss and Jacobi sums, and their applications to density and distribution problems.

Algebraic Cohomology Theories

Development of étale cohomology, crystalline cohomology, and de Rham cohomology in algebraic geometry.

Pell Equations and Continued Fractions

Analysis of solutions to generalized Pell equations using continued fraction expansions and approximation theory.

Class Number Computations and Tables

Computational methods for calculating class numbers of quadratic and number fields with database applications.

Toric Varieties and Lattice Polytopes

Study of toric varieties arising from lattice polytopes with connections to combinatorial and arithmetic geometry.

Group Cohomology and Extensions

Analysis of group cohomology groups, extension groups, and spectral sequences in group-theoretic contexts.

Selmer Groups and Descent Theory

Investigation of Selmer groups, descent methods for computing ranks, and BSD conjecture verification techniques.

Henselian Rings and Strict Henselization

Study of Henselian local rings, Hensel lifting, and strictly Henselian rings in commutative algebra.

Arithmetic Dynamics and Arithmetic Heights

Study of dynamical systems on algebraic varieties with arithmetic metrics and height functions on orbits.

Sieve Methods in Algebraic Contexts

Application of classical and algebraic sieve techniques to problems in prime distribution and polynomial equations.

Algebraic Deformation Theory

Study of versal deformations, tangent spaces, and obstructing classes in deformation-theoretic problems.

Heights and Arithmetic Complexity

Analysis of arithmetic height functions, canonical heights, and lower bounds for non-torsion points on varieties.

Totality of Quadratic Residues

Investigation of quadratic residue patterns, reciprocity laws, and explicit formulas for residue symbols.

Birational Geometry and Rational Maps

Studies rational equivalence of algebraic varieties and the structure of birational automorphism groups over number fields.

Conductor and Discriminant Theory

Analyzes conductors and discriminants of algebraic number fields and their role in determining arithmetic properties of extensions.

Tate Conjectures and Hodge Structures

Investigates connections between Galois representations and Hodge theory through the Tate conjecture for algebraic varieties.

Witt Vectors and Teichmüller Theory

Explores lift structures and deformation theory using Witt vector formalism in characteristic p algebra.

Mahler Measures and Dynamical Systems

Studies algebraic integer heights and arithmetic dynamics through Mahler measure calculations on algebraic integers.

Cohen-Lenstra Heuristics and Statistics

Analyzes probabilistic behavior of class groups and their statistical properties in families of number fields.

Trace Formulas and Spectral Theory

Applies trace formula techniques to study eigenvalues of Hecke operators and spectral properties of automorphic representations.

Stark Conjectures and Regulators

Examines conjectural relationships between L-function values and unit group regulators in algebraic number fields.

Torsion Points on Abelian Varieties

Investigates the distribution and Galois action on torsion points of abelian varieties over number fields.

Fontaine-Laffaille Theory

Studies crystalline and semi-stable representations through Fontaine-Laffaille modules and period rings.

Gross-Zagier Formula Extensions

Generalizes the Gross-Zagier formula relating heights and L-function derivatives in diverse arithmetic contexts.

Arakelov Geometry and Heights

Develops arithmetic intersection theory on arithmetic surfaces and heights in the Arakelov setting.

Brauer Groups and Cohomology

Studies central simple algebras and their classification via Brauer group cohomology over local and global fields.

Cyclotomic Abelian Varieties

Analyzes abelian varieties with endomorphisms by cyclotomic units and their arithmetic properties.

Thue-Siegel-Roth Diophantine Approximation

Applies effective diophantine approximation bounds to solve exponential and polynomial diophantine equations.

Explicit Reciprocity Laws

Develops computational reciprocity laws relating local and global norm residues through explicit formulas.

Kodaira-Spencer and Deformation Spaces

Studies deformation theory of arithmetic objects using Kodaira-Spencer maps and tangent space computations.

Universal Norms and Circular Units

Investigates cyclotomic circular units and universal norms in Zp-extensions and infinite towers of fields.

Modular Jacobians and Rational Points

Studies rational points on Jacobians of modular curves and their relationship to cusp forms.

