Nivolumab in Patients With High-Risk Biochemically Recurrent Prostate Cancer

Inclusion Criteria: * Patients must have signed an informed-consent form indicating that the patient understands the purpose of and procedures required for the study and is willing to participate in the study. * Patients must have a history of prostate adenocarcinoma (adenocarcinoma must be the primary histology; secondary components of variant histologies are acceptable) confirmed on biopsy and treated with primary radical prostatectomy (RP) or definitive radiation (RT). Prior salvage RT is acceptable. * Patients must have experienced biochemical recurrence (BCR) plus have minimum PSA values noted below: * Following primary RP: Any detectable rising PSA after RP (or after salvage RT if performed), minimum PSA 1.0 at time of screening * Following primary RT: PSA rise to ≥2 ng/mL above the nadir * No evidence of metastases on conventional imaging (CT or MRI plus bone scan) * PSA doubling time (PSADT) \<10 months --PSADT: calculated as per Prostate Cancer Working Group 3 (PCWG3) and the Memorial Sloan Kettering Cancer Center calculator: (https://www.mskcc.org/nomograms/prostate/psa\_doubling\_time) With linear regression model of normal logarithm of PSA and time, based on: * At least 3 consecutive PSA values with each value ≥0.2 ng/mL * Interval between first and last PSA values is ≥8 weeks but ≤12 months. -Archival tissue is mandatory, either prostatectomy specimen or (in patients who received primary RT) diagnostic core biopsies. Patients must consent to next-generation sequencing performed on this tissue. * If diagnostic core biopsies are only available tissue, at least 3 cores must be involved by tumor * Easteron Cooperative Oncology Group (ECOG) performance status 0-1 * Age ≥18 years * Adequate organ and marrow function: * System Laboratory Value * Hematological * White blood cell (WBC) ≥ 2000/µL * Absolute Neutrophil Count (ANC) ≥ 1500/μL * Platelets (Plt) ≥ 100 x103/μL * Hemoglobin (Hgb) \> 9.0 g/dL (with or without transfusion) -Renal * Serum Creatinine ≤ 2 x ULN * Hepatic * Bilirubin1 ≤ 1.5× upper limit of normal (ULN) * Aspartate aminotransferase (AST) ≤ 3 × ULN * Alanine aminotransferase (ALT) ≤ 3 × ULN * Except subjects with Gilbert Syndrome, who can have total bilirubin \< 3.0 mg/dL * Baseline testosterone ≥100 ng/dL * Recovery from acute toxicity related to prior therapy, including surgery and radiation, or no treatment-related toxicity ≥ grade 2. * History of prior malignancy or concurrent separate malignancy is not an exclusion criterion so long as the non-prostate malignancy is stable and does not require any treatment. * Able to understand and sign informed consent and adhere to study procedures. * Male patients whose female partners are of reproductive potential must agree to use a contraception during the trial period Exclusion Criteria: * Current use of ADT or plan to initiate ADT during trial period * Major surgery or radiation therapy within 14 days of starting study treatment * Subjects with active autoimmune disease. Patients with a history of autoimmune disease that has not required systemic immunosuppressive therapy or does not threaten vital organ function including central nervous system, heart, lungs, kidneys, skin, and gastrointestinal tract will be allowed. * Known history of immune deficiencies or chronic viral infections including HIV, hepatitis B (HBV), and hepatitis C (HCV) (patients with prior therapy for HBV or HCV is permitted if viral clearance was documented). * Concurrent medical condition requiring use of systemic corticosteroids with prednisone \>10 mg per day or equivalent. Use of inhaled, nasal, and topical steroids (applied to small body areas) is allowed. * Current use (within past 4 weeks) of other prohibited medications including anti-cancer therapies, hormonal therapies, 5-alpha reductase inhibitors, and alternative medications known to alter PSA (e.g. phytoestrogens and saw palmetto). * Prior treatment with immune checkpoint inhibitors. (Prior cancer vaccines are allowed.) * Serious intercurrent medical or psychiatric illness that, in the judgment of the investigator, would interfere with patient's ability to carry out the treatment program