Ostrowski Valuations and Extensions

Analyzes complete classification of valuations on algebraic number fields via Ostrowski theory.

Sato-Tate Distributions

Studies statistical distributions of Frobenius eigenvalues in families of varieties over finite fields.

Green Functions and Arithmetic Curves

Develops Green function theory for arithmetic divisors on curves in the Arakelov setting.

Generalized Fermat Equations

Solves exponential and higher-dimensional Fermat-type equations using modularity and Frey curve techniques.

Module Theory Over Orders

Classifies modules over orders in algebras and their relationship to ideal class groups.

Rational Preperiodic Points

Characterizes preperiodic orbits of algebraic points under rational maps in arithmetic dynamics.

Satake Parameters and Functoriality

Applies Satake isomorphisms to establish functorial relationships between automorphic representations.

Unramified Cohomology

Studies cohomology groups supported at all places simultaneously on arithmetic varieties.

Primitive Roots and Artin Conjecture

Investigates existence of primitive roots modulo primes and generalizations via Artin conjecture.

Rigid Analytic Geometry

Develops p-adic analogs of complex analytic geometry through rigid analytic spaces and their properties.

Picard Curves and Hyperelliptic Jacobians

Studies arithmetic and geometry of genus three Picard curves and their Jacobian varieties.

Wieferich and Wilson Primes

Searches for and analyzes primes satisfying special congruence conditions related to Fermat and Wilson theorems.

Integral Points on Curves Algorithms

Develops computational algorithms for finding integer solutions on curves of higher genus.

Berkovich Spaces and Non-Archimedean Geometry

Constructs and analyzes Berkovich spaces as geometric realizations of non-archimedean analytic geometry.

Fitting Ideals and Determinants

Applies Fitting ideal theory to study annihilators of modules and determinantal ideals in commutative algebra.

ABC Conjecture and Radical Bounds

Explores bounds on integer triples from the ABC conjecture and its implications for diophantine equations.

Geometric Invariant Theory

Studies quotient spaces of algebraic varieties by group actions using invariant theory and stability conditions.

Gross-Hopkins Duality

Analyzes duality theories in chromatic homotopy theory with applications to algebraic number theory.

Norm Subgroups and Ray Class Fields

Characterizes ray class fields by norm subgroups of idele groups in class field theory.

Elliptic Units and Special Values

Studies units arising from elliptic curves and their roles in special value formulas for L-functions.

Quaternion Algebras and Orders

Classifies quaternion orders and their ideal structures in connection with modular forms.

Rank Bounds for Abelian Varieties

Establishes upper and lower bounds on Mordell-Weil ranks using descent and analytic methods.

Radical and Jacobson Rings

Characterizes radical and Jacobson properties of rings and their connections to algebraic geometry.

Atiyah-Segal Completion Theorem

Applies completion theorems in K-theory and cohomology to arithmetic settings.

Genus Theory and Binary Forms

Studies equivalence classes of binary quadratic forms and genus theory in algebraic number fields.

Exponential Diophantine Equations

Solves equations involving exponential terms using linear forms in logarithms and computational methods.

Lubin-Tate Formal Groups

Develops formal group theory over local fields with applications to local class field theory.

Toric Singularities and Resolutions

Analyzes singularities and resolutions of toric varieties using combinatorial lattice methods.

Manin-Mumford Conjecture

Studies finiteness of torsion points on curves embedded in abelian varieties.

Depth Functions and Colength

Analyzes depth filtrations and colength invariants in algebraic structures and their invariants.

Analytic Continuation of Zeta Functions

Develops analytic properties and functional equations of generalized zeta functions in arithmetic geometry.

Arithmetic Surfaces and Effective Bounds

Studies effective computational bounds on rational points and arithmetic invariants of algebraic surfaces defined over number fields.

Motivic Integration and Arc Spaces

Develops motivic integration techniques on arc spaces and their applications to singularities and birational geometry.

Derived Algebraic Geometry Methods

Applies derived categorical and homotopical methods to solve classical problems in algebraic number theory and arithmetic geometry.

Genus Curves and Jacobian Isogenies

Investigates isogeny classes of Jacobians of algebraic curves and their arithmetic properties over global fields.

Algebraic Function Fields Arithmetic

Studies arithmetic properties of function fields over finite fields including divisor class groups and L-functions.

Monsky-Washnitzer Cohomology Theory

Develops rigid analytic cohomology methods for computing zeta functions and L-functions of algebraic varieties over p-adic fields.

Vertex Algebras and Number Theory

Explores connections between vertex operator algebras and arithmetic invariants in algebraic number theory.

Arithmetic Okounkov Bodies

Studies arithmetic intersection theory using Okounkov bodies and their applications to height inequalities.

Equivariant Arakelov Geometry

Develops Arakelov intersection theory with group actions and applications to equivariant arithmetic invariants.

Sato-Tate Conjectures Beyond Curves

Studies generalizations of Sato-Tate distributions for higher-dimensional varieties and their arithmetic applications.

Syntomic Cohomology and Applications

Develops syntomic and p-adic Hodge theory cohomology frameworks for studying arithmetic of varieties over p-adic fields.

Unlikely Intersections and Heights

Investigates unlikely intersection problems using height theory and techniques from transcendental number theory.

Arithmetic of CM Abelian Varieties

Studies complex multiplication theory on abelian varieties and connections to special values of L-functions.

Non-Abelian Chabauty Methods

Applies non-abelian descent and Chabauty-type methods to determine rational points on curves and higher-dimensional varieties.

Sparse Polynomial Systems Solving

Develops efficient algorithms for solving sparse systems of polynomial equations over integers and finite fields.

Motivic Cohomology and Bloch-Kato

Studies motivic cohomology groups and their connection to K-theory through Bloch-Kato conjectures.

Arithmetic Fundamental Groups Galois

Investigates Galois actions on arithmetic fundamental groups and their relationship to rational points.

Generalized Jacobian Varieties

Studies generalized Jacobians of singular curves and their universal properties in arithmetic geometry.

Hasse-Davenport Relations Generalizations

Extends Hasse-Davenport relations and character sum identities to multivariate and higher-dimensional settings.

Integral Points on Surfaces Methods

Develops effective methods for finding and counting integral points on algebraic surfaces using Diophantine techniques.

Quantum Invariants Number Theory

Explores relationships between quantum invariants of knots and algebraic number theory structures.

Birch-Swinnerton-Dyer Conjecture Cases

Investigates special cases and analogues of the Birch-Swinnerton-Dyer conjecture for various classes of curves.

Refined Class Numbers and Iwasawa

Studies refined class number formulas using Iwasawa theory and higher-order derivatives of L-functions.

Toric Variety Zeta Functions

Computes and analyzes zeta functions and L-functions of toric varieties over finite and p-adic fields.

Langlands Functoriality Bases

Studies functoriality transfers in the Langlands program with applications to L-function computations.

Arithmetic Mirror Symmetry Correlates

Explores connections between mirror symmetry and arithmetic invariants of Calabi-Yau varieties.

Moduli Spaces Integral Points

Studies the distribution and structure of integral points on moduli spaces of algebraic varieties.

Densities of Primes Number Fields

Investigates prime ideal densities in number fields through Galois density theorems and sieve methods.

Arithmetic Quantum Unique Ergodicity

Studies quantum unique ergodicity in arithmetic contexts and implications for L-function values.

Rigid Analytic Varieties Geometry

Develops the theory of rigid analytic spaces and their applications to arithmetic problems over p-adic fields.

Hall Algebras Quantum Groups

Studies Hall algebras associated with quivers and their connections to representation theory and number theory.

Ramification Fields and Bounds

Investigates bounds on ramification in field extensions and their applications to class field tower problems.

Definability and Decidability Problems

Studies definability and decidability of arithmetic properties using model theory and logical methods.

Lattice Point Enumeration Algorithms

Develops efficient computational methods for enumerating lattice points in convex polytopes and applications.

Deligne Cohomology Arithmetic

Applies Deligne cohomology and Beilinson-Bloch regulators to study arithmetic of varieties and special values.

Quaternion Orders and Ideal Classes

Studies ideal class groups and unit groups of quaternion algebras over number fields and function fields.

Frobenius Distributions Varieties

Investigates Frobenius element distributions on Galois groups through point-counting on varieties over finite fields.

Transcendence Theory Multiple Values

Studies transcendence and algebraic independence of multiple special values of analytic functions.

Integral Canonical Models Shimura

Develops integral canonical models of Shimura varieties and their reduction modulo primes.

Arithmetic Satake Isomorphism

Studies arithmetic enhancements of the Satake isomorphism and equivariant K-theory methods.

Galois Covers and Dessin Interpretation

Investigates Galois covers of algebraic curves through dessin d''enfants and related combinatorial-arithmetic structures.

Effective Isomorphism Testing Curves

Develops efficient algorithms for testing isomorphism of curves and computing period matrices.

Mahler Measures and Heights

Studies relationships between Mahler measures of polynomials and arithmetic heights of algebraic varieties.

Arithmetic Chow Groups Intersection

Develops arithmetic Chow group theory and applications to intersection multiplicities and arithmetic degrees.

Modular Curves Higher Genus

Studies rational points on modular curves of higher genus and their connection to Diophantine problems.

Cluster Algebras Number Theory

Explores applications of cluster algebra structures to arithmetic invariants and mirror symmetry.

Berkovich Spaces Reduction Theory

Develops theory of Berkovich spaces and their skeleta with applications to reduction of varieties.

Supersingular Isogeny Cryptography

Studies supersingular elliptic curves and isogeny-based cryptographic protocols with arithmetic foundations.

Discriminant Bounds Discriminant Forms

Investigates bounds on discriminants of number fields through discriminant forms and lattice theory.

Perfectoid Geometry and Adic Spaces

Studies perfectoid spaces and adic geometry to understand p-adic analogs of classical algebraic geometry and rigid analytic varieties.

Birational Geometry and Minimal Models

Investigates birational transformations and the minimal model program for algebraic varieties over number fields.

Derived Algebraic Geometry Foundations

Develops derived stack theory and higher categorical methods applied to arithmetic and algebraic geometry problems.

Arakelov Geometry and Heights

Studies arithmetic intersection theory on arithmetic surfaces combining algebraic and differential geometric techniques.

Motivic Cohomology and Cycle Classes

Explores motivic cohomology theories and their connections to arithmetic cycle classes and algebraic K-theory.

Étale Cohomology and Torsion

Analyzes étale cohomology groups and torsion phenomena in Galois cohomology over number fields.

Universal Coverings of Algebraic Curves

Studies universal covering spaces of algebraic curves and their arithmetic properties via uniformization theory.

Anabelian Geometry and Grothendieck Conjecture

Investigates the Grothendieck conjecture on recovering algebraic curves from their absolute Galois groups.

Rational Points on Fano Varieties

Studies the distribution and existence of rational points on Fano varieties using geometric and analytic methods.

Integral Points on Affine Varieties

Analyzes integral solutions to polynomial equations via geometric methods including toroidal compactifications.

Ax-Grothendieck and Model Theory

Applies model theoretic techniques to study algebraic consequences of functional equations over varieties.

Unlikely Intersections and Unlikely Pairs

Studies intersections of subvarieties expected to be empty using height bounds and Diophantine geometry.

Berkovich Analytic Geometry Dynamics

Develops dynamical systems theory on Berkovich spaces with applications to arithmetic geometry.

Rational Canonical Forms and Invariants

Studies canonical forms of matrices and polynomial invariants over number fields and finite rings.

Central Simple Algebras and Brauer Groups

Investigates Brauer groups and central simple algebras over number fields and their local-global principles.

Quaternion Algebras and Ternary Forms

Studies quaternion algebras and representation of integers by ternary quadratic forms.

Lattice Points in Polyhedra Estimation

Develops asymptotic formulas for counting lattice points in families of expanding polyhedra.

Effective Diophantine Approximation Bounds

Establishes effective bounds for Diophantine approximation using transcendence theory and linear forms.

Mahler Measure and Symmetric Functions

Studies Mahler measure of polynomials and connections to special values of L-functions.

Cyclotomic Extensions and Kummer Theory

Analyzes Kummer theory for cyclotomic extensions and connections to unit groups.

Catalan and Pillai Conjecture Approaches

Investigates exponential Diophantine equations related to the Catalan and Pillai conjectures.

Elliptic Divisibility Sequences Dynamics

Studies divisibility properties of sequences arising from elliptic curves and their dynamical behavior.

Thue-Mahler and Generalized Exponential Equations

Applies effective methods to solve Thue-Mahler equations and exponential Diophantine problems.

Fermat Curves and Generalized Fermat Equations

Studies rational and integral points on Fermat-type curves using Frey-Hellegouarch techniques.

Modularity Lifting and Deformation Theory

Develops deformation theory of Galois representations and applications to modularity lifting theorems.

p-adic Hodge Theory and Crystalline Representations

Studies crystalline and semistable p-adic Galois representations and Fontaine theory.

Bloch-Kato Exponential and L-values

Investigates the Bloch-Kato exponential map and conjectures relating L-values to Selmer groups.

Fine Selmer Groups and Kolyvagin Methods

Studies fine Selmer groups using Kolyvagin classes and Heegner point constructions.

Iwasawa Main Conjecture Generalizations

Develops generalizations of the Iwasawa main conjecture to non-ordinary settings and higher rank cases.

Hida Families and Eigenvalue Variations

Analyzes p-adic families of modular forms and variation of Hecke eigenvalues.

Anti-cyclotomic Iwasawa Theory

Studies Iwasawa theory of anti-cyclotomic extensions and applications to Heegner point heights.

Overconvergent Modular Symbols

Develops overconvergent modular symbol machinery for computing p-adic L-functions and zeta elements.

L-invariants and Exceptional Cases

Studies L-invariants of modular forms and applications to exceptional zeros of p-adic L-functions.

Gross-Zagier Type Formulas Generalizations

Develops generalizations of Gross-Zagier formulas relating heights and derivatives of L-functions.

Stark Units and Stickelberger Class Relations

Investigates explicit unit constructions via Stark conjectures and Stickelberger relations.

Galois Cohomology and Descent Obstructions

Studies Galois cohomology groups and their role in Brauer-Manin obstruction to rational points.

Chow Groups and Abel-Jacobi Mappings

Analyzes Chow groups of cycles and Abel-Jacobi mapping properties on algebraic varieties.

Hodge Theory and Period Domains

Studies Hodge structures and period domains in the context of arithmetic varieties.

Crystalline Cohomology and de Rham Theory

Develops crystalline cohomology and comparisons with de Rham cohomology for algebraic varieties.

Syntomic Cohomology and p-adic Comparison

Studies syntomic cohomology theories and comparison theorems in p-adic Hodge theory.

Prismatic Cohomology and Six Functors

Develops prismatic cohomology framework unifying crystalline and de Rham theories.

Character Sums and Kloosterman Sums

Studies bounds and applications of character sums and Kloosterman sums over finite fields.

Exponential Sums and Deligne Estimates

Applies Deligne''s bounds on exponential sums to problems in number theory and cryptography.

Weil Conjectures and Etale Homology

Studies the Weil conjectures and their proof via étale cohomology and l-adic representations.

Arithmetic Codes and Algebraic Geometry

Develops algebraic geometry codes using curves over finite fields with arithmetic properties.

Arithmetic Cryptography Pairing-based Schemes

Studies pairing-based cryptographic schemes and their security from arithmetic geometry perspective.

Post-Quantum Lattice-based Cryptography

Analyzes security of lattice-based cryptographic systems from algebraic number theory viewpoint.

Arithmetic Boolean Circuits and Complexity

Studies computational complexity of arithmetic algorithms using algebraic and number-theoretic methods.

Distribution of Primes in Special Sequences

Investigates prime distribution in sequences defined by recurrence relations and algebraic conditions.

Generalized Riemann Hypothesis Consequences

Explores deep consequences of the generalized Riemann hypothesis for L-functions and class numbers.

Perfectoid Spaces and Tilting Theory

Investigation of perfectoid spaces as a foundational framework for understanding p-adic geometry and their applications to proving results in arithmetic geometry through Scholze''s tilting correspondence and reduction modulo p techniques.